Finite Group Databasebeta

← #29658731

Subgroups — #29658731

Order 15616.

1234 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456760C211yesmaximal, characteristic, complemented
2#21455858C211yesmaximal, characteristic, complemented
3#21455616C211yesmaximal, characteristic, complemented
4#21445012C214yesmaximal, complemented
5#21444946C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C2×C61×#550C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840134C2212yescomplemented
12#16840108C2212yescomplemented
13#16840097—21no—
14#16840062—22no—
15#16840057C2212yescomplemented
16#16840052C2212yescomplemented
17#16840042C2212yescomplemented
18#16839972C2212yescomplemented
19#16839959—22no—
20#16839918—21no—
21#16839913—22no—
22C4×#11758202—22no—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839767—21no—
28#16839506C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C22×#41×C61—21no—
32C61×#550—22no—
33C61×#550—22no—
34C61×#550—22no—
35C61×#550C2214yes—
36C2×#127×C61—21no—
37C2×#127×C61C2211yescharacteristic
38C2×#121×C61—22no—
39C2×#121×C61—21no—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C2×#11758202—22no—
44C2×#11758200—24no—
45C2×#11758194—21no—
46#14519777C2311yescharacteristic
47#14519777C2311yescharacteristic
48#14519769—21no—
49#14519756—21no—
50#14519756—21no—
51C4×Dic122—21no—
52C4×Dic122—21no—
53#14519736—21no—
54#14519710—22no—
55#14519710—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—22no—
62#14519700—22no—
63#14519698—24no—
64#14519698—22no—
65#14519698—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yes—
70C2×#41×C61—24no—
71C2×#41×C61—24no—
72C2×#41×C61—48no—
73C22×Q8×C61D411yescharacteristic
74C22×D4×C61—22no—
75C22×D4×C61—22no—
76C22×D4×C61—21no—
77C22×D4×C61—21no—
78C22×D4×C61D411yescharacteristic
79C22×D4×C61D412yes—
80C23×C4×C61—22no—
81C23×C4×C61—21no—
82#127×C61—24no—
83#127×C61—24no—
84#127×C61—21no—
85#127×C61—21no—
86#127×C61D412yes—
87#127×C61C2312yes—
88#121×C61—24no—
89#121×C61—28no—
90#121×C61—28no—
91#121×C61—24no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61D412yes—
95C4×Q8×C61C2312yes—
96C4×D4×C61—24no—
97C4×D4×C61—24no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61D412yes—
101C4×D4×C61C2312yes—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61—22no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C24×C61—21no—
110C24×C61—22no—
111#41×C61—416no—
112#41×C61—48no—
113#41×C61—48no—
114C2×Q8×C61—24no—
115C2×Q8×C61—21no—
116C2×Q8×C61C2×D412yes—
117C2×D4×C61—28no—
118C2×D4×C61—28no—
119C2×D4×C61—21no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—48no—
128C2×D4×C61—44no—
129C2×D4×C61—42no—
130C2×D4×C61—42no—
131C2×D4×C61—82no—
132C2×D4×C61—84no—
133C2×D4×C61C2×D412yes—
134C2×D4×C61C2×D414yes—
135C22×C4×C61—21no—
136C22×C4×C61—28no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—48no—
142C22×C4×C61—44no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C2×Dic122—21no—
147C2×Dic122—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—28no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2411yescharacteristic
163C22⋊Dic61—41no—
164#11758202—44no—
165#11758200—84no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170#11758196—41no—
171#11758196—41no—
172C4×Dic61—41no—
173#11758194—42no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61—44no—
180C23×C61—42no—
