Finite Group Databasebeta

← #29737627

Subgroups — #29737627

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#21448710C211yesmaximal, characteristic
2#21447386C211yesmaximal, characteristic, complemented
3#21447321C211yesmaximal, characteristic, complemented
4#21447315C211yesmaximal, characteristic, complemented
5#21447144C211yesmaximal, characteristic, complemented
6#21446772C211yesmaximal, characteristic, complemented
7#21446585C211yesmaximal, characteristic, complemented
8#21446408C211yesmaximal, characteristic, complemented
9#21446387C211yesmaximal, characteristic, complemented
10#21446018C211yesmaximal, characteristic
11#21445999C211yesmaximal, characteristic
12#21445911C211yesmaximal, characteristic
13#21445846C211yesmaximal, characteristic
14#21445535C211yesmaximal, characteristic
15C61×#2439C211yesmaximal, characteristic
16C23×#92418C2211yescharacteristic
17C2×#14519736C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25#16839660C2211yescharacteristic
26#16839660C2211yescharacteristic
27#16839656—21no—
28#16839656—21no—
29#16839629—21no—
30#16839622C2211yescharacteristic
31C4×C4⋊Dic61C2211yescharacteristic
32C4×C4⋊Dic61C2211yescharacteristic
33#16839613—21no—
34#16839524C2211yescharacteristic
35#16839524C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×C4⋊Q8×C61C2211yescharacteristic
49C2×#117×C61C2211yescharacteristic
50C61×#396—21no—
51C61×#393—21no—
52C61×#391—21no—
53C61×#389C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C61×#387—21no—
56C61×#382C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C22×#92418—22no—
60C22×#92418—22no—
61C22×#92418—41no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519736—22no—
67#14519736Q814yescomplemented
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79#14519710—22no—
80#14519710—22no—
81#14519710—41no—
82#14519708—41no—
83C4⋊#92418—22no—
84C4⋊#92418—41no—
85C22⋊#92418—22no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91C22×Q8×C61—21no—
92C23×C4×C61C2311yescharacteristic
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61—41no—
95#117×C61—22no—
96#117×C61—41no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61D411yescharacteristic
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111#95×C61—21no—
112#95×C61—21no—
113#95×C61—21no—
114#95×C61—21no—
115#95×C61—21no—
116#95×C61D411yescharacteristic
117#95×C61D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C2×Q8×C61—42no—
120C2×Q8×C61—42no—
121C2×Q8×C61—81no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×Q812yes—
136C22×C4×C61#4112yes—
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—42no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—81no—
152C4⋊C4×C61#4112yes—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—41no—
155C22⋊C4×C61—42no—
156C22⋊C4×C61—81no—
157C42×C61—21no—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—42no—
162C22⋊Dic61—24no—
163C22⋊Dic61—81no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61C2×Q812yes—
171C4⋊Dic61C2×Q812yes—
172#11758196—42no—
173#11758196—42no—
174C4×Dic61—24no—
175C4×Dic61—81no—
176C4×Dic61—81no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—44no—
189C2×#92418#4112yes—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61C22⋊Q812yes—
197C23×C61C22×D411yescharacteristic
198Q8×C61—84no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
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⋊Q812yes—
229C2×C4×C61#13512yes—
230C2×C4×C61C22×Q811yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×Dic61—22no—
237C2×Dic61—22no—
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—
248#92418—22no—
249#92418—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—22no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—42no—
272C4×C61—41no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61#53311yescharacteristic
277C4×C61#54411yescharacteristic
278C4×C61#55111yescharacteristic
279C4×C61#55211yescharacteristic
280Dic61—42no—
281Dic61—42no—
282Dic61—81no—
283Dic61—81no—
284#2439D6111yescharacteristic
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#1735—611no—
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×C8—611no—
308C2×C4⋊Q8C2×D6111yescharacteristic
309C2×#117—611no—
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—21no—
319#393—21no—
