Finite Group Databasebeta

← #29702097

Subgroups — #29702097

Order 15616.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455603C211yesmaximal, characteristic
2#21454765C211yesmaximal, characteristic, complemented
3#21454385C211yesmaximal, characteristic
4#21446798C211yesmaximal, characteristic, complemented
5#21446591C211yesmaximal, characteristic, complemented
6#21446570C211yesmaximal, characteristic, complemented
7#21446564C211yesmaximal, characteristic, complemented
8#21446419C211yesmaximal, characteristic, complemented
9#21446416C211yesmaximal, characteristic, complemented
10#21446092C211yesmaximal, characteristic
11#21445675C211yesmaximal, characteristic, complemented
12#21445666C211yesmaximal, characteristic
13#21445654C211yesmaximal, characteristic
14#21445631C211yesmaximal, characteristic
15C61×#1866C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#16840315C2211yescharacteristic
18#41×Dic61C2211yescharacteristic
19#16840071C411yescharacteristic, complemented
20#16840067C411yescharacteristic, complemented
21#16840064C411yescharacteristic, complemented
22#16840060C411yescharacteristic, complemented
23#16840055C411yescharacteristic, complemented
24#16840050C411yescharacteristic, complemented
25#16840045C411yescharacteristic, complemented
26#16840040C411yescharacteristic, complemented
27#16839891—21no—
28#16839891—21no—
29#16839889—21no—
30#16839886—21no—
31C2×#14519712C2211yescharacteristic
32#16839800C2211yescharacteristic
33#16839798C2211yescharacteristic
34#16839790—21no—
35#16839790C2211yescharacteristic
36#16839790C2211yescharacteristic
37#16839788—21no—
38#16839788C2211yescharacteristic
39#16839785—21no—
40#16839785C2211yescharacteristic
41C2×#14519706—21no—
42C2×#14519706C2211yescharacteristic
43C2×#14519706C2211yescharacteristic
44C2×#14519704C2211yescharacteristic
45#16839758C2211yescharacteristic
46#16839718C2211yescharacteristic
47#16839717C2211yescharacteristic
48M16×Dic61C2211yescharacteristic
49C2×#14519687C2211yescharacteristic
50C2×C8×Dic61C2211yescharacteristic
51#16839608C2211yescharacteristic
52#16839527C2211yescharacteristic
53#16839526C2211yescharacteristic
54#16839523C2211yescharacteristic
55#16839523C2211yescharacteristic
56#16839522C2211yescharacteristic
57#16839522C2211yescharacteristic
58#16839521C2211yescharacteristic
59#16839517C2211yescharacteristic
60C2×#131×C61C2211yescharacteristic
61C2×#108×C61C2211yescharacteristic
62C61×#424C2211yescharacteristic
63C2×#104×C61C2211yescharacteristic
64C61×#416C2211yescharacteristic
65C61×#339C2211yescharacteristic
66C61×#336C2211yescharacteristic
67#14519784—22no—
68#14519784—41no—
69C2×#11758194—21no—
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C22×#92418—21no—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519776C2×C411yescharacteristic
76#14519776C2×C411yescharacteristic
77#14519768C2×C411yescharacteristic
78#14519768C2×C411yescharacteristic
79#14519754—21no—
80#14519754—21no—
81Q8×Dic61C2×C411yescharacteristic
82Q8×Dic61C2×C411yescharacteristic
83C4×Dic122—21no—
84C4×Dic122—21no—
85D4×Dic61—21no—
86D4×Dic61—21no—
87D4×Dic61C2×C411yescharacteristic
88D4×Dic61C2×C411yescharacteristic
89#14519736—21no—
90#14519736C2311yescharacteristic
91#14519736C2311yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519712—41no—
96#14519711C2×C411yescharacteristic
97#14519711C2×C411yescharacteristic
98#14519710C2×C411yescharacteristic
99#14519710C2×C411yescharacteristic
100#14519709C2×C411yescharacteristic
101#14519709C2×C411yescharacteristic
102#14519708C2×C411yescharacteristic
