Finite Group Databasebeta

← #29657275

Subgroups — #29657275

Order 15616.

1024 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456960C211yesmaximal, characteristic, complemented
2C2×#16840108C211yesmaximal, characteristic, complemented
3#21455306C211yesmaximal, characteristic, complemented
4#21445019C214yesmaximal, complemented
5#21445005C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444274C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3508C211yesmaximal, characteristic
10C2×#14519807—21no—
11C2×#14519777C2211yescharacteristic
12#16840158—21no—
13#16840112—21no—
14#16840108—22no—
15#16840108C2214yes—
16#16840090—22no—
17#16840047C2214yescomplemented
18#16839972—21no—
19#16839972C2212yescomplemented
20#16839967—21no—
21#16839967C2212yescomplemented
22#16839815C2211yescharacteristic
23#16839786C2211yescharacteristic
24#16839770—21no—
25C2×#14519698—22no—
26C2×C4×#92418C2211yescharacteristic
27#16839506C2214yes—
28Q8⋊#92418C2212yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2212yescomplemented
31C61×#559—21no—
32C61×#559C2212yes—
33C61×#550—21no—
34C61×#550C2212yes—
35C61×#546—22no—
36C61×#546—22no—
37C61×#540—22no—
38C61×#535—21no—
39C61×#533C2211yescharacteristic
40C61×#532—21no—
41C61×#532—21no—
42C2×#127×C61C2211yescharacteristic
43C61×#518—21no—
44C4×#41×C61C2211yescharacteristic
45C22×Dic122—21no—
46#14519807—44no—
47C2×#11758202—22no—
48C2×#11758200—22no—
49C2×#11758194—21no—
50C22×#92418—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777C2312yes—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57#14519708—24no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—24no—
62#14519700—22no—
63#14519700—22no—
64#14519698—28no—
65C22⋊#92418—21no—
66C4×#92418—21no—
67C4×#92418—21no—
68C4×#92418C2312yes—
69C2×#41×C61—22no—
70C2×#41×C61—22no—
71C2×#41×C61D411yescharacteristic
72C22×D4×C61—21no—
73C22×D4×C61—21no—
74C22×D4×C61D412yes—
75C4⋊Q8×C61D411yescharacteristic
76C4⋊Q8×C61D411yescharacteristic
77#127×C61—22no—
78#127×C61—21no—
79#127×C61—21no—
80#127×C61D412yes—
81#127×C61C2312yes—
82#125×C61—21no—
83#124×C61—22no—
84#124×C61—22no—
85#124×C61—22no—
86#123×C61—24no—
87#123×C61—24no—
88C22⋊Q8×C61—21no—
89C22⋊Q8×C61D412yes—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61—22no—
93#121×C61—24no—
94#121×C61—24no—
95#121×C61—22no—
96#120×C61—24no—
97C4×Q8×C61—21no—
98C4×Q8×C61D411yescharacteristic
99C4×Q8×C61C2312yes—
100C4×Q8×C61C2311yescharacteristic
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61—22no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C4×D4×C61D411yescharacteristic
110C4×D4×C61C2311yescharacteristic
111#117×C61—21no—
112#117×C61—21no—
113#117×C61D411yescharacteristic
114C2×C22⋊C4×C61—24no—
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117C24×C61—22no—
118#41×C61—22no—
119#41×C61—22no—
120#41×C61—84no—
121#41×C61—84no—
122C2×Q8×C61—21no—
123C2×Q8×C61C2×D411yescharacteristic
124C2×Q8×C61C2×D412yes—
125C2×D4×C61—21no—
126C2×D4×C61—24no—
127C2×D4×C61—24no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—44no—
134C2×D4×C61—42no—
135C2×D4×C61—42no—
136C2×D4×C61—82no—
137C2×D4×C61C2×D411yescharacteristic
138C2×D4×C61C2×D414yes—
139C22×C4×C61—24no—
140C22×C4×C61—24no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×Dic61—41no—
148C2×Dic122—22no—
149C2×Dic122—42no—
150C2×Dic122—42no—
151C2×Dic122—41no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C2×D411yescharacteristic
160C4⋊C4×C61C2×D411yescharacteristic
