Finite Group Databasebeta

← #29737674

Subgroups — #29737674

Order 15616.

774 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449521C211yesmaximal, characteristic
2#21448694C211yesmaximal, characteristic
3#21447378C211yesmaximal, characteristic, complemented
4#21447317C211yesmaximal, characteristic, complemented
5#21447317C211yesmaximal, characteristic, complemented
6#21447153C211yesmaximal, characteristic, complemented
7#21446770C211yesmaximal, characteristic, complemented
8#21446583C211yesmaximal, characteristic, complemented
9#21446398C211yesmaximal, characteristic, complemented
10#21446398C211yesmaximal, characteristic, complemented
11#21446014C211yesmaximal, characteristic
12#21445910C211yesmaximal, characteristic
13#21445844C211yesmaximal, characteristic
14C2×#16839525C211yesmaximal, characteristic
15C61×#1850C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519754—21no—
18C22×C4⋊Dic61C2211yescharacteristic
19C22×C4⋊Dic61C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24#16839694—21no—
25#16839683—21no—
26#16839660C2211yescharacteristic
27#16839660C2211yescharacteristic
28#16839660C2211yescharacteristic
29#16839660C2211yescharacteristic
30#16839645—21no—
31#16839645—21no—
32#16839637—21no—
33#16839632C2211yescharacteristic
34#16839622C2211yescharacteristic
35#16839622C2211yescharacteristic
36#16839620—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38C4×C4⋊Dic61C2211yescharacteristic
39C22⋊C4×Dic61—21no—
40#16839525C2211yescharacteristic
41#16839525C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C23×C8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C4×C22⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122—21no—
60C23×Dic61—21no—
61C23×Dic61—21no—
62#14519777—22no—
63#14519777—41no—
64#14519754—44no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—41no—
73C2×C4⋊Dic61Q814yescomplemented
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90#14519709—22no—
91#14519709—22no—
92#14519709—41no—
93#14519709—41no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102C23×C4×C61C2311yescharacteristic
103C22×C8×C61—22no—
104C22×C8×C61—22no—
105C22×C8×C61—41no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C2×C22⋊C4×C61D411yescharacteristic
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C42×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114C4⋊C8×C61—22no—
115C4⋊C8×C61—41no—
116C22⋊C8×C61—22no—
117#95×C61D411yescharacteristic
118#95×C61D411yescharacteristic
119C24×C61C2×D411yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×Q812yes—
130C22×C4×C61#4112yes—
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—24no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—42no—
142C22×Dic61—42no—
143C22×Dic61—81no—
144C22×Dic61—81no—
145C22×Dic61—81no—
146C2×Dic122—42no—
147C2×Dic122—42no—
148C2×Dic122—81no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—44no—
161C2×C8×C61#4112yes—
162C22⋊C4×C61—22no—
163C22⋊C4×C61SD1614yes—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—21no—
168C22⋊Dic61—42no—
169C22⋊Dic61—42no—
170C4⋊Dic61—24no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
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—42no—
186C4⋊Dic61—42no—
187C4⋊Dic61—81no—
188C4⋊Dic61C2×Q812yes—
189C4⋊Dic61C2×Q812yes—
190C4⋊Dic61#4112yes—
191#11758196—42no—
192#11758196—42no—
193#11758196—81no—
194#11758196—81no—
195C4×Dic61—81no—
196C4×Dic61—81no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61C22⋊Q812yes—
202C23×C61C2×SD1612yes—
203C23×C61#13712yes—
204C23×C61C22×D411yescharacteristic
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61C22⋊Q812yes—
222C2×C4×C61C2×SD1612yes—
223C2×C4×C61C2×SD1612yes—
224C2×C4×C61C22×Q811yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×C4×C612−1+411yescharacteristic
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
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×Dic61—84no—
255C2×Dic61—84no—
256Dic122—84no—
257C8×C61—22no—
258C8×C61—41no—
259C8×C61—41no—
260C8×C61—41no—
261#25992—611nomaximal, Sylow
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61#47514yes—
269C22×C61C2×C22⋊Q811yescharacteristic
270C22×C61#54611yescharacteristic
271C22×C61#54711yescharacteristic
272C22×C61C22×SD1611yescharacteristic
273C22×C61C2×#13711yescharacteristic
274C22×C61#57611yescharacteristic
275C22×C61#57711yescharacteristic
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61#53311yescharacteristic
282C4×C61#54411yescharacteristic
283C4×C61#55111yescharacteristic
284C4×C61#55211yescharacteristic
285Dic61—42no—
286Dic61—42no—
287Dic61—42no—
288Dic61—81no—
289Dic61—81no—
290Dic61—81no—
291Dic61—81no—
292Dic61—81no—
293#2440—611no—
294#2345—611no—
295#2099—611no—
296#2088—611no—
297#2088—611no—
298#2058—611no—
299#1988—611no—
300#1948—611no—
301#1919—611no—
302#1919—611no—
303#1853—611no—
304#1850D6111yescharacteristic
305#1824—611no—
306#1807—611no—
307C2×#340—611no—
308C2×C61—24no—
309C2×C61#243811yescharacteristic
310C2×C61C2×#47511yescharacteristic
311C2×C61#308111yescharacteristic
312C2×C61#329511yescharacteristic