103#14519708C2×C411yescharacteristic
104#14519706—21no—
105#14519706—21no—
106#14519706—21no—
107#14519706—21no—
108#14519706—21no—
109#14519706—21no—
110#14519706—21no—
111#14519706—21no—
112#14519706C2×C411yescharacteristic
113#14519706C2×C411yescharacteristic
114#14519706C2×C411yescharacteristic
115#14519706C2×C411yescharacteristic
116#14519706D411yescharacteristic
117#14519706D411yescharacteristic
118#14519706D411yescharacteristic
119#14519706D411yescharacteristic
120#14519704Q814yescomplemented
121#14519701C2×C411yescharacteristic
122#14519701C2×C411yescharacteristic
123#14519698C2×C411yescharacteristic
124#14519698C2×C411yescharacteristic
125#14519688—21no—
126#14519687—21no—
127#14519687—21no—
128#14519687—41no—
129C4×#92418—21no—
130C8×Dic61—21no—
131C8×Dic61—21no—
132C8×Dic61—41no—
133#14519684—21no—
134#14519684—21no—
135C2×#41×C61C2311yescharacteristic
136#131×C61—22no—
137#131×C61—41no—
138C2×M16×C61—21no—
139C2×M16×C61C2311yescharacteristic
140C2×M16×C61C2311yescharacteristic
141C22×C8×C61—21no—
142C22×C8×C61C2311yescharacteristic
143C22×C8×C61C2311yescharacteristic
144#117×C61C2311yescharacteristic
145C2×C42×C61C2311yescharacteristic
146#108×C61—41no—
147C4⋊C8×C61—21no—
148C4⋊C8×C61—21no—
149#104×C61—22no—
150#103×C61C2×C411yescharacteristic
151#103×C61C2×C411yescharacteristic
152#102×C61C2×C411yescharacteristic
153#102×C61C2×C411yescharacteristic
154#41×C61—21no—
155#41×C61—21no—
156#41×C61C2×Q812yes—
157#41×C61#4112yes—
158C2×Q8×C61C22×C411yescharacteristic
159C2×D4×C61—21no—
160C2×D4×C61C22×C411yescharacteristic
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61C2411yescharacteristic
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61#4111yescharacteristic
168C22×Dic61#4111yescharacteristic
169C2×Dic122—21no—
170C2×Dic122—21no—
171M16×C61—21no—
172M16×C61—21no—
173M16×C61—21no—
174M16×C61—21no—
175M16×C61—42no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—42no—
183C2×C8×C61C22×C411yescharacteristic
184C2×C8×C61C22×C411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C2×C8×C61C2×D411yescharacteristic
187C4⋊C4×C61C22×C411yescharacteristic
188C4⋊C4×C61C22×C411yescharacteristic
189C22⋊C4×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C22⋊Dic61—21no—
194C22⋊Dic61—21no—
195C22⋊Dic61—21no—
196C22⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C22×C411yescharacteristic
202C4⋊Dic61C22×C411yescharacteristic
203C4⋊Dic61C22×C411yescharacteristic
204C4⋊Dic61C22×C411yescharacteristic
205#11758196—21no—
206#11758196—21no—
207#11758196—21no—
208#11758196—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61C22×C411yescharacteristic
215C4×Dic61C22×C411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×Q812yes—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194—42no—
223#11758194C2×Q812yes—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—42no—
231C2×#92418C22×C411yescharacteristic
232C2×#92418C22×C411yescharacteristic
233C2×#92418C2×D411yescharacteristic
234C2×#92418C2×D411yescharacteristic
235C23×C61—21no—
236C23×C61C2×#4111yescharacteristic
237Q8×C61—21no—
238Q8×C61C4×Q812yes—
239D4×C61—22no—
240D4×C61—22no—
241D4×C61—21no—
242D4×C61C4×Q812yes—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×Q811yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—42no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—81no—
268C2×Dic61C4×D411yescharacteristic
269C2×Dic61C4×D411yescharacteristic
270C2×Dic61C4×D411yescharacteristic
271C2×Dic61C4×D411yescharacteristic
272Dic122—42no—
273Dic122—81no—
274C8×C61—21no—
275C8×C61—21no—
276C8×C61—21no—