161C4⋊C4×C61C2×D412yes—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—21no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—22no—
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C42×C61C2411yescharacteristic
178#11758202—44no—
179#11758200—44no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—41no—
183C4⋊Dic61—41no—
184#11758194—42no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—42no—
189C23×C61—24no—
190C23×C61—21no—
191C23×C61—44no—
192C23×C61—82no—
193C23×C61—82no—
194C23×C61#12011yescharacteristic
195Q8×C61—21no—
196Q8×C61—21no—
197Q8×C61—24no—
198Q8×C61—42no—
199D4×C61—21no—
200D4×C61—21no—
201D4×C61—44no—
202D4×C61—42no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61—44no—
206D4×C61—44no—
207D4×C61—44no—
208D4×C61—82no—
209D4×C61—84no—
210D4×C61—84no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—84no—
222C2×C4×C61—84no—
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61#13512yes—
227C2×C4×C61AGL(1,ℤ/8)12yes—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×Dic61—42no—
232C2×Dic61—82no—
233Dic122—42no—
234Dic122—42no—
235Dic122—82no—
236#92418—42no—
237#92418—42no—
238#92418—42no—
239#18634—611nomaximal, Sylow
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—41no—
243C22×C61—44no—
244C22×C61—41no—
245C22×C61—44no—
246C22×C61—41no—
247C22×C61—44no—
248C22×C61—164no—
249C22×C61#44912yes—
250C22×C61C2×#12011yescharacteristic
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—41no—
254C4×C61—41no—
255C4×C61—42no—
256C4×C61—41no—
257C4×C61—42no—
258C4×C61#44912yes—
259C4×C61#45512yes—
260C4×C61C2×#13511yescharacteristic
261C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C4×C61#57712yes—
263Dic61—82no—
264#3508D6111yescharacteristic
265#3229—611no—
266C2×#486—611no—
267#3039—611no—
268#1642—614no—
269#1640—614no—
270#1545—611no—
271#1493—611no—
272#1476—611no—
273C2×C61—22no—
274C2×C61—81no—
275C2×C61—84no—
276C2×C61C2×#44911yescharacteristic
277C2×C61#300911yescharacteristic
278C2×C61C2×#45511yescharacteristic
279C2×#137—1221no—
280#559—21no—
281#559C2×D6112yes—
282#550—21no—
283#550C2×D6112yes—
284#546—22no—
285#546—22no—
286#540—22no—
287#535—21no—
288#533C2×D6111yescharacteristic
289#532—21no—
290#532—21no—
291C2×C4⋊Q8—611no—
292C2×#127C2×D6111yescharacteristic
293#518—21no—
294C4×#41C2×D6111yescharacteristic
295#498—1221no—
296#487—1221no—
297#486—614no—
298#486—1222no—
299#483—1222no—
300#475—614no—
301#462—612no—
302#462—1221no—
303#461—612no—
304#461—1221no—
305#431—611no—
306#423—611no—
307#419—1221no—
308C2×#102—1222no—
309C2×C4×C8—611no—
310#331—614no—
311Q8⋊C8—612no—
312Q8⋊C8—614no—
313D4⋊C8—612no—
314C61#1863411yescharacteristic, complemented, Sylow
315C2×#41—22no—
316C2×#41—22no—
317C2×#41C22⋊D6111yescharacteristic
318C22×Q8—1221no—
319C22×D4—21no—
320C22×D4—21no—
321C22×D4C22⋊D6112yes—
322#137—2444no—
323C2×Q16—1222no—
324C2×SD16—1222no—
325C2×M16—1221no—
326C22×C8—1221no—
327C4⋊Q8—612no—
328C4⋊Q8—1221no—
329C4⋊Q8—1221no—
330C4⋊Q8—1221no—
331C4⋊Q8C22⋊D6111yescharacteristic
332C4⋊Q8C22⋊D6111yescharacteristic
333#127—22no—
334#127—21no—
335#127—21no—
336#127C22⋊D6112yes—
337#127C22×D6112yes—
338#125—21no—
339#124—22no—
340#124—22no—
341#124—22no—
342#123—24no—
343#123—24no—
344C22⋊Q8—21no—
345C22⋊Q8C22⋊D6112yes—
346#121—22no—
347#121—22no—
348#121—22no—
349#121—24no—
350#121—24no—
351#121—22no—
352#120—24no—
353C4×Q8—21no—
354C4×Q8C22⋊D6111yescharacteristic
355C4×Q8C22×D6111yescharacteristic
356C4×Q8C22×D6112yes—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4C22⋊D6111yescharacteristic