181C23×C61—44no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—44no—
185C23×C61#12011yescharacteristic
186Q8×C61—24no—
187Q8×C61—22no—
188Q8×C61AGL(1,ℤ/8)14yes—
189D4×C61—22no—
190D4×C61—22no—
191D4×C61—44no—
192D4×C61—416no—
193D4×C61—44no—
194D4×C61—42no—
195D4×C61—44no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61—44no—
199D4×C61—82no—
200D4×C61—88no—
201C2×C4×C61—24no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—416no—
208C2×C4×C61—44no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—44no—
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61#12011yescharacteristic
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#13512yes—
216C2×C4×C61#13712yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×Dic61—41no—
221C2×Dic61—41no—
222Dic122—42no—
223Dic122—42no—
224#92418—42no—
225#92418—42no—
226#92418—42no—
227#18771—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—42no—
231C22×C61—44no—
232C22×C61—44no—
233C22×C61—42no—
234C22×C61—44no—
235C22×C61—42no—
236C22×C61—84no—
237C22×C61—88no—
238C22×C61#45212yes—
239C22×C61C2×#12011yescharacteristic
240C4×C61—24no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61#45212yes—
247C4×C61C2×#13511yescharacteristic
248C4×C61C2×AGL(1,ℤ/8)12yes—
249C4×C61C2×#13711yescharacteristic
250Dic61—81no—
251Dic61—81no—
252C2×#550D6111yescharacteristic
253#3208—611no—
254#3104—611no—
255#3077—611no—
256#1641—614no—
257#1631—614no—
258#1545—611no—
259#1487—611no—
260#1478—611no—
261C2×C61—22no—
262C2×C61—82no—
263C2×C61—84no—
264C2×C61C2×#45211yescharacteristic
265C2×C61#300311yescharacteristic
266C2×C61#301911yescharacteristic
267C22×#41—21no—
268#550—22no—
269#550—22no—
270#550—22no—
271#550C2×D6114yes—
272#533—611no—
273C2×#127—21no—
274C2×#127C2×D6111yescharacteristic
275C2×#121—22no—
276C2×#121—21no—
277C2×C4×Q8C2×D6111yescharacteristic
278C2×C4×D4—21no—
279C2×C4×D4C2×D6111yescharacteristic
280#492—612no—
281#486—612no—
282#484—1221no—
283#478—1222no—
284#477—612no—
285#476—612no—
286#474—612no—
287#462—612no—
288#460—1222no—
289#451—1221no—
290#450—1222no—
291C4×Q16—1222no—
292#431—611no—
293#431—611no—
294#428—1221no—
295#423—611no—
296#418—1221no—
297#331—614no—
298Q8⋊C8—614no—
299D4⋊C8—614no—
300C61#1877111yescharacteristic, complemented, Sylow
301C2×#41—24no—
302C2×#41—24no—
303C2×#41—48no—
304C22×Q8C22⋊D6111yescharacteristic
305C22×D4—22no—
306C22×D4—22no—
307C22×D4—21no—
308C22×D4—21no—
309C22×D4C22⋊D6111yescharacteristic
310C22×D4C22⋊D6112yes—
311C23×C4—22no—
312C23×C4—21no—
313C2×Q16—1222no—
314C2×SD16—1224no—
315C2×M16—1221no—
316C4⋊Q8—611no—
317C4⋊Q8—611no—
318#127—24no—
319#127—24no—
320#127—21no—
321#127—21no—
322#127C22⋊D6112yes—
323#127C22×D6112yes—
324#125—1221no—
325C22⋊Q8—1221no—
326C22⋊Q8—1221no—
327#121—24no—
328#121—28no—
329#121—28no—
330#121—24no—
331C4×Q8—21no—
332C4×Q8—21no—
333C4×Q8—1221no—
334C4×Q8—1221no—
335C4×Q8C22⋊D6112yes—
336C4×Q8C22×D6112yes—
337C4×D4—24no—
338C4×D4—24no—
339C4×D4—21no—
340C4×D4—21no—
341C4×D4C22⋊D6112yes—
342C4×D4C22×D6112yes—
343#117—1221no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4C22⋊D6111yescharacteristic
346C2×C4⋊C4C22⋊D6111yescharacteristic
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C42—21no—
350C2×C42C22×D6111yescharacteristic
351#107—1222no—
352#107—1222no—
353#106—1222no—
354C4⋊C8—612no—
355C4⋊C8—612no—
356C4⋊C8—1221no—