277C8×C61—21no—
278C8×C61—41no—
279C8×C61—41no—
280#92418—21no—
281#92418—21no—
282#92418—21no—
283#92418—21no—
284#92418—22no—
285#92418—41no—
286#22984—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—22no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61C2×C4×D411yescharacteristic
293C22×C61#55211yescharacteristic
294C22×C61#55411yescharacteristic
295C4×C61—22no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—81no—
299C4×C61C2×C4×Q811yescharacteristic
300C4×C61#52011yescharacteristic
301C4×C61#54611yescharacteristic
302C4×C61D4×Q811yescharacteristic
303Dic61—21no—
304Dic61—21no—
305Dic61—21no—
306Dic61—21no—
307Dic61—42no—
308Dic61—41no—
309Dic61—41no—
310Dic61—41no—
311Dic61—41no—
312#3076—611no—
313#2962—611no—
314#2906—611no—
315#1995—611no—
316#1950—611no—
317#1946—611no—
318#1945—611no—
319#1922—611no—
320#1922—611no—
321#1866—611no—
322#1866D6111yescharacteristic
323#1768—611no—
324#1765—611no—
325#1764—611no—
326#1759—611no—
327C2×C61—21no—
328C2×C61—21no—
329C2×C61—42no—
330C2×C61#236011yescharacteristic
331C2×C61#295511yescharacteristic
332C2×C61#295711yescharacteristic
333C2×#131—611no—
334C2×#131C2×D6111yescharacteristic
335#533—611no—
336C4×#41—611no—
337#479—611no—
338#479—611no—
339#478—611no—
340#478—611no—
341#477—611no—
342#476—611no—
343#475—611no—
344#474—611no—
345#446—1221no—
346#446—1221no—
347#446—1221no—
348C8×Q8—1221no—
349C2×#108—611no—
350C2×#108C2×D6111yescharacteristic
351#427—611no—
352#427—611no—
353#424—611no—
354#424—611no—
355#424—611no—
356#424—1221no—
357#424—1221no—
358#424C2×D6111yescharacteristic
359#423—611no—
360#423—1221no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—611no—
363C2×C4⋊C8—1221no—
364C2×#104—611no—
365C2×#104C2×D6111yescharacteristic
366#416—611no—
367#416C2×D6111yescharacteristic
368#405—611no—
369#405—611no—
370C4×M16—611no—
371C2×#97—611no—
372C2×C4×C8—611no—
373#376—611no—
374#341—611no—
375#341—611no—
376#339—611no—
377#339—611no—
378#339—611no—
379#339—611no—
380#339—611no—
381#339C2×D6111yescharacteristic
382#336—611no—
383#336C2×D6111yescharacteristic
384C61#2298411yescharacteristic, complemented, Sylow
385C2×#41C22×D6111yescharacteristic
386#131—22no—
387#131—41no—
388#131—1222no—
389#131—2441no—
390C2×M16—21no—
391C2×M16—611no—
392C2×M16—611no—
393C2×M16—1221no—
394C2×M16C22×D6111yescharacteristic
395C2×M16C22×D6111yescharacteristic
396C22×C8—21no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—1221no—
400C22×C8C22×D6111yescharacteristic
401C22×C8C22×D6111yescharacteristic
402C4⋊Q8—611no—
403C4⋊Q8—611no—
404#125—611no—
405#125—611no—
406C22⋊Q8—1221no—
407C22⋊Q8—1221no—
408C4×Q8—611no—
409C4×Q8—611no—
410C4×Q8—1221no—
411C4×Q8—1221no—
412C4×D4—611no—
413C4×D4—611no—
414C4×D4—1221no—
415C4×D4—1221no—
416#117—611no—
417#117—611no—
418#117—1221no—
419#117C22×D6111yescharacteristic
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—1221no—
423C2×C42C22×D6111yescharacteristic
424#108—41no—
425#108—2441no—
426#107—611no—
427#107—611no—
428#107—611no—
429#107—611no—
430#106—611no—
431#106—611no—
432#106—611no—
433#106—611no—
434C4⋊C8—21no—
435C4⋊C8—21no—
436C4⋊C8—611no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—611no—
441C4⋊C8—611no—
442C4⋊C8—611no—
443C4⋊C8—611no—
444C4⋊C8—1221no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447C4⋊C8—1221no—
448C4⋊C8—1221no—
449C4⋊C8—1221no—
450C4⋊C8—1221no—
451C4⋊C8—1221no—
452#104—22no—
453#104—614no—
454#103—611no—
455#103—611no—
456#103C4×D6111yescharacteristic
457#103C4×D6111yescharacteristic
458#102—611no—
459#102—611no—
