Finite Group Databasebeta

← #29737520

Subgroups — #29737520

Order 15616.

988 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840062C211yesmaximal, characteristic, complemented
2C2×#16839763C211yesmaximal, characteristic
3#21449543C211yesmaximal, characteristic, complemented
4#21447367C211yesmaximal, characteristic, complemented
5#21447214C211yesmaximal, characteristic, complemented
6#21447214C211yesmaximal, characteristic, complemented
7#21446970C211yesmaximal, characteristic, complemented
8#21446066C211yesmaximal, characteristic, complemented
9#21446066C211yesmaximal, characteristic, complemented
10#21446006C211yesmaximal, characteristic, complemented
11#21446004C211yesmaximal, characteristic, complemented
12#21445850C211yesmaximal, characteristic, complemented
13#21445829C211yesmaximal, characteristic, complemented
14#21445759C211yesmaximal, characteristic
15C61×#2339C211yesmaximal, characteristic
16C22×#11758194C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19#16840062—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26#16839763—24no—
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×C22⋊#92418C2211yescharacteristic
36#16839704—21no—
37#16839696—21no—
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839634—21no—
41#16839628C2211yescharacteristic
42#16839622C2211yescharacteristic
43#16839620—21no—
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C22×#41×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C61×#391C2211yescharacteristic
54C61×#391C2211yescharacteristic
55C61×#386—21no—
56C61×#386—21no—
57C61×#381—21no—
58C61×#380C2211yescharacteristic
59C4×C22⋊C4×C61C2211yescharacteristic
60C2×#95×C61—21no—
61C23×Dic61—21no—
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194—41no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519769—22no—
70#14519769—41no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—41no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83#14519710—22no—
84#14519710—41no—
85#14519710—41no—
86#14519708—22no—
87C4⋊#92418—22no—
88C4⋊#92418—41no—
89#14519701—22no—
90#14519701—22no—
91#14519701—41no—
92#14519701—41no—
93#14519698—22no—
94#14519698—22no—
95#14519698—22no—
96#14519698D414yes—
97C22⋊#92418—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C2×#41×C61—22no—
104C2×#41×C61—22no—
105C2×#41×C61—22no—
106C2×#41×C61—41no—
107C2×#41×C61—41no—
108C2×#41×C61Q814yes—
109C22×Q8×C61C2311yescharacteristic
110C22×D4×C61—21no—
111C22×D4×C61C2311yescharacteristic
112C23×C4×C61—21no—
113C23×C4×C61—21no—
114C23×C4×C61—21no—
115C23×C4×C61C2311yescharacteristic
116C4×D4×C61—22no—
117C4×D4×C61—22no—
118C4×D4×C61—41no—
119C4×D4×C61—41no—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61D411yescharacteristic
126C2×C22⋊C4×C61D411yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132#95×C61—21no—
133#95×C61—21no—
134#95×C61D411yescharacteristic
135#95×C61D411yescharacteristic
136C24×C61—21no—
137C24×C61C2×D411yescharacteristic
138#41×C61—28no—
139#41×C61—44no—
140#41×C61—44no—
141#41×C61—44no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×Q812yes—
148C2×D4×C61—24no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—42no—
153C2×D4×C61—42no—
154C2×D4×C61C2×D412yes—
155C2×D4×C61C2×Q812yes—
156C2×D4×C61#4112yes—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—24no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—42no—
166C22×C4×C61—42no—
167C22×C4×C61—42no—
168C22×C4×C61—42no—
169C22×C4×C61—42no—
170C22×C4×C61—42no—
171C22×C4×C61—81no—
172C22×C4×C61—81no—
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×Q812yes—
179C22×C4×C61#4112yes—
180C22×C4×C61C2411yescharacteristic
181C22×Dic61—21no—
182C22×Dic61—21no—
183C22×Dic61—21no—
184C22×Dic61—21no—
185C22×Dic61—42no—
186C22×Dic61—42no—
187C22×Dic61—81no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C22⋊C4×C61—41no—
193C22⋊C4×C61—41no—
194C22⋊C4×C61—42no—
195C22⋊C4×C61—42no—
196C22⋊C4×C61—81no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—21no—
201C22⋊Dic61—42no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—42no—
210C4⋊Dic61C2×D412yes—
211#11758196—42no—
212#11758196—42no—
213#11758196—81no—
214#11758196—81no—
215#11758194—44no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418C2×D412yes—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—21no—
231C23×C61—22no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61—21no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61—41no—
