Finite Group Databasebeta

← #29731801

Subgroups — #29731801

Order 15616.

812 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840172C211yesmaximal, characteristic, complemented
2#21450772C211yesmaximal, characteristic
3#21448919C211yesmaximal, characteristic, complemented
4#21447298C211yesmaximal, characteristic, complemented
5#21447293C211yesmaximal, characteristic, complemented
6#21447203C211yesmaximal, characteristic
7#21447189C211yesmaximal, characteristic
8#21446766C211yesmaximal, characteristic, complemented
9#21446579C211yesmaximal, characteristic, complemented
10#21446574C211yesmaximal, characteristic, complemented
11#21446540C211yesmaximal, characteristic, complemented
12#21446449C211yesmaximal, characteristic
13#21446442C211yesmaximal, characteristic
14#21445644C211yesmaximal, characteristic
15C2×C61×#478C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519756—21no—
18C2×#14519756—21no—
19#16840172—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29#16839698—21no—
30#16839698—21no—
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839661—21no—
34#16839661C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839648—21no—
37#16839648—21no—
38#16839631C2211yescharacteristic
39#16839631C2211yescharacteristic
40C4⋊C4×Dic61—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#125×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C61×#478—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519777—22no—
65#14519777—22no—
66#14519756—44no—
67#14519756—44no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519710—22no—
91#14519710—22no—
92#14519708—22no—
93#14519708—22no—
94#14519706—22no—
95#14519706—22no—
96#14519698—22no—
97#14519698Q814yescomplemented
98#14519688—22no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106C22×D4×C61C2311yescharacteristic
107C23×C4×C61—21no—
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110#125×C61—22no—
111#125×C61D414yes—
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#107×C61—22no—
122#106×C61—22no—
123C4⋊C8×C61—22no—
124#102×C61—22no—
125#102×C61—22no—
126C24×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61C2×Q812yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×Dic122—42no—
151C2×Dic122—42no—
152C2×Dic122—42no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C2×D412yes—
166C4⋊C4×C61C2×D412yes—
167C22⋊C4×C61—42no—
168C42×C61C2×D412yes—
169C42×C61#4112yes—
170C22⋊Dic61—42no—
171C22⋊Dic61—42no—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174C4⋊Dic61—21no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—42no—
184C4⋊Dic61—42no—
185C4⋊Dic61—81no—
186C4⋊Dic61C2×Q812yes—
187#11758196—42no—
188#11758196—42no—
189#11758196—42no—
190#11758196—42no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—42no—
196C4×Dic61—81no—
197C4×Dic61—81no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C2×Q812yes—
203C2×#92418#4112yes—
204C23×C61—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C22×D411yescharacteristic
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—41no—
215D4×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—84no—
230C2×C4×C61#12112yes—
231C2×C4×C61#12112yes—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×Q811yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×C4×C612−1+411yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61AGL(1,ℤ/8)12yes—
261C2×Dic61AGL(1,ℤ/8)12yes—
262Dic122—84no—
263C8×C61—41no—
264C8×C61—41no—
265#92418—22no—
266#92418—22no—
267#25465—611nomaximal, Sylow
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61C2×#12111yescharacteristic
276C22×C61#54711yescharacteristic
277C22×C61D4×Q811yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C61#57711yescharacteristic
281C22×C61#57811yescharacteristic
282C4×C61—22no—
283C4×C61—22no—
284C4×C61—42no—
285C4×C61—42no—
286C4×C61—81no—
287C4×C61#53311yescharacteristic
288C4×C61#54111yescharacteristic
289C4×C61#54811yescharacteristic
290C4×C61D4×Q811yescharacteristic
291Dic61—22no—
292Dic61—22no—
293Dic61—41no—
294Dic61—41no—
295Dic61—42no—
296Dic61—81no—
297Dic61—81no—
298Dic61—81no—
299C2×#501—611no—
300C2×#478D6111yescharacteristic
301#2542—611no—
302#2370—611no—
303#2084—611no—
304#2083—611no—
305#2067—611no—
306#2065—611no—
307#1987—611no—
308#1948—611no—
309#1947—611no—
310#1940—611no—
311#1926—611no—
312#1925—611no—
313#1762—611no—
314C2×C61—82no—
315C2×C61#242911yescharacteristic
316C2×C61#303811yescharacteristic
317C2×C61#304611yescharacteristic
