Finite Group Databasebeta

← #29738821

Subgroups — #29738821

Order 15616.

774 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839803C211yesmaximal, characteristic
2#21449539C211yesmaximal, characteristic
3#21449528C211yesmaximal, characteristic
4#21447304C211yesmaximal, characteristic, complemented
5#21447298C211yesmaximal, characteristic, complemented
6#21447154C211yesmaximal, characteristic, complemented
7#21447154C211yesmaximal, characteristic, complemented
8#21446558C211yesmaximal, characteristic, complemented
9#21446547C211yesmaximal, characteristic, complemented
10#21446017C211yesmaximal, characteristic
11#21446010C211yesmaximal, characteristic
12#21445986C211yesmaximal, characteristic, complemented
13#21445853C211yesmaximal, characteristic
14#21445602C211yesmaximal, characteristic
15C2×C61×#499C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×#14519735C2211yescharacteristic
18C2×C4⋊C4×D61C2211yescharacteristic
19#16839803—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28#16839700—21no—
29#16839697—21no—
30#16839695—21no—
31#16839684—21no—
32#16839661C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839658—21no—
37#16839643—21no—
38#16839633—21no—
39#16839631C2211yescharacteristic
40#16839627C2211yescharacteristic
41#16839619—21no—
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44C4×#11758198C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#125×C61C2211yescharacteristic
54C2×#125×C61C2211yescharacteristic
55C61×#499—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#106×C61C2211yescharacteristic
59C2×#106×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C2×#11758189—22no—
64C2×#11758189—22no—
65C2×#11758189Q814yes—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519735—22no—
69#14519735—41no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C4⋊C4×D61—22no—
82C4⋊C4×D61—41no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198D411yescharacteristic
87C2×#11758198D411yescharacteristic
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90#14519710—22no—
91#14519710—41no—
92#14519709—22no—
93#14519709—22no—
94#14519708—22no—
95#14519708—41no—
96#14519689Q814yes—
97#14519687—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110#125×C61—22no—
111#125×C61—22no—
112#125×C61—41no—
113#125×C61—41no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
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#107×C61D414yes—
125#106×C61—22no—
126#106×C61—22no—
127C4×C8×C61—22no—
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×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61C2×D412yes—
155C2×C8×C61C2×Q812yes—
156C2×C8×C61C2×Q812yes—
157C2×C8×C61#4112yes—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61—42no—
167C4⋊C4×C61—42no—
168C4⋊C4×C61C2×D412yes—
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—21no—
185C4⋊Dic61—42no—
186C4⋊Dic61—81no—
187#11758196—42no—
188#11758196—42no—
189#11758196—81no—
190#11758196—81no—
191#11758196—81no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—42no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418C2×Q812yes—
200C2×#92418C2×Q812yes—
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—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
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—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—41no—
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—
260C8×C61—22no—
261C8×C61C22⋊Q814yes—
262#92418—22no—
263#92418—22no—
264#26075—611nomaximal, Sylow
265C22×C61C2×C4⋊Q811yescharacteristic
266C22×C61D4211yescharacteristic
267C22×C61#54711yescharacteristic
268C22×C61#57511yescharacteristic
269C22×C61#57511yescharacteristic
270C22×C61#57711yescharacteristic
271C22×C61#57711yescharacteristic
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C4×C61—42no—
280C4×C61—41no—
281C4×C61—42no—
282C4×C61—41no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61C2×C22⋊Q811yescharacteristic
286C4×C61#54411yescharacteristic
287C4×C61D4×Q811yescharacteristic
288C4×C61#55411yescharacteristic
289Dic61—22no—
290Dic61—41no—
291Dic61—41no—
292Dic61—81no—
293Dic61—81no—
294Dic61—81no—
295Dic61—81no—
296C2×#499D6111yescharacteristic, complemented
297C2×#428—611no—
298#2442—611no—
299#2441—611no—
300#2085—611no—
301#2084—611no—
302#2059—611no—
303#2059—611no—
304#1943—611no—
305#1941—611no—
306#1854—611no—
307#1853—611no—
308#1847—611no—
309#1809—611no—
310#1753—611no—
311C2×C61#244011yescharacteristic
312C2×C61#316311yescharacteristic
313C2×C61#316611yescharacteristic
314C2×C61#326711yescharacteristic
315C2×C61#327111yescharacteristic
316C2×C61#335111yescharacteristic
317C2×C61#335111yescharacteristic
318D61—42no—
319C22×M16—611no—
320C2×#125C2×D6111yescharacteristic
321C2×#125C2×D6111yescharacteristic
322C2×#117—611no—
323C22×C4⋊C4—611no—
324#499—24no—