460#102C4×D6111yescharacteristic
461#102C4×D6111yescharacteristic
462#97—1221no—
463#97—1221no—
464#97—1221no—
465#97—2441no—
466C4×C8—1221no—
467C4×C8—1221no—
468C4×C8—1221no—
469C4×C8—2441no—
470#95—1221no—
471#95—1221no—
472#41—21no—
473#41—21no—
474#41#1175822412yes—
475#41Q8×D6112yes—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8C2×C4×D6111yescharacteristic
479C2×D4—21no—
480C2×D4C2×C4×D6111yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4C23×D6111yescharacteristic
489M16—21no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—42no—
494M16—612no—
495M16—1221no—
496M16—1221no—
497M16—1221no—
498M16—2442no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—42no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—2442no—
517C2×C8C2×C4×D6111yescharacteristic
518C2×C8C2×C4×D6111yescharacteristic
519C2×C8D4×D6111yescharacteristic
520C2×C8D4×D6111yescharacteristic
521C4⋊C4—611no—
522C4⋊C4—611no—
523C4⋊C4—611no—
524C4⋊C4—611no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4C2×C4×D6111yescharacteristic
534C4⋊C4C2×C4×D6111yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—612no—
548C42—1221no—
549C42—1221no—
550C42—1221no—
551C42—1221no—
552C42—1221no—
553C23—21no—
554C23#41×D6111yescharacteristic
555Q8—21no—
556Q8—4881no—
557Q8—2442no—
558Q8#1451974212yes—
559D4—22no—
560D4—22no—
561D4—21no—
562D4#1451974212yes—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—611no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—4881no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2442no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4C22×C4×D6111yescharacteristic
587C2×C4C2×D4×D6111yescharacteristic
588C2×C4C2×Q8×D6111yescharacteristic
589C2×C4C2×#1175822411yescharacteristic
590C2×C4#1451982211yescharacteristic
591C2×C4#1451982411yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—41no—
597C8—41no—
598C8—1221no—
599C8—1221no—
600C8—1221no—
601C8—1221no—
602C8—1222no—
603C8—2441no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C4×D4×D6111yescharacteristic
610C22#1684044911yescharacteristic
611C22#1684046611yescharacteristic
612C4—22no—
613C4—41no—
614C4—41no—
615C4—81no—
616C4—1221no—
617C4—1221no—
618C4—1221no—
619C4—1221no—
620C4—2442no—
621C4—2441no—
622C4—2441no—
623C4—2441no—
624C4—2441no—
625C4C2×#1451974211yescharacteristic
626C4#1684024111yescharacteristic
627C4#1684039911yescharacteristic
628C4#1684042511yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#2144881911yescharacteristic
633C2#2145471711yescharacteristic
634C2#2145473311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1866C2
socleC22×C61C2×C4×D4
prime radicalC2×#41×C61C23
cosocleC22×C4×C61C24
hypercenter#1866D61
hypocenterC61#22984
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

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

NH ≅ G/Ntype
2. #21454765C2semidirect
4. #21446798C2semidirect
5. #21446591C2semidirect
6. #21446570C2semidirect
7. #21446564C2semidirect
8. #21446419C2semidirect
9. #21446416C2semidirect
11. #21445675C2semidirect
19. #16840071C4semidirect
20. #16840067C4semidirect
21. #16840064C4semidirect
22. #16840060C4semidirect
23. #16840055C4semidirect
24. #16840050C4semidirect
25. #16840045C4semidirect
26. #16840040C4semidirect
120. #14519704Q8semidirect
384. C61#22984semidirect

Maximal subgroups

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

subgroupindex
#214556032
#214547652
#214543852
#214467982
#214465912
#214465702
#214465642
#214464192
#214464162
#214460922
#214456752
#214456662
#214456542
#214456312
C61×#18662
#2298461