Finite Group Databasebeta

← #29658851

Subgroups — #29658851

Order 15616.

1170 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456970C211yesmaximal, characteristic, complemented
2#21455827C211yesmaximal, characteristic, complemented
3#21455616C211yesmaximal, characteristic, complemented
4#21445054C214yesmaximal, complemented
5#21444970C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C2×C61×#547C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840165C2212yescomplemented
12#16840114C2212yescomplemented
13#16840097—22no—
14#16840092—21no—
15#16840062—22no—
16#16840054—22no—
17#16840052C2212yescomplemented
18#16840052C2212yescomplemented
19#16840042C2212yescomplemented
20#16839967C2212yescomplemented
21#16839913—21no—
22C4×#11758200—22no—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839767—21no—
28#16839508C2214yescomplemented
29D4⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C22×#41×C61—21no—
32C61×#547—22no—
33C61×#547—22no—
34C61×#547—22no—
35C61×#547C2214yes—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#123×C61—22no—
38C2×C22⋊Q8×C61—21no—
39C2×#121×C61—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C22×C4⋊C4×C61—21no—
43C2×#11758200—22no—
44C2×#11758194—21no—
45#14519777C2311yescharacteristic
46#14519777C2311yescharacteristic
47#14519769—21no—
48#14519756—21no—
49#14519756—21no—
50C4×Dic122—21no—
51C4×Dic122—21no—
52#14519736—21no—
53#14519710—22no—
54#14519710—24no—
55#14519708—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519700—22no—
62#14519700—24no—
63#14519698—22no—
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×C61—21no—
74C22×D4×C61—22no—
75C22×D4×C61D411yescharacteristic
76C22×D4×C61D411yescharacteristic
77C23×C4×C61—22no—
78C23×C4×C61—21no—
79C23×C4×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61D412yes—
83C4⋊Q8×C61C2312yes—
84#123×C61—28no—
85#123×C61—28no—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—24no—
88#121×C61—24no—
89#121×C61—24no—
90C4×D4×C61—21no—
91C4×D4×C61—21no—
92C4×D4×C61—21no—
93C4×D4×C61—21no—
94C4×D4×C61D412yes—
95C4×D4×C61D412yes—
96C4×D4×C61C2312yes—
97C4×D4×C61C2312yes—
98C2×C4⋊C4×C61—24no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—24no—
102C2×C4⋊C4×C61D412yes—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61—22no—
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C24×C61—21no—
110C24×C61—21no—
111#41×C61—416no—
112#41×C61—48no—
113#41×C61—48no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—44no—
117C2×D4×C61—24no—
118C2×D4×C61—28no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—48no—
124C2×D4×C61C2×D412yes—
125C2×D4×C61C2×D412yes—
126C22×C4×C61—28no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—44no—
135C22×C4×C61—48no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C2×Dic122—21no—
142C2×Dic122—21no—
143C4⋊C4×C61—28no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C2×D412yes—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D414yes—
152C22⋊C4×C61—28no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2411yescharacteristic
161C22⋊Dic61—41no—
162#11758200—44no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167#11758196—41no—
168#11758196—41no—
169C4×Dic61—41no—
170#11758194—42no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C23×C61—24no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61#12011yescharacteristic
183Q8×C61—44no—
184Q8×C61—44no—
185D4×C61—24no—
186D4×C61—22no—
187D4×C61—22no—
188D4×C61—22no—
189D4×C61—416no—
190D4×C61—42no—
191D4×C61AGL(1,ℤ/8)14yes—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—24no—
196C2×C4×C61—41no—
197C2×C4×C61—44no—
198C2×C4×C61—42no—
199C2×C4×C61—44no—
200C2×C4×C61—416no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—44no—
204C2×C4×C61—42no—
205C2×C4×C61—44no—
206C2×C4×C61#12011yescharacteristic
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61#13512yes—
210C2×C4×C61#13712yes—
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×Dic61—41no—
215C2×Dic61—41no—
216Dic122—42no—
217Dic122—42no—
218#92418—42no—
219#92418—42no—
220#92418—42no—
221#18783—611nomaximal, Sylow
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—24no—
225C22×C61—48no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—84no—
232C22×C61#45212yes—
233C22×C61C2×#12011yescharacteristic
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—42no—
238C4×C61—42no—
239C4×C61—44no—
240C4×C61#45212yes—
241C4×C61C2×#13511yescharacteristic
242C4×C61C2×AGL(1,ℤ/8)12yes—
243C4×C61C2×#13711yescharacteristic
244Dic61—81no—
245Dic61—81no—
246C2×#547D6111yescharacteristic
247#3230—611no—
248#3101—611no—
249#3077—611no—
250#1647—614no—
251#1635—614no—
252#1549—611no—
253#1486—611no—
254#1478—611no—
255C2×C61—22no—
256C2×C61—44no—
257C2×C61—82no—
258C2×C61C2×#45211yescharacteristic
259C2×C61#300311yescharacteristic
260C2×C61#302011yescharacteristic