325#428—1224no—
326C2×#107—611no—
327C2×#107—611no—
328C2×#107C2×D6111yescharacteristic
329C2×#107C2×D6111yescharacteristic
330C2×#106—611no—
331C2×#106—611no—
332C2×#106—611no—
333C2×#106—611no—
334C2×#106C2×D6111yescharacteristic
335C2×#106C2×D6111yescharacteristic
336C2×C22⋊C8—611no—
337C2×#97—611no—
338C2×C4×C8C2×D6111yescharacteristic
339#400—1221no—
340#399—1221no—
341#398—1221no—
342#396—1221no—
343#389—611no—
344#389—611no—
345#389—611no—
346#389—611no—
347#389—1221no—
348#387—1221no—
349#385—1221no—
350#384—611no—
351#382—611no—
352#380—611no—
353#380—611no—
354#380—1221no—
355C4×C22⋊C4—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#336—611no—
363#336—611no—
364C61#2607511yescharacteristic, complemented, Sylow
365C23×C4—611no—
366C23×C4—1221no—
367C2×M16—614no—
368C2×M16—1222no—
369C2×M16—1222no—
370C22×C8—611no—
371C22×C8—611no—
372C22×C8C22×D6111yescharacteristic
373C22×C8C22×D6111yescharacteristic
374#125—22no—
375#125—22no—
376#125—41no—
377#125—41no—
378#117—1222no—
379#117—2441no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—1222no—
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—1221no—
396C2×C4⋊C4—2441no—
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C22⋊C4—611no—
402C2×C22⋊C4—611no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C42—611no—
407C2×C42—1221no—
408C2×C42C22×D6111yescharacteristic
409#107—22no—
410#107—1222no—
411#107—2441no—
412#107C22⋊D6114yes—
413#106—22no—
414#106—22no—
415#106—1222no—
416#106—1222no—
417#106—1222no—
418#106—2441no—
419C22⋊C8—614no—
420#97—1222no—
421C4×C8—22no—
422#95—611no—
423#95—611no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432C24—611no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—611no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—4881no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456M16—1228no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—612no—
460C2×C8—612no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8Q8×D6112yes—
466C2×C8Q8×D6112yes—
467C2×C8#1175822612yes—
468C2×C8C2×C22⋊D6112yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4881no—
488C4⋊C4—4881no—
489C4⋊C4—4881no—
490C4⋊C4—4881no—
491C4⋊C4—2442no—
492C4⋊C4—2442no—
493C4⋊C4—2442no—
494C4⋊C4C2×C22⋊D6112yes—
495C4⋊C4C2×C22⋊D6112yes—
496C22⋊C4—4881no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C42—21no—
502C42—21no—
503C42—1221no—
504C42—1221no—
505C42—2442no—
506C23—612no—
507C23—1221no—
508C23—1221no—
509C23—1221no—
510C23—1221no—
511C23—1221no—
512C23—1221no—
513C23C2×D4×D6111yescharacteristic
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
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—612no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—4884no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
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×C4C22×C22⋊D6111yescharacteristic
559C2×C4C2×Q8×D6111yescharacteristic
560C2×C4C2×Q8×D6111yescharacteristic
561C2×C4C2×#1175822611yescharacteristic
562C2×C4#1451982211yescharacteristic
563C2×C4#1451982511yescharacteristic
564C8—22no—
565C8—1222no—
566C8—1222no—
567C8#1451975314yes—
568C22—1222no—
569C22—1222no—
570C22—1222no—
571C22—2441no—
572C22—2441no—
573C22—2441no—
574C22—2441no—
575C22C4⋊Q8×D6111yescharacteristic
576C22D4×C22⋊D6111yescharacteristic
577C22#1684041511yescharacteristic
578C22#1684055911yescharacteristic
579C22#1684055911yescharacteristic
580C22#1684056911yescharacteristic
581C22#1684056911yescharacteristic
582C4—41no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—1222no—
589C4—4881no—
590C4—4881no—
591C4—4881no—
592C4—4881no—
593C4—2441no—
594C4—2441no—
595C4C2×#1451975311yescharacteristic
596C4#1684038911yescharacteristic
597C4Q8×C22⋊D6111yescharacteristic
598C4#1684047311yescharacteristic
599C2—2442no—
600C2#2144951711yescharacteristic
601C2#2145634811yescharacteristic
602C2#2145636711yescharacteristic
603C2#2145734411yescharacteristic
604C2#2145740411yescharacteristic
605C2#2145837811yescharacteristic
606C2#2145837811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#499C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#499D61
hypocenterC61#26075
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#499

Complements

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

NH ≅ G/Ntype
4. #21447304C2semidirect
5. #21447298C2semidirect
6. #21447154C2semidirect
7. #21447154C2semidirect
8. #21446558C2semidirect
9. #21446547C2semidirect
12. #21445986C2semidirect
15. C2×C61×#499C2semidirect
296. C2×#499D61semidirect
364. C61#26075semidirect

Maximal subgroups

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

subgroupindex
C2×#168398032
#214495392
#214495282
#214473042
#214472982
#214471542
#214465582
#214465472
#214460172
#214460102
#214459862
#214458532
#214456022
C2×C61×#4992
#2607561