318C2×C61#334311yescharacteristic
319C2×C61#334511yescharacteristic
320C2×C61#338411yescharacteristic
321C2×C61#338611yescharacteristic
322C2×C4⋊Q8—611no—
323C2×#125C2×D6111yescharacteristic
324C2×C22⋊Q8—1221no—
325C2×C22⋊Q8—1221no—
326C2×C4×D4C2×D6111yescharacteristic
327#501—1224no—
328#478—24no—
329C2×#107—611no—
330C2×#107—611no—
331C2×#107C2×D6111yescharacteristic
332C2×#106—611no—
333C2×#106—611no—
334C2×#106C2×D6111yescharacteristic
335C2×C4⋊C8—611no—
336C2×C4⋊C8—611no—
337C2×C4⋊C8C2×D6111yescharacteristic
338C2×#102—611no—
339C2×#102—611no—
340C2×#102C2×D6111yescharacteristic
341C2×#102C2×D6111yescharacteristic
342C2×#97—611no—
343#399—1221no—
344#399—1221no—
345#390—611no—
346#390—611no—
347#389—611no—
348#389—611no—
349#389—1221no—
350#387—1221no—
351#387—1221no—
352#384—611no—
353#384—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—1221no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#336—611no—
366#336—611no—
367C61#2546511yescharacteristic, complemented, Sylow
368C22×Q8—1221no—
369C22×D4C22×D6111yescharacteristic
370C23×C4—21no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C4⋊Q8—1222no—
376C4⋊Q8—1222no—
377#125—22no—
378#125C22⋊D6114yes—
379C22⋊Q8—2444no—
380C22⋊Q8—2444no—
381C4×D4—22no—
382C4×D4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42C22×D6111yescharacteristic
412#107—22no—
413#107—1222no—
414#107—1222no—
415#106—22no—
416#106—1222no—
417#106—1222no—
418C4⋊C8—22no—
419C4⋊C8—1222no—
420C4⋊C8—1222no—
421#102—22no—
422#102—22no—
423#102—614no—
424#102—1222no—
425#97—1222no—
426#95—611no—
427#95—611no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433C24—21no—
434C2×Q8—2442no—
435C2×Q8—2442no—
436C2×Q8—2442no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4Q8×D6112yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—612no—
467C2×C8—612no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—612no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—4881no—
489C4⋊C4—2442no—
490C4⋊C4—2442no—
491C4⋊C4—2442no—
492C4⋊C4—2442no—
493C4⋊C4—2442no—
494C4⋊C4—2442no—
495C4⋊C4C2×C22⋊D6112yes—
496C4⋊C4C2×C22⋊D6112yes—
497C22⋊C4—42no—
498C22⋊C4—2442no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C22⋊C4—2442no—
502C42—1221no—
503C42—1221no—
504C42—1221no—
505C42—1221no—
506C42—4881no—
507C42—4881no—
508C42—2442no—
509C42#1175822412yes—
510C42C2×C22⋊D6112yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×D4×D6111yescharacteristic
519Q8—4884no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1222no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4#1451974912yes—
563C2×C4#1451974912yes—
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4C2×Q8×D6111yescharacteristic
566C2×C4#41×D6111yescharacteristic
567C2×C4C2×#1175822411yescharacteristic
568C2×C4#1451982411yescharacteristic
569C2×C4#1451982511yescharacteristic
570C8—41no—
571C8—41no—
572C8—1222no—
573C8—1222no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×#1451974911yescharacteristic
582C22#1684041411yescharacteristic
583C22#1684042511yescharacteristic
584C22AGL(1,ℤ/8)×D6111yescharacteristic
585C22AGL(1,ℤ/8)×D6111yescharacteristic
586C22#1684056911yescharacteristic
587C22#1684057611yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—42no—
592C4—81no—
593C4—1222no—
594C4—1222no—
595C4—4881no—
596C4—4881no—
597C4—4881no—
598C4—2442no—
599C4—2441no—
600C4—2441no—
601C4#1684031411yescharacteristic
602C4#1684037211yescharacteristic
603C4#1684041911yescharacteristic
604C4Q8×C22⋊D6111yescharacteristic
605C2—82no—
606C2#2144942011yescharacteristic
607C2#2145529711yescharacteristic
608C2#2145536711yescharacteristic
609C2#2145826611yescharacteristic
610C2#2145828611yescharacteristic
611C2#2145872111yescharacteristic
612C2#2145873511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#478C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#478D61
hypocenterC61#25465
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. C2×#16840172C2semidirect
3. #21448919C2semidirect
4. #21447298C2semidirect
5. #21447293C2semidirect
8. #21446766C2semidirect
9. #21446579C2semidirect
10. #21446574C2semidirect
11. #21446540C2semidirect
97. #14519698Q8semidirect
367. C61#25465semidirect

Maximal subgroups

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

subgroupindex
C2×#168401722
#214507722
#214489192
#214472982
#214472932
#214472032
#214471892
#214467662
#214465792
#214465742
#214465402
#214464492
#214464422
#214456442
C2×C61×#4782
#2546561