240C23×C61—41no—
241C23×C61C22⋊Q812yes—
242C23×C61C22×D411yescharacteristic
243Q8×C61—22no—
244Q8×C61—22no—
245Q8×C61—41no—
246Q8×C61—41no—
247D4×C61—22no—
248D4×C61—22no—
249D4×C61—22no—
250D4×C61—48no—
251D4×C61C22⋊Q814yes—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61—21no—
269C2×C4×C61—21no—
270C2×C4×C61—21no—
271C2×C4×C61—48no—
272C2×C4×C61—41no—
273C2×C4×C61—41no—
274C2×C4×C61—41no—
275C2×C4×C61—41no—
276C2×C4×C61—41no—
277C2×C4×C61—41no—
278C2×C4×C61—41no—
279C2×C4×C61—41no—
280C2×C4×C61—84no—
281C2×C4×C61—84no—
282C2×C4×C61C22⋊Q812yes—
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C22×Q811yescharacteristic
285C2×C4×C61C2×#4111yescharacteristic
286C2×C4×C61C2×#4111yescharacteristic
287C2×C4×C61H(5,𝔽2)11yescharacteristic
288C2×C4×C61H(5,𝔽2)11yescharacteristic
289C2×Dic61—22no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61—41no—
297C2×Dic61—41no—
298C2×Dic61—41no—
299C2×Dic61—41no—
300C2×Dic61—41no—
301C2×Dic61—84no—
302#92418—22no—
303#92418—41no—
304#92418—41no—
305#92418—41no—
306#25980—611nomaximal, Sylow
307C22×C61—22no—
308C22×C61—22no—
309C22×C61—22no—
310C22×C61—24no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—41no—
314C22×C61—42no—
315C22×C61—41no—
316C22×C61—42no—
317C22×C61—41no—
318C22×C61—42no—
319C22×C61—42no—
320C22×C61—41no—
321C22×C61C2×C22⋊Q811yescharacteristic
322C22×C61D4211yescharacteristic
323C22×C61#54611yescharacteristic
324C22×C61#57511yescharacteristic
325C22×C61#57511yescharacteristic
326C22×C61#57611yescharacteristic
327C22×C61#57811yescharacteristic
328C4×C61—22no—
329C4×C61—42no—
330C4×C61—41no—
331C4×C61—41no—
332C4×C61—41no—
333C4×C61—41no—
334C4×C61—81no—
335C4×C61—81no—
336C4×C61C2×C22⋊Q811yescharacteristic
337C4×C61#54311yescharacteristic
338C4×C61#54611yescharacteristic
339C4×C61#55111yescharacteristic
340Dic61—42no—
341Dic61—81no—
342Dic61—81no—
343Dic61—81no—
344C2×#478—611no—
345C2×#417—611no—
346#2442—611no—
347#2339D6111yescharacteristic
348#2097—611no—
349#2068—611no—
350#2068—611no—
351#2026—611no—
352#1862—611no—
353#1862—611no—
354#1852—611no—
355#1851—611no—
356#1809—611no—
357#1804—611no—
358#1788—611no—
359C2×C61—42no—
360C2×C61—44no—
361C2×C61#243711yescharacteristic
362C2×C61#308011yescharacteristic
363C2×C61#308311yescharacteristic
364C2×C61#326911yescharacteristic
365C2×C61#327311yescharacteristic
366C2×C61#331511yescharacteristic
367C2×C61#332211yescharacteristic
368C22×#41C2×D6111yescharacteristic
369C22×M16—611no—
370C2×#125—611no—
371C2×C4×D4C2×D6111yescharacteristic
372C2×C4×D4C2×D6111yescharacteristic
373C22×C4⋊C4—611no—
374#478—1224no—
375C2×#107—611no—
376C2×#107—611no—
377C2×#107—611no—
378C2×#106—611no—
379C2×C4⋊C8—611no—
380C2×C4⋊C8—611no—
381#417—1224no—
382C2×#103—611no—
383C2×#103—611no—
384C2×#103—611no—
385C2×#103—611no—
386C2×#102—611no—
387C2×#102—611no—
388C2×#102—611no—
389C2×#102—611no—
390C2×C22⋊C8—611no—
391#400—1221no—
392#398—1221no—
393#391C2×D6111yescharacteristic
394#391C2×D6111yescharacteristic
395#389—611no—
396#389—611no—
397#386—21no—
398#386—21no—
399#385—1221no—
400#382—611no—
401#381—21no—
402#380—611no—
403#380—1221no—
404#380C2×D6111yescharacteristic
405C4×C22⋊C4C2×D6111yescharacteristic
406C2×#95—21no—
407#340—611no—
408#340—611no—
409#340—611no—
410#340—611no—
411#336—611no—
412#336—611no—
413C61#2598011yescharacteristic, complemented, Sylow
414C2×#41—22no—
415C2×#41—22no—
416C2×#41—22no—
417C2×#41—41no—
418C2×#41—41no—
419C2×#41Dic12214yes—
420C22×Q8C22×D6111yescharacteristic
421C22×D4—21no—
422C22×D4C22×D6111yescharacteristic
423C23×C4—21no—
424C23×C4—21no—
425C23×C4—21no—
426C23×C4—1221no—
427C23×C4C22×D6111yescharacteristic
428C2×M16—1222no—
429C2×M16—1222no—
430C2×M16—2441no—
431C22×C8—611no—
432C22×C8—611no—
433C22×C8—611no—
434C22×C8—611no—
435#125—1222no—
436#125—2441no—
437C4×D4—22no—
438C4×D4—22no—
439C4×D4—41no—
440C4×D4—41no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—1222no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—2441no—
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C22⋊C4—21no—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4C22⋊D6111yescharacteristic
459C2×C22⋊C4C22⋊D6111yescharacteristic
460C2×C42C22×D6111yescharacteristic
461C2×C42C22×D6111yescharacteristic
462#107—1222no—
463#107—2441no—
464#107—2441no—
465#106—1222no—
466C4⋊C8—1222no—
