Finite Group Databasebeta

← #29740062

Subgroups — #29740062

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839959C211yesmaximal, characteristic
2C2×C4×#11758200C211yesmaximal, characteristic, complemented
3C4⋊C4×Dic122C211yesmaximal, characteristic, complemented
4#21447352C211yesmaximal, characteristic, complemented
5#21447320C211yesmaximal, characteristic, complemented
6#21447222C211yesmaximal, characteristic, complemented
7#21447050C211yesmaximal, characteristic
8#21446505C211yesmaximal, characteristic, complemented
9#21446378C211yesmaximal, characteristic, complemented
10#21446177C211yesmaximal, characteristic, complemented
11#21446177C211yesmaximal, characteristic, complemented
12#21446177C211yesmaximal, characteristic, complemented
13#21446086C211yesmaximal, characteristic, complemented
14#21446080C211yesmaximal, characteristic, complemented
15C61×#2445C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21#16839959—24no—
22C4×#11758200—24no—
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839659—21no—
38#16839659—21no—
39#16839632C2211yescharacteristic
40#16839631—21no—
41C4⋊C4×Dic61—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×#127×C61C2211yescharacteristic
51C2×#123×C61—21no—
52C2×C4×D4×C61C2211yescharacteristic
53C61×#394—21no—
54C61×#394—21no—
55C61×#390C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#386—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758200—22no—
63C2×#11758200—22no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200—22no—
67C2×#11758200D414yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C4×Dic122—22no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C4×Dic122—41no—
77C4×Dic122—41no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91#14519708—22no—
92#14519708—41no—
93C4⋊#92418—22no—
94#14519700—22no—
95#14519700—22no—
96#14519698—22no—
97#14519698—22no—
98#14519698—22no—
99#14519698—22no—
100#14519698—22no—
101#14519698—41no—
102C4×#92418—22no—
103#14519684—21no—
104#14519684—21no—
105C22×D4×C61—21no—
106C22×D4×C61—21no—
107C22×D4×C61C2311yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C23×C4×C61—21no—
110#127×C61—22no—
111#127×C61—41no—
112#123×C61—44no—
113C4×D4×C61—22no—
114C4×D4×C61—41no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129C2×C42×C61C2311yescharacteristic
130#95×C61—21no—
131#95×C61D411yescharacteristic
132#95×C61D411yescharacteristic
133C24×C61—21no—
134C24×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—42no—
147C2×D4×C61—42no—
148C2×D4×C61—42no—
149C2×D4×C61—42no—
150C2×D4×C61—81no—
151C2×D4×C61—81no—
152C2×D4×C61C2×D412yes—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—42no—
156C22×C4×C61—42no—
157C22×C4×C61—81no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2411yescharacteristic
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122C2×D412yes—
173C2×Dic122#4112yes—
174C2×Dic122#4112yes—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—42no—
181C4⋊C4×C61SD1614yescomplemented
182C22⋊C4×C61—42no—
183C22⋊C4×C61—42no—
184C22⋊C4×C61—42no—
185C22⋊C4×C61—42no—
186C22⋊C4×C61—81no—
187C22⋊C4×C61—81no—
188C22⋊C4×C61—81no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195#11758200—28no—
196#11758200—44no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—42no—
203C4⋊Dic61—81no—
204C4⋊Dic61#4112yes—
205#11758196—24no—
206#11758196—42no—
207#11758196—42no—
208#11758196—81no—
209#11758196—81no—
210C4×Dic61—24no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
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×#92418C2×D412yes—
225C23×C61—22no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61C22×D411yescharacteristic
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61—41no—
247D4×C61—41no—
248D4×C61—84no—
