Finite Group Databasebeta

← #27153343

Subgroups — #27153343

Order 12416.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690836C211yesmaximal, characteristic, complemented
2C2×#15690835C211yesmaximal, characteristic, complemented
3#20261381C211yesmaximal, characteristic, complemented
4#20261117C211yesmaximal, characteristic, complemented
5#20260983C211yesmaximal, characteristic, complemented
6#20260979C211yesmaximal, characteristic, complemented
7#20260950C211yesmaximal, characteristic, complemented
8#20260947C211yesmaximal, characteristic, complemented
9#20260897C211yesmaximal, characteristic, complemented
10#20260895C211yesmaximal, characteristic, complemented
11#20260869C211yesmaximal, characteristic, complemented
12#20260868C211yesmaximal, characteristic, complemented
13C4×#13095496C211yesmaximal, characteristic, complemented
14#20260813C211yesmaximal, characteristic, complemented
15#546×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×D388C2211yescharacteristic, complemented
18#15690836—24no—
19#15690835—24no—
20C2×#13095501C2211yescharacteristic, complemented
21C2×#13095499C2211yescharacteristic, complemented
22C2×#13095499C2211yescharacteristic, complemented
23C2×#13095498C2211yescharacteristic, complemented
24C2×#13095498C2211yescharacteristic, complemented
25C2×#13095496C2211yescharacteristic
26C2×#13095496C2211yescharacteristic
27C2×#13095496C2211yescharacteristic
28C2×#13095496C2211yescharacteristic
29C2×#13095495C2211yescharacteristic, complemented
30C2×#13095495C2211yescharacteristic, complemented
31#15690798C2211yescharacteristic, complemented
32#15690795—21no—
33#15690789C2211yescharacteristic, complemented
34#15690784C2211yescharacteristic
35#15690783—21no—
36#15690778—21no—
37C4×D388C2211yescharacteristic, complemented
38C22⋊C4×D97—21no—
39#15690743C2211yescharacteristic, complemented
40C4⋊#11715895C2211yescharacteristic, complemented
41#15690734C2211yescharacteristic, complemented
42#15690733C2211yescharacteristic, complemented
43#15690731C2211yescharacteristic, complemented
44#15690731C2211yescharacteristic, complemented
45#15690730C2211yescharacteristic, complemented
46#15690730C2211yescharacteristic, complemented
47C22⋊#11715895C2211yescharacteristic, complemented
48C22⋊#11715895C2211yescharacteristic, complemented
49C4×#11715895C2211yescharacteristic, complemented
50C2×#41×C97C2211yescharacteristic
51C22×D4×C97C2211yescharacteristic, complemented
52#124×C97C2211yescharacteristic, complemented
53#123×C97—21no—
54C22⋊Q8×C97C2211yescharacteristic, complemented
55#121×C97—21no—
56#121×C97C2211yescharacteristic
57#120×C97—21no—
58C4×D4×C97C2211yescharacteristic
59C4×D4×C97C2211yescharacteristic, complemented
60C2×C22⋊C4×C97—21no—
61C23×D97—21no—
62#13095511—22no—
63#13095511—41no—
64C2×Dic194C2311yescharacteristic
65C2×D388—22no—
66C2×D388—21no—
67C2×D388—41no—
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74#13095501—41no—
75#13095499—22no—
76#13095499—22no—
77#13095498—22no—
78#13095498—22no—
79#13095496—22no—
80#13095496—22no—
81#13095496—41no—
82#13095496D414yescomplemented
83#13095495—22no—
84#13095495—22no—
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C2×#11715895C2311yescharacteristic
89C4⋊Dic97C2311yescharacteristic
90#13095490—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96C24×C97—21no—
97#41×C97—22no—
98#41×C97—41no—
99C2×Q8×C97C2311yescharacteristic
100C2×D4×C97—22no—
101C2×D4×C97—21no—
102C2×D4×C97—21no—
103C2×D4×C97—21no—
104C2×D4×C97—41no—
105C2×D4×C97D411yescharacteristic
106C2×D4×C97C2311yescharacteristic
107C2×D4×C97C2311yescharacteristic
108C22×C4×C97—21no—
109C22×C4×C97—21no—
110C22×C4×C97C2311yescharacteristic
111C22×C4×C97C2311yescharacteristic
112C4⋊C4×C97—21no—
113C4⋊C4×C97D411yescharacteristic
114C4⋊C4×C97C2311yescharacteristic
115C22⋊C4×C97—21no—
116C22⋊C4×C97—21no—
117C22⋊C4×C97—21no—
118C22⋊C4×C97—21no—
119C22⋊C4×C97—21no—
120C22⋊C4×C97D411yescharacteristic
121C22⋊C4×C97D411yescharacteristic
122C42×C97C2311yescharacteristic
123C23×C97—21no—
124C23×C97—21no—
125C23×C97—42no—
126C23×C97—42no—
127C23×C97—81no—
128C23×C97C2×D411yescharacteristic
129C23×C97C2×D411yescharacteristic
130C22×D97—21no—
131C22×D97—21no—
132C22×D97—42no—
133C22×D97—42no—
134C22×D97—81no—
135Q8×C97—21no—
136Q8×C97—21no—
137D4×C97—21no—
138D4×C97—21no—
139D4×C97—21no—
140D4×C97—41no—
141D4×C97—42no—
142D4×C97—42no—