320#391—21no—
321#389—611no—
322#389—611no—
323#389C2×D6111yescharacteristic
324#389C2×D6111yescharacteristic
325#388—1221no—
326#388—1221no—
327#387—21no—
328#383—1221no—
329#382C2×D6111yescharacteristic
330#380—611no—
331#380C2×D6111yescharacteristic
332C4×C4⋊C4—611no—
333C4×C4⋊C4—611no—
334C4×C4⋊C4—1221no—
335C4×C22⋊C4C2×D6111yescharacteristic
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#340—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#336—611no—
349#336—611no—
350C61#2598911yescharacteristic, complemented, Sylow
351C22×Q8—21no—
352C23×C4C22×D6111yescharacteristic
353C22×C8—611no—
354C22×C8—611no—
355C22×C8—611no—
356C22×C8—611no—
357C22×C8—1222no—
358C22×C8—1222no—
359C22×C8—2441no—
360C4⋊Q8—22no—
361C4⋊Q8—41no—
362#117—22no—
363#117—41no—
364#117—614no—
365#117—1222no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
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—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4C22⋊D6111yescharacteristic
384C2×C22⋊C4C22⋊D6111yescharacteristic
385C2×C42—21no—
386C2×C42—1221no—
387C2×C42—1221no—
388C2×C42—1221no—
389C2×C42C22×D6111yescharacteristic
390C2×C42C22×D6111yescharacteristic
391#107—1222no—
392#107—1222no—
393#107—2441no—
394#106—2441no—
395C4⋊C8—1222no—
396C4⋊C8—2441no—
397C22⋊C8—1222no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408#95C22⋊D6111yescharacteristic
409#95C22⋊D6111yescharacteristic
410C24C2×C22⋊D6111yescharacteristic
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—81no—
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—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4Q8×D6112yes—
427C22×C4#1175822612yes—
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C23×D6111yescharacteristic
434C2×C8—612no—
435C2×C8—1221no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—2444no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—612no—
458C4⋊C4—612no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—2442no—
466C4⋊C4—2442no—
467C4⋊C4#1175822412yes—
468C22⋊C4—22no—
469C22⋊C4—41no—
470C22⋊C4—42no—
471C22⋊C4—81no—
472C22⋊C4—1224no—
473C22⋊C4—4881no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—42no—
479C42—1224no—
480C42—4881no—
481C42—4881no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1451975312yes—
489C23C22×C22⋊D6111yescharacteristic
490Q8—84no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—1222no—
521C2×C4—1222no—
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#1451975312yes—
533C2×C4#1451979712yes—
534C2×C4C2×Q8×D6111yescharacteristic
535C2×C4#41×D6111yescharacteristic
536C2×C4C2×#1175822411yescharacteristic
537C2×C4C2×#1175822611yescharacteristic
538C2×C4#1451982211yescharacteristic
539C2×C4#1451982411yescharacteristic
540C8—1222no—
541C8—2441no—
542C8—2441no—
543C8—2441no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#1451975311yescharacteristic
552C22#1684040611yescharacteristic
553C22#1684041111yescharacteristic
554C22C2×#1451979711yescharacteristic
555C22#1684056011yescharacteristic
556C22#1684056511yescharacteristic
557C22#1684057711yescharacteristic
558C4—22no—
559C4—22no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—4881no—
568C4—4881no—
569C4—2442no—
570C4—2442no—
571C4#1684031411yescharacteristic
572C4#1684039111yescharacteristic
573C4#1684043911yescharacteristic
574C4#1684045411yescharacteristic
575C2—42no—
576C2#2144949111yescharacteristic
577C2#2145560111yescharacteristic
578C2#2145566611yescharacteristic
579C2#2145766911yescharacteristic
580C2#2145788511yescharacteristic
581C2#2145824511yescharacteristic
582C2#2145831911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2439C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2439D61
hypocenterC61#25989
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
2. #21447386C2semidirect
3. #21447321C2semidirect
4. #21447315C2semidirect
5. #21447144C2semidirect
6. #21446772C2semidirect
7. #21446585C2semidirect
8. #21446408C2semidirect
9. #21446387C2semidirect
67. #14519736Q8semidirect
350. C61#25989semidirect

Maximal subgroups

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

subgroupindex
#214487102
#214473862
#214473212
#214473152
#214471442
#214467722
#214465852
#214464082
#214463872
#214460182
#214459992
#214459112
#214458462
#214455352
C61×#24392
#2598961