Finite Group Databasebeta

← #29667920

Subgroups — #29667920

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456769C211yesmaximal, characteristic, complemented
2#21456374C211yesmaximal, characteristic, complemented
3#21456212C211yesmaximal, characteristic, complemented
4#21456193C211yesmaximal, characteristic, complemented
5C2×#16840108C211yesmaximal, characteristic, complemented
6#21445478C212yesmaximal, complemented
7#21445457C212yesmaximal, complemented
8#21445430C212yesmaximal, complemented
9#21445424C212yesmaximal, complemented
10#21444100C211yesmaximal, characteristic, complemented
11C61×#3205C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840158C2211yescharacteristic, complemented
15#16840142C2212yescomplemented
16#16840134C2211yescharacteristic, complemented
17#16840126—22no—
18#16840121—22no—
19#16840114C2211yescharacteristic, complemented
20#16840113C2212yescomplemented
21#16840112C2212yescomplemented
22#16840108—22no—
23#16840108C2212yes—
24#16840108C2212yes—
25#16840108C2212yes—
26#16840108C2212yes—
27#16840108C2211yescharacteristic, complemented
28#16840108C2212yescomplemented
29#16839989—22no—
30#16839973—22no—
31#16839920—22no—
32C4×#11758200—22no—
33#16839767—22no—
34#16839763—22no—
35C2×#14519698—22no—
36C4×#11758194C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839484C2212yescomplemented
39C8×#92418C2212yescomplemented
40C61×#532C2211yescharacteristic
41C2×#127×C61C2211yescharacteristic
42C61×#493C2212yes—
43C61×#486C2212yes—
44C61×#459—22no—
45C61×#447—22no—
46C61×#431C2211yescharacteristic
47C22×Dic122—21no—
48C2×Dic244—24no—
49C2×#11758200—22no—
50C2×#11758194D412yes—
51C22×#92418D412yes—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2312yes—
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519769—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63#14519736—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66#14519708—22no—
67#14519702—24no—
68#14519701—24no—
69#14519700—22no—
70#14519698—22no—
71#14519698—24no—
72#14519698—24no—
73#14519698—24no—
74#14519698—24no—
75#14519698—24no—
76#14519688D412yes—
77C4×#92418D412yes—
78C4×#92418C2311yescharacteristic
79C4×#92418C2311yescharacteristic
80C4×#92418C2312yes—
81C2×#41×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C2×D8×C61—24no—
86C4⋊Q8×C61C2311yescharacteristic
87#127×C61—21no—
88#127×C61—21no—
89#127×C61—21no—
90#127×C61C2312yes—
91#127×C61C2311yescharacteristic
92#124×C61—21no—
93#121×C61—21no—
94#121×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61D412yes—
99#102×C61—24no—
100C22⋊C8×C61D412yes—
101C4×C8×C61C2312yes—
102C24×C61—41no—
103#41×C61—44no—
104C2×Q8×C61—21no—
105C2×D4×C61—22no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—44no—
112C2×D4×C61—41no—
113C2×D4×C61—41no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—44no—
117C2×D4×C61—82no—
118C22×C4×C61—41no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×Dic61—41no—
123C2×Dic122—22no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×Dic122—41no—
127C2×Dic122—82no—
128D8×C61—84no—
129C2×C8×C61C2×D414yes—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C22⋊C4×C61—41no—
133C22⋊C4×C61—41no—
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137C42×C61C2411yescharacteristic
138C22⋊Dic61—41no—
139#11758200—44no—
