Finite Group Databasebeta

← #29738888

Subgroups — #29738888

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840170C211yesmaximal, characteristic, complemented
2C2×#16839801C211yesmaximal, characteristic
3#21449541C211yesmaximal, characteristic
4#21449513C211yesmaximal, characteristic
5#21447302C211yesmaximal, characteristic, complemented
6#21447159C211yesmaximal, characteristic, complemented
7#21447146C211yesmaximal, characteristic, complemented
8#21446556C211yesmaximal, characteristic, complemented
9#21446549C211yesmaximal, characteristic, complemented
10#21446019C211yesmaximal, characteristic
11#21446005C211yesmaximal, characteristic
12#21445981C211yesmaximal, characteristic, complemented
13#21445851C211yesmaximal, characteristic
14#21445599C211yesmaximal, characteristic
15C61×#2085C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×C4⋊C4×D61C2211yescharacteristic
21#16840170—24no—
22#16839801—24no—
23C2×#14519711C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519709C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31C2×#14519689C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
34#16839702—21no—
35#16839692—21no—
36#16839686—21no—
37#16839675—21no—
38#16839666—21no—
39#16839661C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839644—21no—
43#16839636—21no—
44#16839628C2211yescharacteristic
45#16839619—21no—
46#16839619C2211yescharacteristic
47#16839619C2211yescharacteristic
48C4×#11758198C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#382C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×C4×D61—21no—
63C22×C4×D61C2311yescharacteristic
64C2×#11758189—22no—
65C2×#11758189—22no—
66C2×#11758189Q814yescomplemented
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519776—22no—
70#14519776—41no—
71#14519768—22no—
72#14519768—41no—
73#14519735—22no—
74#14519735—41no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61—41no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198D411yescharacteristic
92C2×#11758198D411yescharacteristic
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95#14519711—22no—
96#14519710—22no—
97#14519710—41no—
98#14519710—41no—
99#14519710D414yes—
100#14519709—22no—
101#14519708—22no—
102#14519708—22no—
103#14519689Q814yescomplemented
104#14519687—22no—
105C4×#92418—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123#107×C61—22no—
124#106×C61—22no—
125#95×C61—21no—
126#95×C61—21no—
127#95×C61—21no—
128C23×D61C2×D411yescharacteristic
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61—81no—
150C2×C4×D61C2×Q812yes—
151C2×C4×D61C2×Q812yes—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—81no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61C2×Q812yes—
160C2×C8×C61C2×Q812yes—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—81no—
169C4⋊C4×C61C2×D412yes—
170C42×C61—21no—
171C42×C61—21no—
172#11758198—41no—
173#11758198—41no—
174#11758198—42no—
175#11758198—42no—
176#11758198—81no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—42no—
185C4⋊Dic61—81no—
186C4⋊Dic61C2×D412yes—
187#11758196—42no—
188#11758196—42no—
189#11758196—42no—
190#11758196—42no—
191#11758196—81no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—42no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418C2×D412yes—
198C2×#92418C2×Q812yes—
199C2×#92418C2×Q812yes—
200C2×#92418#4112yes—
201#11758189—28no—
202C23×C61C22×D411yescharacteristic
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61C4⋊Q812yes—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
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×C61C22×D411yescharacteristic
229C2×C4×C61C22×Q811yescharacteristic
230C2×C4×C61C22×Q811yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×C4×C612−1+411yescharacteristic
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61C4⋊Q812yes—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—84no—
260Dic122—84no—
261C8×C61—22no—
262C8×C61—22no—
263#92418—22no—
264#92418C22⋊Q814yes—
265#26079—611nomaximal, Sylow
266C22×C61C2×C4⋊Q811yescharacteristic
267C22×C61D4211yescharacteristic
268C22×C61#54711yescharacteristic
269C22×C61#57511yescharacteristic
270C22×C61#57511yescharacteristic
271C22×C61#57611yescharacteristic
272C22×C61#57811yescharacteristic
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61C2×C22⋊Q811yescharacteristic
287C4×C61#54411yescharacteristic
288C4×C61D4×Q811yescharacteristic
289C4×C61#55411yescharacteristic
290Dic61—22no—
291Dic61—41no—
292Dic61—42no—
293Dic61—41no—
294Dic61—81no—
295Dic61—81no—
296Dic61—81no—
297C2×#501—611no—
298C2×#428—611no—
299#2442—611no—
300#2439—611no—
301#2085—611no—
302#2085D6111yescharacteristic, complemented
303#2059—611no—
304#2057—611no—
305#1943—611no—
306#1942—611no—
307#1854—611no—
308#1852—611no—
309#1846—611no—
310#1809—611no—
311#1753—611no—
312C2×C61#244011yescharacteristic
313C2×C61#316411yescharacteristic
314C2×C61#316611yescharacteristic
315C2×C61#327311yescharacteristic
316C2×C61#327711yescharacteristic
317C2×C61#334711yescharacteristic
318C2×C61#334911yescharacteristic
319D61—42no—