366C4×D4C22×D6111yescharacteristic
367#117—21no—
368#117—21no—
369#117C22⋊D6111yescharacteristic
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C22⋊C4—24no—
373C2×C42—21no—
374C2×C42C22×D6111yescharacteristic
375#106—1224no—
376C4⋊C8—612no—
377C4⋊C8—612no—
378C4⋊C8—1221no—
379#103—1224no—
380#103—1222no—
381#103—1222no—
382#102—1228no—
383C22⋊C8—1221no—
384C4×C8—612no—
385C4×C8—1221no—
386C4×C8—1221no—
387C24—22no—
388#41—22no—
389#41—22no—
390#41—84no—
391#41—84no—
392C2×Q8—21no—
393C2×Q8—1222no—
394C2×Q8—2442no—
395C2×Q8—2442no—
396C2×Q8—2441no—
397C2×Q8C2×C22⋊D6112yes—
398C2×Q8C2×C22⋊D6111yescharacteristic
399C2×D4—21no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—82no—
411C2×D4C2×C22⋊D6111yescharacteristic
412C2×D4C2×C22⋊D6114yes—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—2441no—
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421C22×C4C2×C22⋊D6111yescharacteristic
422Q16—2444no—
423SD16—2444no—
424M16—2442no—
425C2×C8—1222no—
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8—2442no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4C2×C22⋊D6111yescharacteristic
441C4⋊C4C2×C22⋊D6111yescharacteristic
442C4⋊C4C2×C22⋊D6112yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—21no—
448C22⋊C4—22no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C42—21no—
452C42—22no—
453C42—21no—
454C42—21no—
455C42C2×C22⋊D6111yescharacteristic
456C42C2×C22⋊D6111yescharacteristic
457C42C2×C22⋊D6111yescharacteristic
458C42C23×D6111yescharacteristic
459C23—24no—
460C23—21no—
461C23—44no—
462C23—82no—
463C23—82no—
464C23#1451974611yescharacteristic
465Q8—21no—
466Q8—21no—
467Q8—24no—
468Q8—42no—
469Q8—4882no—
470Q8—2442no—
471Q8—2442no—
472D4—21no—
473D4—21no—
474D4—44no—
475D4—42no—
476D4—44no—
477D4—42no—
478D4—42no—
479D4—44no—
480D4—44no—
481D4—82no—
482D4—84no—
483D4—84no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—4882no—
497C2×C4—2442no—
498C2×C4#1451974611yescharacteristic
499C2×C4#1451974611yescharacteristic
500C2×C4#1451974611yescharacteristic
501C2×C4#1451979712yes—
502C2×C4#1451980212yes—
503C2×C4C22×C22⋊D6111yescharacteristic
504C2×C4C22×C22⋊D6111yescharacteristic
505C2×C4C22×C22⋊D6111yescharacteristic
506C8—2442no—
507C8—2442no—
508C8—2442no—
509C22—21no—
510C22—21no—
511C22—41no—
512C22—44no—
513C22—41no—
514C22—44no—
515C22—41no—
516C22—44no—
517C22—164no—
518C22#1683990712yes—
519C22C2×#1451974611yescharacteristic
520C4—21no—
521C4—21no—
522C4—41no—
523C4—42no—
524C4—41no—
525C4—41no—
526C4—42no—
527C4—4882no—
528C4#1683990712yes—
529C4#1683994012yes—
530C4C2×#1451979711yescharacteristic
531C4C2×#1451980211yescharacteristic
532C4#1684057312yes—
533C2—22no—
534C2—84no—
535C2—81no—
536C2C2×#1683990711yescharacteristic
537C2#2145509611yescharacteristic
538C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3508C2
socleC22×C61C2×#120
hypercenter#3508D61
hypocenterC61#18634
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
1. #21456960C2semidirect
2. C2×#16840108C2semidirect
3. #21455306C2semidirect
4. #21445019C2semidirect
5. #21445005C2semidirect
6. #21444487C2semidirect
7. #21444274C2semidirect
8. #21444201C2semidirect
17. #16840047C22semidirect
19. #16839972C22semidirect
21. #16839967C22semidirect
28. Q8⋊#92418C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
314. C61#18634semidirect

Maximal subgroups

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

subgroupindex
#214569602
C2×#168401082
#214553062
#214450192
#214450052
#214444872
#214442742
#214442012
C61×#35082
#1863461