Finite Group Databasebeta

← #29737631

Subgroups — #29737631

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449504C211yesmaximal, characteristic
2#21448712C211yesmaximal, characteristic
3#21447392C211yesmaximal, characteristic, complemented
4#21447323C211yesmaximal, characteristic, complemented
5#21447314C211yesmaximal, characteristic, complemented
6#21447146C211yesmaximal, characteristic, complemented
7#21446769C211yesmaximal, characteristic, complemented
8#21446590C211yesmaximal, characteristic, complemented
9#21446410C211yesmaximal, characteristic, complemented
10#21446393C211yesmaximal, characteristic, complemented
11#21446018C211yesmaximal, characteristic
12#21445997C211yesmaximal, characteristic
13#21445914C211yesmaximal, characteristic
14#21445848C211yesmaximal, characteristic
15C61×#1735C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519711C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24#16839685—21no—
25#16839674—21no—
26#16839668—21no—
27#16839661C2211yescharacteristic
28#16839661C2211yescharacteristic
29#16839661C2211yescharacteristic
30#16839658C2211yescharacteristic
31#16839655—21no—
32#16839655—21no—
33#16839646—21no—
34#16839630—21no—
35#16839627C2211yescharacteristic
36#16839622C2211yescharacteristic
37#16839622C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4×C4⋊Dic61—21no—
41C4×C22⋊Dic61C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C23×C8×C61C2211yescharacteristic
53C2×#117×C61C2211yescharacteristic
54C2×#117×C61C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519776—22no—
63#14519776—41no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61D411yescharacteristic
76C2×C22⋊Dic61D411yescharacteristic
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2311yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81#14519711—22no—
82#14519711—41no—
83#14519710—22no—
84#14519708—41no—
85#14519706—22no—
86#14519706—41no—
87C22⋊#92418—22no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684D411yescharacteristic
99#14519684D411yescharacteristic
100C23×C4×C61C2311yescharacteristic
101C22×C8×C61—22no—
102C22×C8×C61—22no—
103C22×C8×C61—41no—
104C22×C8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106#117×C61—22no—
107#117×C61—22no—
108#117×C61—41no—
109#117×C61Q814yescomplemented
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61—21no—
118C24×C61C2×D411yescharacteristic
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×Q812yes—
129C22×C4×C61#4112yes—
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C2×Dic122—42no—
140C2×Dic122—42no—
141C2×Dic122—81no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—44no—
148C2×C8×C61#4112yes—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61C2×Q812yes—
155C4⋊C4×C61C2×Q812yes—
156C4⋊C4×C61#4112yes—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—42no—
159C42×C61—24no—
160C42×C61—42no—
161C22⋊Dic61—22no—
162C22⋊Dic61—41no—
163C22⋊Dic61—81no—
164C22⋊Dic61—81no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—42no—
174C4⋊Dic61—42no—
175C4⋊Dic61—81no—
176#11758196—42no—
177#11758196—42no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—81no—
183C4×Dic61—81no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61C22⋊Q812yes—
197C23×C61C22×D411yescharacteristic
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61C22⋊Q812yes—
217C2×C4×C61C22×Q811yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C612−1+411yescharacteristic
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
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#13512yes—
247Dic122—84no—
248C8×C61—22no—
249C8×C61—41no—
250#92418—41no—
251#92418—41no—
252#25989—611nomaximal, Sylow
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61C2×C22⋊Q811yescharacteristic
261C22×C61#54611yescharacteristic
262C22×C61#54711yescharacteristic
263C22×C61C2×#13511yescharacteristic
264C22×C61#57511yescharacteristic
265C22×C61#57611yescharacteristic
266C22×C61#57811yescharacteristic
267C4×C61—22no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—81no—
272C4×C61#53311yescharacteristic
273C4×C61#54411yescharacteristic
274C4×C61#55111yescharacteristic
275C4×C61#55211yescharacteristic
276Dic61—22no—
277Dic61—41no—
278Dic61—41no—
279Dic61—41no—
280Dic61—81no—
281Dic61—81no—
282Dic61—81no—
283Dic61—81no—
284#2439—611no—
285#2347—611no—
286#2101—611no—
287#2088—611no—
288#2087—611no—
289#2057—611no—
290#1988—611no—
291#1949—611no—
292#1921—611no—
293#1918—611no—
294#1854—611no—
295#1850—611no—
296#1825—611no—
297#1808—611no—
298#1735D6111yescharacteristic
299C2×C61—42no—
300C2×C61#243811yescharacteristic
301C2×C61#307611yescharacteristic
302C2×C61#308311yescharacteristic
303C2×C61#329711yescharacteristic