261C22×#41—21no—
262#547—22no—
263#547—22no—
264#547—22no—
265#547C2×D6114yes—
266#533—611no—
267C2×C4⋊Q8C2×D6111yescharacteristic
268C2×#123—22no—
269C2×C22⋊Q8—21no—
270C2×#121—21no—
271C2×C4×D4C2×D6111yescharacteristic
272C2×C4×D4C2×D6111yescharacteristic
273C22×C4⋊C4—21no—
274#500—612no—
275#487—612no—
276#484—1222no—
277#483—1221no—
278#478—1222no—
279#477—1222no—
280#476—612no—
281#476—612no—
282#474—612no—
283#461—612no—
284#450—1221no—
285C4×SD16—1222no—
286#431—611no—
287#431—611no—
288#428—1221no—
289#423—611no—
290#418—1221no—
291#332—614no—
292D4⋊C8—614no—
293D4⋊C8—614no—
294C61#1878311yescharacteristic, complemented, Sylow
295C2×#41—24no—
296C2×#41—24no—
297C2×#41—48no—
298C22×Q8—21no—
299C22×D4—22no—
300C22×D4C22⋊D6111yescharacteristic
301C22×D4C22⋊D6111yescharacteristic
302C23×C4—22no—
303C23×C4—21no—
304C23×C4—21no—
305C2×SD16—1222no—
306C2×M16—1221no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—611no—
310C4⋊Q8—611no—
311C4⋊Q8C22⋊D6112yes—
312C4⋊Q8C22×D6112yes—
313#125—1221no—
314#123—28no—
315#123—28no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—1221no—
319C22⋊Q8—1221no—
320#121—24no—
321#121—24no—
322C4×Q8—1221no—
323C4×Q8—1221no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4C22⋊D6112yes—
329C4×D4C22⋊D6112yes—
330C4×D4C22×D6112yes—
331C4×D4C22×D6112yes—
332#117—1221no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4C22⋊D6111yescharacteristic
338C2×C4⋊C4C22⋊D6111yescharacteristic
339C2×C4⋊C4C22⋊D6112yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C42C22×D6111yescharacteristic
344#107—1222no—
345#107—1224no—
346#106—1222no—
347#106—1222no—
348C4⋊C8—612no—
349C4⋊C8—612no—
350C4⋊C8—1221no—
351C4⋊C8—1221no—
352#103—1222no—
353#103—1224no—
354#102—1222no—
355#102—1222no—
356#102—1222no—
357C22⋊C8—1221no—
358C22⋊C8—1221no—
359C4×C8—612no—
360C4×C8—1221no—
361C24—21no—
362C24—21no—
363#41—416no—
364#41—48no—
365#41—48no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—44no—
369C2×Q8—1221no—
370C2×Q8—1221no—
371C2×D4—24no—
372C2×D4—28no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—48no—
378C2×D4C2×C22⋊D6112yes—
379C2×D4C2×C22⋊D6112yes—
380C22×C4—28no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C2×C22⋊D6111yescharacteristic
393C22×C4C2×C22⋊D6111yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395SD16—2444no—
396M16—2442no—
397C2×C8—1222no—
398C2×C8—1222no—
399C2×C8—1222no—
400C4⋊C4—28no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4—2441no—
411C4⋊C4—2441no—
412C4⋊C4C2×C22⋊D6112yes—
413C4⋊C4C2×C22⋊D6114yes—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—28no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—2441no—
421C42—2441no—
422C42C2×C22⋊D6111yescharacteristic
423C42C2×C22⋊D6111yescharacteristic
424C42C2×C22⋊D6111yescharacteristic
425C42C23×D6111yescharacteristic
426C23—24no—
427C23—22no—
428C23—22no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23#1451974611yescharacteristic
435Q8—44no—
436Q8—44no—
437Q8—2442no—
438Q8—2442no—
439D4—24no—
440D4—22no—
441D4—22no—
442D4—22no—
443D4—416no—
444D4—42no—
445D4#1451980314yes—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—416no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4#1451974611yescharacteristic
463C2×C4#1451974611yescharacteristic
464C2×C4#1451974611yescharacteristic
465C2×C4#1451979712yes—
466C2×C4#1451980712yes—
467C2×C4C22×C22⋊D6111yescharacteristic
468C2×C4C22×C22⋊D6111yescharacteristic
469C2×C4C22×C22⋊D6111yescharacteristic
470C8—2442no—
471C8—2442no—
472C8—2442no—
473C22—21no—
474C22—21no—
475C22—24no—
476C22—48no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—84no—
483C22#1683992412yes—
484C22C2×#1451974611yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—44no—
491C4—4881no—
492C4—4881no—
493C4#1683992412yes—
494C4C2×#1451979711yescharacteristic
495C4C2×#1451980312yes—
496C4C2×#1451980711yescharacteristic
497C2—22no—
498C2—44no—
499C2—82no—
500C2C2×#1683992411yescharacteristic
501C2#2145504611yescharacteristic
502C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC2×C61×#547C2
socleC22×C61C2×#120
hypercenterC2×#547D61
hypocenterC61#18783
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #21456970C2semidirect
2. #21455827C2semidirect
3. #21455616C2semidirect
4. #21445054C2semidirect
5. #21444970C2semidirect
6. #21444504C2semidirect
7. #21444244C2semidirect
8. #21444210C2semidirect
11. #16840165C22semidirect
12. #16840114C22semidirect
17. #16840052C22semidirect
18. #16840052C22semidirect
19. #16840042C22semidirect
20. #16839967C22semidirect
28. #16839508C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
294. C61#18783semidirect

Maximal subgroups

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

subgroupindex
#214569702
#214558272
#214556162
#214450542
#214449702
#214445042
#214442442
#214442102
C2×C61×#5472
#1878361