140C4⋊Dic61—22no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—41no—
147C4⋊Dic61—41no—
148#11758196—41no—
149#11758196—41no—
150C4×Dic61—41no—
151#11758194—24no—
152C2×#92418—42no—
153C2×#92418C2×D412yes—
154C2×#92418C2×D412yes—
155C2×#92418C2×D414yes—
156Dic244—84no—
157C23×C61—41no—
158C23×C61—42no—
159C23×C61—82no—
160C23×C61—82no—
161C23×C61—82no—
162C23×C61#12711yescharacteristic
163Q8×C61—42no—
164D4×C61—44no—
165D4×C61—42no—
166D4×C61—42no—
167D4×C61—44no—
168D4×C61—42no—
169D4×C61—44no—
170D4×C61—82no—
171D4×C61—82no—
172D4×C61—82no—
173D4×C61—82no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—41no—
177C2×C4×C61—82no—
178C2×C4×C61#12711yescharacteristic
179C2×C4×C61#12711yescharacteristic
180C2×C4×C61#12711yescharacteristic
181C2×C4×C61#13514yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—82no—
189Dic122—42no—
190Dic122—42no—
191Dic122—44no—
192Dic122—82no—
193C8×C61—24no—
194#92418—24no—
195#92418—22no—
196#92418—22no—
197#19627—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—81no—
201C22×C61—82no—
202C22×C61—82no—
203C22×C61—81no—
204C22×C61—82no—
205C22×C61—81no—
206C22×C61—164no—
207C22×C61#49512yes—
208C22×C61C2×#12711yescharacteristic
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—81no—
212C4×C61#49512yes—
213C4×C61C2×#13512yes—
214C4×C61#57512yes—
215C4×C61#57612yes—
216Dic61—81no—
217Dic61—82no—
218Dic61—81no—
219#3209—611no—
220#3205D6111yescharacteristic
221#3167—611no—
222#3141—611no—
223#3138—611no—
224C2×#486—611no—
225#1720—612no—
226#1717—612no—
227#1711—612no—
228#1709—612no—
229C8×M16—611no—
230C2×C61—22no—
231C2×C61—161no—
232C2×C61—162no—
233C2×C61C2×#49511yescharacteristic
234C2×C61#315311yescharacteristic
235C2×C61#315511yescharacteristic
236#533—611no—
237#532C2×D6111yescharacteristic
238C2×C4⋊Q8—611no—
239C2×#127C2×D6111yescharacteristic
240#498—611no—
241#494—612no—
242#493C2×D6112yes—
243#492—611no—
244#490—1222no—
245#489—1222no—
246#487—612no—
247#487—611no—
248#487—612no—
249#486—612no—
250#486—612no—
251#486—612no—
252#486—612no—
253#486—612no—
254#486—611no—
255#486—1222no—
256#486C2×D6112yes—
257#465—1222no—
258#462—1222no—
259#459—22no—
260#451—1222no—
261#447—22no—
262C4×SD16—1222no—
263#431C2×D6111yescharacteristic
264#418—1222no—
265#417—1222no—
266C2×#102—1222no—
267C4×M16—611no—
268C2×C4×C8—611no—
269#322—612no—
270C82—612no—
271C61#1962711yescharacteristic, complemented, Sylow
272C2×#41—21no—
273C22×Q8—1221no—
274C22×D4—21no—
275C22×D4—21no—
276C22×D4—21no—
277C2×Q16—1224no—
278C2×SD16—1222no—
279C2×D8—24no—
280C2×M16—612no—
281C22×C8—612no—
282C4⋊Q8—612no—
283C4⋊Q8—611no—
284C4⋊Q8—611no—
285C4⋊Q8—1221no—
286C4⋊Q8—1221no—
287C4⋊Q8—1221no—
288C4⋊Q8C22×D6111yescharacteristic
289#127—21no—
290#127—21no—
291#127—21no—
292#127C22×D6112yes—
293#127C22×D6111yescharacteristic
294#125—1221no—
295#124—21no—
296C22⋊Q8—1221no—
297C22⋊Q8—1221no—
298#121—21no—
299#121—21no—
300C4×Q8—1221no—
301C4×Q8—1221no—
302C4×D4—21no—
303C4×D4—21no—
304#117—1221no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4—1221no—
307C2×C42C22×D6111yescharacteristic
308#106—1222no—
309C4⋊C8C22⋊D6112yes—
310#103—1222no—
311#103—1224no—
312#103—1224no—
313#102—24no—
314#102—1224no—
315#102—1224no—
316#102—1224no—
317#102—1224no—
318#102—1224no—
319#102—1222no—