357C4⋊C8—1221no—
358#103—1222no—
359#103—1222no—
360#102—1222no—
361#102—1222no—
362#102—1224no—
363C22⋊C8—1221no—
364C22⋊C8—1221no—
365C4×C8—612no—
366C4×C8—1221no—
367C24—21no—
368C24—22no—
369#41—416no—
370#41—48no—
371#41—48no—
372C2×Q8—24no—
373C2×Q8—21no—
374C2×Q8—1221no—
375C2×Q8—1221no—
376C2×Q8C2×C22⋊D6112yes—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—48no—
388C2×D4—44no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—82no—
392C2×D4—84no—
393C2×D4C2×C22⋊D6112yes—
394C2×D4C2×C22⋊D6114yes—
395C22×C4—21no—
396C22×C4—28no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—48no—
402C22×C4—44no—
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406Q16—2444no—
407SD16—4884no—
408M16—2442no—
409C2×C8—1222no—
410C2×C8—1222no—
411C2×C8—1222no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—1221no—
417C4⋊C4—1221no—
418C4⋊C4—1221no—
419C4⋊C4—1221no—
420C4⋊C4—2441no—
421C4⋊C4—2441no—
422C4⋊C4C2×C22⋊D6112yes—
423C4⋊C4C2×C22⋊D6112yes—
424C22⋊C4—28no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—2441no—
428C42—22no—
429C42—22no—
430C42—2441no—
431C42C2×C22⋊D6111yescharacteristic
432C42C2×C22⋊D6111yescharacteristic
433C42C2×C22⋊D6111yescharacteristic
434C42C23×D6111yescharacteristic
435C23—24no—
436C23—22no—
437C23—44no—
438C23—42no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23—44no—
443C23#1451974611yescharacteristic
444Q8—24no—
445Q8—22no—
446Q8—2442no—
447Q8—2442no—
448Q8#1451980314yes—
449D4—22no—
450D4—22no—
451D4—44no—
452D4—416no—
453D4—44no—
454D4—42no—
455D4—44no—
456D4—42no—
457D4—42no—
458D4—44no—
459D4—82no—
460D4—88no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—416no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4#1451974611yescharacteristic
475C2×C4#1451974611yescharacteristic
476C2×C4#1451974611yescharacteristic
477C2×C4#1451979712yes—
478C2×C4#1451980712yes—
479C2×C4C22×C22⋊D6111yescharacteristic
480C2×C4C22×C22⋊D6111yescharacteristic
481C2×C4C22×C22⋊D6111yescharacteristic
482C8—2442no—
483C8—2442no—
484C8—2442no—
485C22—21no—
486C22—21no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22—84no—
494C22—88no—
495C22#1683992412yes—
496C22C2×#1451974611yescharacteristic
497C4—21no—
498C4—21no—
499C4—24no—
500C4—21no—
501C4—42no—
502C4—42no—
503C4—4881no—
504C4—4881no—
505C4#1683992412yes—
506C4C2×#1451979711yescharacteristic
507C4C2×#1451980312yes—
508C4C2×#1451980711yescharacteristic
509C2—22no—
510C2—84no—
511C2—82no—
512C2C2×#1683992411yescharacteristic
513C2#2145504611yescharacteristic
514C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC2×C61×#550C2
socleC22×C61C2×#120
hypercenterC2×#550D61
hypocenterC61#18771
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #21456760C2semidirect
2. #21455858C2semidirect
3. #21455616C2semidirect
4. #21445012C2semidirect
5. #21444946C2semidirect
6. #21444487C2semidirect
7. #21444248C2semidirect
8. #21444210C2semidirect
11. #16840134C22semidirect
12. #16840108C22semidirect
15. #16840057C22semidirect
16. #16840052C22semidirect
17. #16840042C22semidirect
18. #16839972C22semidirect
28. #16839506C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
300. C61#18771semidirect

Maximal subgroups

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

subgroupindex
#214567602
#214558582
#214556162
#214450122
#214449462
#214444872
#214442482
#214442102
C2×C61×#5502
#1877161