313C2×C61#332011yescharacteristic
314C2×C61#333511yescharacteristic
315C2×C61#334611yescharacteristic
316C23×C8C2×D6111yescharacteristic
317C2×C4⋊Q8—611no—
318C2×C22⋊Q8—1221no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—611no—
321C2×#106—611no—
322C2×#106—611no—
323C2×#106—611no—
324C2×#106—611no—
325C2×C4⋊C8C2×D6111yescharacteristic
326C2×C4⋊C8C2×D6111yescharacteristic
327C2×C22⋊C8C2×D6111yescharacteristic
328#398—1221no—
329#395—1221no—
330#389—611no—
331#389—611no—
332#389—611no—
333#389—611no—
334#387—1221no—
335#387—1221no—
336#385—1221no—
337#384—611no—
338#380—611no—
339#380—611no—
340#380—1221no—
341C4×C4⋊C4—611no—
342C4×C4⋊C4—611no—
343C4×C22⋊C4—1221no—
344C4×C22⋊C4C2×D6111yescharacteristic
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#336C2×D6111yescharacteristic
358#336C2×D6111yescharacteristic
359C61#2599211yescharacteristic, complemented, Sylow
360C22×Q8—1221no—
361C23×C4—1221no—
362C23×C4—1221no—
363C23×C4C22×D6111yescharacteristic
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—41no—
367C22×C8C22×D6111yescharacteristic
368C22×C8C22×D6111yescharacteristic
369C22×C8C22×D6111yescharacteristic
370C22×C8C22×D6111yescharacteristic
371C4⋊Q8—1222no—
372C4⋊Q8—2441no—
373C22⋊Q8—2444no—
374C2×C4⋊C4—614no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
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—2441no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4D24411yescharacteristic
398C2×C22⋊C4D24411yescharacteristic
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42C22×D6111yescharacteristic
402C2×C42C22×D6111yescharacteristic
403#106—1222no—
404#106—1222no—
405#106—2441no—
406#106—2441no—
407C4⋊C8—22no—
408C4⋊C8—41no—
409C22⋊C8—22no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95D24411yescharacteristic
419#95D24411yescharacteristic
420C24C2×D24411yescharacteristic
421C2×Q8—4881no—
422C2×Q8—2442no—
423C2×Q8—2442no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—1224no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—4881no—
438C22×C4—4881no—
439C22×C4—4881no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4C2×D24411yescharacteristic
443C22×C4C2×D24411yescharacteristic
444C22×C4C2×D24411yescharacteristic
445C22×C4C2×D24411yescharacteristic
446C22×C4C2×D24411yescharacteristic
447C22×C4#1175822412yes—
448C22×C4Q8×D6112yes—
449C22×C4C23×D6111yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—44no—
462C2×C8#1175822412yes—
463C4⋊C4—612no—
464C4⋊C4—612no—
465C4⋊C4—612no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1224no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—4881no—
481C4⋊C4—4881no—
482C4⋊C4—4881no—
483C4⋊C4—2442no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C4⋊C4—2442no—
488C22⋊C4—22no—
489C22⋊C4—2442no—
490C22⋊C4—2442no—
491C22⋊C4#1175819014yes—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—4881no—
497C42—4881no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23#1451975712yes—
503C23C2×#1175819012yes—
504C23#1451981012yes—
505C23C22×D24411yescharacteristic
506Q8—4884no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—4884no—
527C2×C4—4884no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4#1451975712yes—
550C2×C4C2×#1175819012yes—
551C2×C4C2×#1175819012yes—
552C2×C4C2×Q8×D6111yescharacteristic
553C2×C4#41×D6111yescharacteristic
554C2×C4C2×#1175822411yescharacteristic
555C2×C4C2×#1175822411yescharacteristic
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982411yescharacteristic
558C8—22no—
559C8—41no—
560C8—41no—
561C8—41no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22#1684004814yes—
569C22C2×#1451975711yescharacteristic
570C22#1684040711yescharacteristic
571C22#1684041611yescharacteristic
572C22C22×#1175819011yescharacteristic
573C22C2×#1451981011yescharacteristic
574C22#1684056711yescharacteristic
575C22#1684057411yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—4881no—
582C4—4881no—
583C4—4881no—
584C4—4881no—
585C4—4881no—
586C4—2442no—
587C4—2442no—
588C4—2442no—
589C4#1684031411yescharacteristic
590C4#1684039111yescharacteristic
591C4#1684044311yescharacteristic
592C4#1684044911yescharacteristic
593C2—24no—
594C2#2144949311yescharacteristic
595C2C2×#1684004811yescharacteristic
596C2#2145565111yescharacteristic
597C2#2145764411yescharacteristic
598C2#2145799211yescharacteristic
599C2#2145817711yescharacteristic
600C2#2145830611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1850C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1850D61
hypocenterC61#25992
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
3. #21447378C2semidirect
4. #21447317C2semidirect
5. #21447317C2semidirect
6. #21447153C2semidirect
7. #21446770C2semidirect
8. #21446583C2semidirect
9. #21446398C2semidirect
10. #21446398C2semidirect
73. C2×C4⋊Dic61Q8semidirect
359. C61#25992semidirect

Maximal subgroups

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

subgroupindex
#214495212
#214486942
#214473782
#214473172
#214471532
#214467702
#214465832
#214463982
#214460142
#214459102
#214458442
C2×#168395252
C61×#18502
#2599261