Finite Group Databasebeta

← #29737614

Subgroups — #29737614

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449521C211yesmaximal, characteristic
2#21448686C211yesmaximal, characteristic
3#21447378C211yesmaximal, characteristic, complemented
4#21447307C211yesmaximal, characteristic, complemented
5#21447307C211yesmaximal, characteristic, complemented
6#21447148C211yesmaximal, characteristic, complemented
7#21446770C211yesmaximal, characteristic, complemented
8#21446578C211yesmaximal, characteristic, complemented
9#21446391C211yesmaximal, characteristic, complemented
10#21446391C211yesmaximal, characteristic, complemented
11#21446009C211yesmaximal, characteristic
12#21445906C211yesmaximal, characteristic
13#21445848C211yesmaximal, characteristic
14C2×#16839525C211yesmaximal, characteristic
15C61×#1850C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519754—21no—
18C2×#14519733—21no—
19C22×C4⋊Dic61C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519711C2211yescharacteristic
25#16839694—21no—
26#16839683—21no—
27#16839660C2211yescharacteristic
28#16839660C2211yescharacteristic
29#16839660C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839647—21no—
32#16839647—21no—
33#16839637—21no—
34#16839632C2211yescharacteristic
35#16839623—21no—
36#16839622C2211yescharacteristic
37#16839622C2211yescharacteristic
38C4×C4⋊Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
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—
65#14519733—44no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—41no—
74C2×C4⋊Dic61Q814yescomplemented
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91#14519711—22no—
92#14519711—22no—
93#14519711—41no—
94#14519711—41no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C23×C4×C61C2311yescharacteristic
104C22×C8×C61—22no—
105C22×C8×C61—22no—
106C22×C8×C61—41no—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C42×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115C4⋊C8×C61—22no—
116C4⋊C8×C61—41no—
117C22⋊C8×C61—22no—
118#95×C61D411yescharacteristic
119#95×C61D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×Q812yes—
131C22×C4×C61#4112yes—
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—24no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61—81no—
147C2×Dic122—42no—
148C2×Dic122—42no—
149C2×Dic122—81no—
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—21no—
161C2×C8×C61—44no—
162C2×C8×C61#4112yes—
163C22⋊C4×C61—22no—
164C22⋊C4×C61D814yes—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—21no—
168C42×C61—21no—
169C22⋊Dic61—42no—
170C22⋊Dic61—42no—
171C4⋊Dic61—24no—
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—21no—
185C4⋊Dic61—42no—
186C4⋊Dic61—42no—
187C4⋊Dic61—42no—
188C4⋊Dic61—81no—
189C4⋊Dic61C2×Q812yes—
190C4⋊Dic61C2×Q812yes—
191C4⋊Dic61#4112yes—
192#11758196—42no—
193#11758196—42no—
194#11758196—42no—
195#11758196—42no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61C22⋊Q812yes—
203C23×C61C2×D812yes—
204C23×C61#13712yes—
205C23×C61C22×D411yescharacteristic
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61C22⋊Q812yes—
223C2×C4×C61C2×D812yes—
224C2×C4×C61C2×D812yes—
225C2×C4×C61C22×Q811yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—48no—
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—41no—
255C2×Dic61—41no—
256C2×Dic61—84no—
257Dic122—84no—
258C8×C61—22no—
259C8×C61—41no—
260C8×C61—41no—
261C8×C61—41no—
262#25988—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61#47414yes—
270C22×C61C2×C22⋊Q811yescharacteristic
271C22×C61#54611yescharacteristic
272C22×C61#54711yescharacteristic
273C22×C61C22×D811yescharacteristic
274C22×C61C2×#13711yescharacteristic
275C22×C61#57711yescharacteristic
276C22×C61#57811yescharacteristic
277C4×C61—22no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61#53311yescharacteristic
283C4×C61#54411yescharacteristic
284C4×C61#55111yescharacteristic
285C4×C61#55211yescharacteristic
286Dic61—42no—
287Dic61—42no—
288Dic61—42no—
289Dic61—81no—
290Dic61—81no—
291Dic61—81no—
292Dic61—81no—
293Dic61—81no—
294#2440—611no—
295#2344—611no—
296#2099—611no—
297#2086—611no—
298#2086—611no—
299#2057—611no—
300#1988—611no—
301#1947—611no—
302#1918—611no—
303#1918—611no—
304#1852—611no—
305#1850D6111yescharacteristic
306#1823—611no—
307#1808—611no—
308C2×#340—611no—
309C2×C61—24no—
310C2×C61#243811yescharacteristic
311C2×C61C2×#47411yescharacteristic
312C2×C61#308211yescharacteristic
313C2×C61#329411yescharacteristic
314C2×C61#331211yescharacteristic
315C2×C61#333411yescharacteristic
316C2×C61#334411yescharacteristic
317C23×C8C2×D6111yescharacteristic
318C2×C4⋊Q8—611no—
319C2×C22⋊Q8—1221no—
320C2×#117—1221no—
321C22×C4⋊C4—611no—
322C22×C4⋊C4—611no—
323C2×#107—611no—
324C2×#107—611no—
325C2×#107—611no—
326C2×#107—611no—
327C2×C4⋊C8C2×D6111yescharacteristic
328C2×C4⋊C8C2×D6111yescharacteristic