467C4⋊C8—2441no—
468#103—1222no—
469#103—1222no—
470#103—2441no—
471#103—2441no—
472#102—614no—
473#102—1222no—
474#102—1222no—
475#102—1222no—
476C22⋊C8—1222no—
477#95—21no—
478#95—21no—
479#95—21no—
480#95—21no—
481#95—21no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487#95C22⋊D6111yescharacteristic
488#95C22⋊D6111yescharacteristic
489C24—21no—
490C24C2×C22⋊D6111yescharacteristic
491#41—28no—
492#41—44no—
493#41—44no—
494#41—44no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8C2×Dic12212yes—
501C2×D4—24no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—21no—
505C2×D4—42no—
506C2×D4—42no—
507C2×D4C2×Dic12212yes—
508C2×D4D4×D6112yes—
509C2×D4#1175822412yes—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—24no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—42no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—42no—
522C22×C4—42no—
523C22×C4—42no—
524C22×C4—81no—
525C22×C4—81no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—4881no—
531C22×C4—2442no—
532C22×C4—2442no—
533C22×C4C2×Dic12212yes—
534C22×C4#1175822212yes—
535C22×C4C2×C22⋊D6111yescharacteristic
536C22×C4C2×C22⋊D6111yescharacteristic
537C22×C4C2×C22⋊D6111yescharacteristic
538C22×C4C2×C22⋊D6111yescharacteristic
539C22×C4C2×C22⋊D6111yescharacteristic
540C22×C4C23×D6111yescharacteristic
541M16—2444no—
542C2×C8—612no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—1221no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—612no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4—1221no—
565C4⋊C4—1221no—
566C4⋊C4—4881no—
567C4⋊C4—4881no—
568C4⋊C4—2442no—
569C4⋊C4—2442no—
570C4⋊C4—2442no—
571C22⋊C4—41no—
572C22⋊C4—41no—
573C22⋊C4—42no—
574C22⋊C4—42no—
575C22⋊C4—81no—
576C22⋊C4—2442no—
577C42—21no—
578C42—21no—
579C42—21no—
580C42—21no—
581C23—22no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—22no—
587C23—21no—
588C23—22no—
589C23—21no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23#1451975612yes—
595C23C22×C22⋊D6111yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—41no—
599Q8—41no—
600D4—22no—
601D4—22no—
602D4—22no—
603D4—48no—
604D4#1451975414yes—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—48no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—1222no—
636C2×C4—4884no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4#1451975612yes—
649C2×C4C2×D4×D6111yescharacteristic
650C2×C4C22×Dic12211yescharacteristic
651C2×C4C2×#1175822411yescharacteristic
652C2×C4C2×#1175822211yescharacteristic
653C2×C4#1451982211yescharacteristic
654C2×C4#1451982211yescharacteristic
655C8—1222no—
656C8—2441no—
657C8—2441no—
658C8—2441no—
659C22—22no—
660C22—24no—
661C22—22no—
662C22—22no—
663C22—42no—
664C22—41no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—41no—
669C22—41no—
670C22—42no—
671C22—41no—
672C22—42no—
673C22C2×#1451975611yescharacteristic
674C22D4×C22⋊D6111yescharacteristic
675C22#1684040611yescharacteristic
676C22#1684056011yescharacteristic
677C22#1684056011yescharacteristic
678C22#1684056511yescharacteristic
679C22#1684057711yescharacteristic
680C4—22no—
681C4—42no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—81no—
687C4—81no—
688C4—4881no—
689C4—4881no—
690C4—4881no—
691C4—2442no—
692C4C2×#1451975411yescharacteristic
693C4#1684038811yescharacteristic
694C4#1684040211yescharacteristic
695C4#1684043811yescharacteristic
696C2—42no—
697C2—44no—
698C2#2144948011yescharacteristic
699C2#2145564111yescharacteristic
700C2#2145567011yescharacteristic
701C2#2145737011yescharacteristic
702C2#2145742611yescharacteristic
703C2#2145791511yescharacteristic
704C2#2145802011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2339C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenter#2339D61
hypocenterC61#25980
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2339

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840062C2semidirect
3. #21449543C2semidirect
4. #21447367C2semidirect
5. #21447214C2semidirect
6. #21447214C2semidirect
7. #21446970C2semidirect
8. #21446066C2semidirect
9. #21446066C2semidirect
10. #21446006C2semidirect
11. #21446004C2semidirect
12. #21445850C2semidirect
13. #21445829C2semidirect
413. C61#25980semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168400622
C2×#168397632
#214495432
#214473672
#214472142
#214469702
#214460662
#214460062
#214460042
#214458502
#214458292
#214457592
C61×#23392
#2598061