249D4×C61—84no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
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—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—84no—
272C2×C4×C61C2×SD1612yes—
273C2×C4×C61C2×SD1612yes—
274C2×C4×C61C2×SD1612yes—
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×C4×C61C2×#4111yescharacteristic
279C2×C4×C61H(5,𝔽2)11yescharacteristic
280C2×C4×C612−1+411yescharacteristic
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—22no—
284C2×Dic61—22no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293Dic122—22no—
294Dic122—22no—
295Dic122—22no—
296Dic122#12314yescomplemented
297#92418—22no—
298#92418—41no—
299#92418—41no—
300#92418—41no—
301#26163—611nomaximal, Sylow
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61—81no—
312C22×C61—81no—
313C22×C61—81no—
314C22×C61—81no—
315C22×C61—81no—
316C22×C61#52511yescharacteristic
317C22×C61D4211yescharacteristic
318C22×C61#54811yescharacteristic
319C22×C61C22×SD1611yescharacteristic
320C22×C61#57511yescharacteristic
321C22×C61#57611yescharacteristic
322C22×C61#57711yescharacteristic
323C4×C61—22no—
324C4×C61—22no—
325C4×C61—22no—
326C4×C61—41no—
327C4×C61—41no—
328C4×C61—41no—
329C4×C61—81no—
330C4×C61—81no—
331C4×C61C2×#12311yescharacteristic
332C4×C61#53911yescharacteristic
333C4×C61#54711yescharacteristic
334C4×C61#55311yescharacteristic
335Dic61—24no—
336Dic61—42no—
337Dic61—81no—
338Dic61—81no—
339C2×#460—611no—
340C2×C4×SD16—611no—
341#2445D6111yescharacteristic
342Q8×C4⋊C4—611no—
343#2094—611no—
344#2088—611no—
345#2071—611no—
346#2041—611no—
347#1934—611no—
348#1916—611no—
349#1880—611no—
350#1880—611no—
351#1880—611no—
352#1865—611no—
353#1864—611no—
354C2×C61—82no—
355C2×C61—161no—
356C2×C61#244311yescharacteristic
357C2×C61#310311yescharacteristic
358C2×C61#311911yescharacteristic
359C2×C61#326811yescharacteristic
360C2×C61D4×SD1611yescharacteristic
361C2×C61#336211yescharacteristic
362C2×C61#337111yescharacteristic
363C22×SD16—611no—
364C22×SD16—611no—
365C2×#127C2×D6111yescharacteristic
366C2×#123—21no—
367C2×C4×Q8—611no—
368C2×C4×Q8—611no—
369C2×C4×Q8—611no—
370C2×C4×D4C2×D6111yescharacteristic
371#460—1224no—
372C4×SD16—1224no—
373C2×#106—611no—
374C2×#106—611no—
375C2×C4⋊C8—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×#102—611no—
383C2×#102—611no—
384C2×C4×C8—611no—
385#394—21no—
386#394—21no—
387#390C2×D6111yescharacteristic
388#390C2×D6111yescharacteristic
389#390C2×D6111yescharacteristic
390#389—611no—
391#389—611no—
392#389—1221no—
393#389—1221no—
394#389C2×D6111yescharacteristic
395#386—21no—
396#384—611no—
397#384—1221no—
398C4×C4⋊C4—1221no—
399C4×C4⋊C4C2×D6111yescharacteristic
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404#336—611no—
405#336—611no—
406#336—611no—
407#336—611no—
408C61#2616311yescharacteristic, complemented, Sylow
409C22×Q8—611no—
410C22×D4—21no—
411C22×D4—21no—
412C22×D4C22×D6111yescharacteristic
413C22×D4C22×D6111yescharacteristic
414C23×C4—21no—
415C2×SD16—614no—
416C2×SD16—1222no—
417C2×SD16—1222no—
418C2×SD16—1222no—
419C2×SD16—1222no—
420C2×SD16—1222no—
421C22×C8—611no—
422C22×C8—611no—
423C22×C8—611no—
424C22×C8—611no—
425#127—22no—
426#127—41no—
427#123—44no—
428C4×Q8—1222no—
429C4×Q8—1222no—
430C4×Q8—1222no—
431C4×Q8—1222no—
432C4×Q8—2441no—
433C4×Q8—2441no—
434C4×D4—22no—
435C4×D4—41no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4C22⋊D6111yescharacteristic
448C2×C4⋊C4C22⋊D6111yescharacteristic
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—21no—
458C2×C42—1221no—
459C2×C42—1221no—
460C2×C42—1221no—
461C2×C42C22×D6111yescharacteristic
462C2×C42C22×D6111yescharacteristic
463C2×C42C22×D6111yescharacteristic
464#106—1222no—
465#106—2441no—
466C4⋊C8—1222no—
467#103—1222no—
468#103—1222no—
469#102—1222no—
470#102—1222no—
471#102—1222no—