329C2×C22⋊C8C2×D6111yescharacteristic
330#398—1221no—
331#395—1221no—
332#389—611no—
333#389—611no—
334#389—611no—
335#389—611no—
336#387—1221no—
337#387—1221no—
338#385—1221no—
339#384—611no—
340#381—1221no—
341#380—611no—
342#380—611no—
343C4×C4⋊C4—611no—
344C4×C4⋊C4—611no—
345C4×C22⋊C4C2×D6111yescharacteristic
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#340—611no—
358#336C2×D6111yescharacteristic
359#336C2×D6111yescharacteristic
360C61#2598811yescharacteristic, complemented, Sylow
361C22×Q8—1221no—
362C23×C4—1221no—
363C23×C4—1221no—
364C23×C4C22×D6111yescharacteristic
365C22×C8—22no—
366C22×C8—22no—
367C22×C8—41no—
368C22×C8C22×D6111yescharacteristic
369C22×C8C22×D6111yescharacteristic
370C22×C8C22×D6111yescharacteristic
371C22×C8C22×D6111yescharacteristic
372C4⋊Q8—1222no—
373C4⋊Q8—2441no—
374C22⋊Q8—2444no—
375#117—2444no—
376C2×C4⋊C4—614no—
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—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1221no—
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—1221no—
396C2×C4⋊C4—2441no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4D24411yescharacteristic
400C2×C22⋊C4D24411yescharacteristic
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42C22×D6111yescharacteristic
404C2×C42C22×D6111yescharacteristic
405#107—1222no—
406#107—1222no—
407#107—2441no—
408#107—2441no—
409C4⋊C8—22no—
410C4⋊C8—41no—
411C22⋊C8—22no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95D24411yescharacteristic
421#95D24411yescharacteristic
422C24C2×D24411yescharacteristic
423C2×Q8—4881no—
424C2×Q8—2442no—
425C2×Q8—2442no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1224no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—4881no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4—2442no—
443C22×C4—2442no—
444C22×C4C2×D24411yescharacteristic
445C22×C4C2×D24411yescharacteristic
446C22×C4C2×D24411yescharacteristic
447C22×C4C2×D24411yescharacteristic
448C22×C4C2×D24411yescharacteristic
449C22×C4#1175822412yes—
450C22×C4Q8×D6112yes—
451C22×C4C23×D6111yescharacteristic
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—21no—
462C2×C8—21no—
463C2×C8—44no—
464C2×C8#1175822412yes—
465C4⋊C4—612no—
466C4⋊C4—612no—
467C4⋊C4—612no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1224no—
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—1221no—
481C4⋊C4—1221no—
482C4⋊C4—4881no—
483C4⋊C4—2442no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C4⋊C4—2442no—
488C4⋊C4—2442no—
489C4⋊C4—2442no—
490C22⋊C4—22no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C22⋊C4D48814yes—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—2442no—
499C42—2442no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23#1451975712yes—
505C23C2×D48812yes—
506C23#1451981012yes—
507C23C22×D24411yescharacteristic
508Q8—4884no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—4884no—
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—2448no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4#1451975712yes—
552C2×C4C2×D48812yes—
553C2×C4C2×D48812yes—
554C2×C4C2×Q8×D6111yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C2×C4C2×#1175822411yescharacteristic
557C2×C4C2×#1175822411yescharacteristic
558C2×C4#1451982211yescharacteristic
559C2×C4#1451982411yescharacteristic
560C8—22no—
561C8—41no—
562C8—41no—
563C8—41no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22#1684004314yes—
571C22C2×#1451975711yescharacteristic
572C22#1684040711yescharacteristic
573C22#1684041611yescharacteristic
574C22C22×D48811yescharacteristic
575C22C2×#1451981011yescharacteristic
576C22#1684057411yescharacteristic
577C22#1684057911yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—41no—
582C4—41no—
583C4—4881no—
584C4—4881no—
585C4—4881no—
586C4—4881no—
587C4—4881no—
588C4—2442no—
589C4—2442no—
590C4—2442no—
591C4#1684031411yescharacteristic
592C4#1684039111yescharacteristic
593C4#1684044311yescharacteristic
594C4#1684044911yescharacteristic
595C2—24no—
596C2#2144949311yescharacteristic
597C2C2×#1684004311yescharacteristic
598C2#2145566211yescharacteristic
599C2#2145763311yescharacteristic
600C2#2145787211yescharacteristic
601C2#2145816711yescharacteristic
602C2#2145827611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1850C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1850D61
hypocenterC61#25988
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. #21447307C2semidirect
5. #21447307C2semidirect
6. #21447148C2semidirect
7. #21446770C2semidirect
8. #21446578C2semidirect
9. #21446391C2semidirect
10. #21446391C2semidirect
74. C2×C4⋊Dic61Q8semidirect
360. C61#25988semidirect

Maximal subgroups

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

subgroupindex
#214495212
#214486862
#214473782
#214473072
#214471482
#214467702
#214465782
#214463912
#214460092
#214459062
#214458482
C2×#168395252
C61×#18502
#2598861