320C22×M16—611no—
321C2×C4⋊Q8—611no—
322C2×#125—611no—
323C2×#117—611no—
324C22×C4⋊C4—611no—
325#501—1224no—
326#428—1224no—
327C2×#107—611no—
328C2×#107—611no—
329C2×#107—611no—
330C2×#107—611no—
331C2×#107—611no—
332C2×#107C2×D6111yescharacteristic
333C2×#106—611no—
334C2×#106—611no—
335C2×#106—611no—
336C2×#106C2×D6111yescharacteristic
337C2×C22⋊C8—611no—
338C2×#97—611no—
339C2×C4×C8—611no—
340#400—1221no—
341#398—1221no—
342#396—1221no—
343#393—1221no—
344#391—1221no—
345#389—611no—
346#389—611no—
347#389—611no—
348#389C2×D6111yescharacteristic
349#387—1221no—
350#385—1221no—
351#382—611no—
352#382C2×D6111yescharacteristic
353#380—611no—
354#380—611no—
355#380—1221no—
356C4×C22⋊C4—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340C2×D6111yescharacteristic
362#340C2×D6111yescharacteristic
363#336—611no—
364#336C2×D6111yescharacteristic
365C61#2607911yescharacteristic, complemented, Sylow
366C22×Q8—1221no—
367C23×C4—611no—
368C23×C4—1221no—
369C2×M16—614no—
370C2×M16—1222no—
371C2×M16—1222no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8C22×D6111yescharacteristic
375C22×C8C22×D6111yescharacteristic
376C4⋊Q8—1222no—
377C4⋊Q8—2441no—
378#125—1222no—
379#125—2441no—
380#117—1222no—
381#117—2441no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—2441no—
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—611no—
401C2×C22⋊C4—611no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C42—611no—
406C2×C42—1221no—
407C2×C42C22×D6111yescharacteristic
408#107—22no—
409#107—614no—
410#107—1222no—
411#107—1222no—
412#107—2441no—
413#107—2441no—
414#106—22no—
415#106—1222no—
416#106—1222no—
417#106—1222no—
418C22⋊C8—614no—
419#97—1222no—
420C4×C8—1222no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—611no—
425#95—611no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433C24—611no—
434C2×Q8—4881no—
435C2×Q8—2442no—
436C2×Q8—2442no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—611no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—4881no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460M16—1228no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—612no—
466C2×C8—612no—
467C2×C8—612no—
468C2×C8—612no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8Q8×D6112yes—
472C2×C8Q8×D6112yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—81no—
481C4⋊C4—612no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—4881no—
490C4⋊C4—4881no—
491C4⋊C4—2442no—
492C4⋊C4—2442no—
493C4⋊C4—2442no—
494C4⋊C4—2442no—
495C4⋊C4—2442no—
496C4⋊C4D4×D6112yes—
497C22⋊C4—4881no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C22⋊C4—2442no—
501C22⋊C4—2442no—
502C42—21no—
503C42—21no—
504C42—1221no—
505C42—1221no—
506C42—2442no—
507C23—612no—
508C23—1221no—
509C23—1221no—
510C23—1221no—
511C23—1221no—
512C23—1221no—
513C23—1221no—
514C23C2×D4×D6111yescharacteristic
515Q8—4884no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—612no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1222no—
548C2×C4—4884no—
549C2×C4—2441no—
550C2×C4—2441no—
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×C4C2×D4×D6111yescharacteristic
561C2×C4C2×Q8×D6111yescharacteristic
562C2×C4C2×Q8×D6111yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C2×C4#1451982311yescharacteristic
565C2×C4#1451982411yescharacteristic
566C8—22no—
567C8—22no—
568C8—614no—
569C8—1222no—
570C22—1222no—
571C22—1222no—
572C22—1222no—
573C22—2441no—
574C22—2441no—
575C22—2441no—
576C22—2441no—
577C22C4⋊Q8×D6111yescharacteristic
578C22#1684039611yescharacteristic
579C22#1684041011yescharacteristic
580C22#1684055911yescharacteristic
581C22#1684055911yescharacteristic
582C22#1684056611yescharacteristic
583C22#1684057811yescharacteristic
584C4—42no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—1222no—
591C4—4881no—
592C4—4881no—
593C4—4881no—
594C4—2441no—
595C4—2442no—
596C4—2441no—
597C4C22⋊Q8×D6111yescharacteristic
598C4#1684039211yescharacteristic
599C4#1684042511yescharacteristic
600C4#1684046811yescharacteristic
601C2—2442no—
602C2#2144951611yescharacteristic
603C2#2145635411yescharacteristic
604C2#2145636711yescharacteristic
605C2#2145742511yescharacteristic
606C2#2145747111yescharacteristic
607C2#2145832311yescharacteristic
608C2#2145834711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2085C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2085D61
hypocenterC61#26079
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2085

Complements

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

NH ≅ G/Ntype
1. C2×#16840170C2semidirect
5. #21447302C2semidirect
6. #21447159C2semidirect
7. #21447146C2semidirect
8. #21446556C2semidirect
9. #21446549C2semidirect
12. #21445981C2semidirect
15. C61×#2085C2semidirect
66. C2×#11758189Q8semidirect
103. #14519689Q8semidirect
302. #2085D61semidirect
365. C61#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#168401702
C2×#168398012
#214495412
#214495132
#214473022
#214471592
#214471462
#214465562
#214465492
#214460192
#214460052
#214459812
#214458512
#214455992
C61×#20852
#2607961