472#102—1222no—
473#102—1222no—
474#102—2441no—
475C4×C8—1222no—
476#95—21no—
477#95—1221no—
478#95—1221no—
479#95C22⋊D6111yescharacteristic
480#95C22⋊D6111yescharacteristic
481C24—21no—
482C24—21no—
483C2×Q8—612no—
484C2×Q8—612no—
485C2×Q8—612no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—42no—
501C2×D4—42no—
502C2×D4—42no—
503C2×D4—42no—
504C2×D4—81no—
505C2×D4—81no—
506C2×D4D4×D6112yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—81no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4C2×C22⋊D6111yescharacteristic
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C2×C22⋊D6111yescharacteristic
519C22×C4C2×C22⋊D6111yescharacteristic
520C22×C4C2×C22⋊D6111yescharacteristic
521C22×C4C2×C22⋊D6111yescharacteristic
522C22×C4C23×D6111yescharacteristic
523SD16—1228no—
524SD16—2444no—
525C2×C8—612no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—41no—
542C4⋊C4—42no—
543C4⋊C4—612no—
544C4⋊C4—1224no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—4881no—
551C4⋊C4—4881no—
552C4⋊C4—4881no—
553C4⋊C4—2442no—
554C4⋊C4—2442no—
555C4⋊C4—2442no—
556C4⋊C4#1175820014yescomplemented
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—81no—
562C22⋊C4—81no—
563C22⋊C4—81no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—1224no—
571C42—2442no—
572C42—2442no—
573C23—22no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23C22×C22⋊D6111yescharacteristic
588Q8—614no—
589Q8—1222no—
590Q8—1222no—
591Q8—1222no—
592D4—22no—
593D4—22no—
594D4—41no—
595D4—41no—
596D4—41no—
597D4—41no—
598D4—41no—
599D4—41no—
600D4—84no—
601D4—84no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—84no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4C2×#1175820012yes—
637C2×C4C2×#1175820012yes—
638C2×C4C2×#1175820012yes—
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4#41×D6111yescharacteristic
641C2×C4#41×D6111yescharacteristic
642C2×C4#41×D6111yescharacteristic
643C2×C4#1451982211yescharacteristic
644C2×C4#1451982411yescharacteristic
645C8—1222no—
646C8—2441no—
647C8—2441no—
648C8—2441no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22#1684026911yescharacteristic
664C22D4×C22⋊D6111yescharacteristic
665C22#1684041911yescharacteristic
666C22C22×#1175820011yescharacteristic
667C22#1684056011yescharacteristic
668C22#1684056511yescharacteristic
669C22#1684057311yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—81no—
677C4—81no—
678C4—1224no—
679C4—4881no—
680C4—4881no—
681C4—2442no—
682C4#123×D6111yescharacteristic
683C4#1684035511yescharacteristic
684C4#1684041011yescharacteristic
685C4#1684045711yescharacteristic
686C2—82no—
687C2—161no—
688C2#2144955111yescharacteristic
689C2#2145584611yescharacteristic
690C2#2145601211yescharacteristic
691C2#2145735511yescharacteristic
692C2D4×#1175820011yescharacteristic
693C2#2145848611yescharacteristic
694C2#2145858111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2445C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenter#2445D61
hypocenterC61#26163
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2445

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758200C2semidirect
3. C4⋊C4×Dic122C2semidirect
4. #21447352C2semidirect
5. #21447320C2semidirect
6. #21447222C2semidirect
8. #21446505C2semidirect
9. #21446378C2semidirect
10. #21446177C2semidirect
11. #21446177C2semidirect
12. #21446177C2semidirect
13. #21446086C2semidirect
14. #21446080C2semidirect
181. C4⋊C4×C61SD16semidirect
296. Dic122#123semidirect
408. C61#26163semidirect
556. C4⋊C4#11758200semidirect

Maximal subgroups

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

subgroupindex
C2×#168399592
C2×C4×#117582002
C4⋊C4×Dic1222
#214473522
#214473202
#214472222
#214470502
#214465052
#214463782
#214461772
#214460862
#214460802
C61×#24452
#2616361