320C22⋊C8C22⋊D6112yes—
321#97—612no—
322C4×C8—612no—
323C4×C8—611no—
324C4×C8—612no—
325C4×C8—611no—
326C4×C8C22×D6112yes—
327C24—41no—
328#41—44no—
329C2×Q8—21no—
330C2×Q8—1221no—
331C2×Q8—1222no—
332C2×Q8—1221no—
333C2×Q8—4882no—
334C2×Q8—2441no—
335C2×D4—21no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4—21no—
341C2×D4—44no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—44no—
347C2×D4—82no—
348C22×C4—41no—
349C22×C4—2441no—
350C22×C4D4×D6111yescharacteristic
351C22×C4D4×D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353Q16—4884no—
354SD16—2444no—
355D8—84no—
356M16—1224no—
357C2×C8—612no—
358C2×C8—612no—
359C2×C8—614no—
360C2×C8—2442no—
361C2×C8C2×C22⋊D6114yes—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—1222no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C4⋊C4—1222no—
368C4⋊C4—1221no—
369C4⋊C4—1221no—
370C4⋊C4—2441no—
371C4⋊C4—2441no—
372C4⋊C4—2441no—
373C4⋊C4—2441no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—2441no—
377C42—2441no—
378C42D4×D6111yescharacteristic
379C42D4×D6111yescharacteristic
380C42C2×C22⋊D6111yescharacteristic
381C42C23×D6111yescharacteristic
382C23—41no—
383C23—42no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23#1451977311yescharacteristic
388Q8—42no—
389Q8—4882no—
390Q8—2442no—
391Q8—2442no—
392Q8—2444no—
393D4—44no—
394D4—42no—
395D4—42no—
396D4—44no—
397D4—42no—
398D4—44no—
399D4—82no—
400D4—82no—
401D4—82no—
402D4—82no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—82no—
407C2×C4—4882no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4—2442no—
411C2×C4#1451977311yescharacteristic
412C2×C4#1451977311yescharacteristic
413C2×C4#1451977311yescharacteristic
414C2×C4#1451979614yes—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4C2×D4×D6111yescharacteristic
417C2×C4C22×C22⋊D6111yescharacteristic
418C8—24no—
419C8—1224no—
420C8—1222no—
421C8—1222no—
422C22—21no—
423C22—21no—
424C22—81no—
425C22—82no—
426C22—82no—
427C22—81no—
428C22—82no—
429C22—81no—
430C22—164no—
431C22#1684014512yes—
432C22C2×#1451977311yescharacteristic
433C4—21no—
434C4—21no—
435C4—81no—
436C4—4881no—
437C4—4882no—
438C4—4881no—
439C4#1684014512yes—
440C4C2×#1451979612yes—
441C4#1684055812yes—
442C4#1684056512yes—
443C2—22no—
444C2—162no—
445C2—161no—
446C2C2×#1684014511yescharacteristic
447C2#2145628111yescharacteristic
448C2#2145630211yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3205C2
socleC22×C61C2×#127
hypercenter#3205D61
hypocenterC61#19627
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #21456769C2semidirect
2. #21456374C2semidirect
3. #21456212C2semidirect
4. #21456193C2semidirect
5. C2×#16840108C2semidirect
6. #21445478C2semidirect
7. #21445457C2semidirect
8. #21445430C2semidirect
9. #21445424C2semidirect
10. #21444100C2semidirect
14. #16840158C22semidirect
15. #16840142C22semidirect
16. #16840134C22semidirect
19. #16840114C22semidirect
20. #16840113C22semidirect
21. #16840112C22semidirect
27. #16840108C22semidirect
28. #16840108C22semidirect
38. #16839484C22semidirect
39. C8×#92418C22semidirect
271. C61#19627semidirect

Maximal subgroups

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

subgroupindex
#214567692
#214563742
#214562122
#214561932
C2×#168401082
#214454782
#214454572
#214454302
#214454242
#214441002
C61×#32052
#1962761