304C2×C61#331311yescharacteristic
305C2×C61#334211yescharacteristic
306C2×C61#334711yescharacteristic
307C23×C8C2×D6111yescharacteristic
308C2×C4⋊Q8—611no—
309C2×#117C2×D6111yescharacteristic
310C2×#117C2×D6111yescharacteristic
311C2×#107—611no—
312C2×#107—611no—
313C2×#107—611no—
314C2×#106—611no—
315C2×C4⋊C8—611no—
316C2×C4⋊C8—611no—
317C2×C22⋊C8—611no—
318#396—1221no—
319#393—1221no—
320#391—1221no—
321#389—611no—
322#389—611no—
323#389—611no—
324#389—611no—
325#388—1221no—
326#388—1221no—
327#387—1221no—
328#383—1221no—
329#382—611no—
330#380—611no—
331#380—611no—
332C4×C4⋊C4—611no—
333C4×C4⋊C4—611no—
334C4×C4⋊C4—1221no—
335C4×C22⋊C4—611no—
336#340—611no—
337#340—611no—
338#340—611no—
339#340—611no—
340#340—611no—
341#340—611no—
342#340—611no—
343#340—611no—
344#340C2×D6111yescharacteristic
345#340C2×D6111yescharacteristic
346#340C2×D6111yescharacteristic
347#340C2×D6111yescharacteristic
348#336—611no—
349#336—611no—
350C61#2598911yescharacteristic, complemented, Sylow
351C22×Q8—1221no—
352C23×C4C22×D6111yescharacteristic
353C22×C8—22no—
354C22×C8—22no—
355C22×C8—41no—
356C22×C8—611no—
357C22×C8—611no—
358C22×C8C22×D6111yescharacteristic
359C22×C8C22×D6111yescharacteristic
360C4⋊Q8—1222no—
361C4⋊Q8—2441no—
362#117—22no—
363#117—22no—
364#117—41no—
365#117Dic12214yescomplemented
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4C22×D6111yescharacteristic
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—611no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—1221no—
390C2×C42—1221no—
391#107—1222no—
392#107—1222no—
393#107—2441no—
394#106—2441no—
395C4⋊C8—1222no—
396C4⋊C8—2441no—
397C22⋊C8—1222no—
398#95—611no—
399#95—611no—
400#95—1221no—
401#95—1221no—
402#95—1221no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410C24D4×D6111yescharacteristic
411C2×Q8—4881no—
412C2×Q8—2442no—
413C2×Q8—2442no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4C2×Dic12212yes—
431C22×C4D4×D6111yescharacteristic
432C22×C4#1175822412yes—
433C22×C4C23×D6111yescharacteristic
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—44no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8#1175822212yes—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—4881no—
461C4⋊C4—2442no—
462C4⋊C4—2442no—
463C4⋊C4—2442no—
464C4⋊C4—2442no—
465C4⋊C4C2×Dic12212yes—
466C4⋊C4C2×Dic12212yes—
467C4⋊C4#1175822212yes—
468C22⋊C4—24no—
469C22⋊C4—42no—
470C22⋊C4—1222no—
471C22⋊C4—4881no—
472C22⋊C4—4881no—
473C22⋊C4—2441no—
474C42—24no—
475C42—42no—
476C42—1221no—
477C42—1221no—
478C42—1221no—
479C42—1221no—
480C42—4881no—
481C42—4881no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1451975412yes—
489C23C2×D4×D6111yescharacteristic
490Q8—4884no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—612no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4#1451975412yes—
534C2×C4C22×Dic12211yescharacteristic
535C2×C4C2×#1175822411yescharacteristic
536C2×C4C2×#1175822211yescharacteristic
537C2×C4C2×#1175822211yescharacteristic
538C2×C4#1451982211yescharacteristic
539C2×C4#1451982511yescharacteristic
540C8—22no—
541C8—41no—
542C8—2441no—
543C8—2441no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#1451975411yescharacteristic
552C22#1684040211yescharacteristic
553C22#1684041311yescharacteristic
554C22#135×D6111yescharacteristic
555C22#1684055911yescharacteristic
556C22#1684056411yescharacteristic
557C22#1684057611yescharacteristic
558C4—22no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—81no—
563C4—1222no—
564C4—4881no—
565C4—4881no—
566C4—4881no—
567C4—4881no—
568C4—2441no—
569C4—2441no—
570C4—2441no—
571C4#1684031911yescharacteristic
572C4#1684039311yescharacteristic
573C4#1684043811yescharacteristic
574C4#1684044711yescharacteristic
575C2—42no—
576C2#2144949511yescharacteristic
577C2#2145560311yescharacteristic
578C2#2145566811yescharacteristic
579C2#2145767011yescharacteristic
580C2#2145788111yescharacteristic
581C2#2145824711yescharacteristic
582C2#2145831511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1735C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1735D61
hypocenterC61#25989
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
3. #21447392C2semidirect
4. #21447323C2semidirect
5. #21447314C2semidirect
6. #21447146C2semidirect
7. #21446769C2semidirect
8. #21446590C2semidirect
9. #21446410C2semidirect
10. #21446393C2semidirect
109. #117×C61Q8semidirect
350. C61#25989semidirect
365. #117Dic122semidirect

Maximal subgroups

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

subgroupindex
#214495042
#214487122
#214473922
#214473232
#214473142
#214471462
#214467692
#214465902
#214464102
#214463932
#214460182
#214459972
#214459142
#214458482
C61×#17352
#2598961