143D4×C97—81no—
144D4×C97C2×D412yes—
145C2×C4×C97—21no—
146C2×C4×C97—21no—
147C2×C4×C97—21no—
148C2×C4×C97—21no—
149C2×C4×C97—21no—
150C2×C4×C97—21no—
151C2×C4×C97—42no—
152C2×C4×C97—81no—
153C2×C4×C97C2×D411yescharacteristic
154C2×C4×C97C2×D411yescharacteristic
155C2×C4×C97C2×D411yescharacteristic
156C2×C4×C97C2×D411yescharacteristic
157C2×C4×C97C2411yescharacteristic
158C22⋊D97—42no—
159C22⋊D97—81no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162D388—21no—
163D388—21no—
164D388—21no—
165D388—42no—
166D388C2×D412yes—
167C4×D97—42no—
168C4×D97—81no—
169Dic194—21no—
170Dic194—21no—
171#11715895—21no—
172#11715895—21no—
173#11715895—21no—
174#11715895C2×D412yes—
175C22×C97—21no—
176C22×C97—21no—
177C22×C97—22no—
178C22×C97—21no—
179C22×C97—21no—
180C22×C97—21no—
181C22×C97—21no—
182C22×C97—41no—
183C22×C97—41no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97—84no—
188C22×C97C22×D411yescharacteristic
189C2×D97—22no—
190C2×D97—41no—
191C2×D97—41no—
192C2×D97—41no—
193C2×D97—41no—
194C2×D97—41no—
195C2×D97—84no—
196C4×C97—21no—
197C4×C97—21no—
198C4×C97—21no—
199C4×C97—21no—
200C4×C97—41no—
201C4×C97—41no—
202C4×C97C22×D411yescharacteristic
203C4×C97H(5,𝔽2)11yescharacteristic
204Dic97—41no—
205Dic97—41no—
206C2×C97—41no—
207C2×C97—41no—
208C2×C97—42no—
209C2×C97—81no—
210C2×C97D4211yescharacteristic
211C2×C97#57511yescharacteristic
212C2×C97#57611yescharacteristic
213D97—42no—
214D97—81no—
215#3273—971nomaximal, Sylow
216C97#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—971no—
218C2×AGL(1,ℤ/8)—971no—
219#546—971no—
220#546D9711yescharacteristic, complemented
221#495—971no—
222#468—971no—
223#468—971no—
224#463—971no—
225#463—971no—
226#452—971no—
227#452—971no—
228#447—971no—
229#447—971no—
230C4×D8—971no—
231#436—971no—
232C2×#41—971no—
233C2×#41C2×D9711yescharacteristic
234C22×D4—971no—
235C22×D4C2×D9711yescharacteristic, complemented
236AGL(1,ℤ/8)—1944no—
237AGL(1,ℤ/8)—1944no—
238C2×Q16—971no—
239C2×SD16—971no—
240C2×SD16—971no—
241C2×SD16—971no—
242C2×SD16—971no—
243C2×D8—971no—
244C2×D8—971no—
245C2×D8—971no—
246C2×D8—971no—
247C2×M16—971no—
248C2×M16—971no—
249#124—971no—
250#124C2×D9711yescharacteristic, complemented
251#123—21no—
252#123—1941no—
253C22⋊Q8—971no—
254C22⋊Q8C2×D9711yescharacteristic, complemented
255#121—21no—
256#121—971no—
257#121—1941no—
258#121C2×D9711yescharacteristic
259#120—21no—
260#120—1941no—
261C4×D4—971no—
262C4×D4C2×D9711yescharacteristic
263C4×D4C2×D9711yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—1941no—
266#107—971no—
267C4⋊C8—971no—
268#103—971no—
269#103—971no—
270#102—971no—
271#102—971no—
272#102—971no—
273#102—971no—
274C22⋊C8—971no—
275C22⋊C8—971no—
276C4×C8—971no—
277C24—21no—
278C24—1941no—
279#41—22no—
280#41—41no—
281#41—3881no—
282#41—1942no—
283C2×Q8—971no—
284C2×Q8C22×D9711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—971no—
291C2×D4—971no—
292C2×D4—3881no—
293C2×D4—1942no—
294C2×D4—1941no—
295C2×D4—1941no—
296C2×D4—1941no—
297C2×D4C22⋊D9711yescharacteristic
298C2×D4C22×D9711yescharacteristic
299C2×D4C22×D9711yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—1941no—
303C22×C4—1941no—
304C22×C4C22×D9711yescharacteristic
305C22×C4C22×D9711yescharacteristic
306Q16—3881no—
307SD16—1942no—
308SD16—1942no—
309SD16—1942no—
310SD16—1942no—
311D8—974no—
312D8—3881no—
313D8—1942no—
314D8—1942no—
315M16—1942no—
316M16—1942no—
317C2×C8—971no—
318C2×C8—971no—
319C2×C8—971no—
320C2×C8—971no—
321C4⋊C4—21no—
322C4⋊C4—971no—
323C4⋊C4—1941no—
324C4⋊C4C22⋊D9711yescharacteristic
325C4⋊C4C22×D9711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—1941no—
332C22⋊C4—1941no—
333C22⋊C4—1941no—
334C22⋊C4—1941no—
335C22⋊C4—1941no—
336C22⋊C4C22⋊D9711yescharacteristic
337C22⋊C4C22⋊D9711yescharacteristic
338C42C22×D9711yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—7761no—
345C23—3882no—
346C23—3882no—
347C23—1941no—
348C23—1941no—
349C23C2×C22⋊D9711yescharacteristic
350C23C2×C22⋊D9711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—1941no—
354Q8—1941no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—7761no—
363D4—972no—
364D4—3882no—
365D4—3882no—
366D4—1941no—
367D4—1941no—
368D4—1941no—
369D4D4×D9712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—7761no—
379C2×C4—3882no—
380C2×C4—1941no—
381C2×C4—1941no—
382C2×C4C2×C22⋊D9711yescharacteristic
383C2×C4C2×C22⋊D9711yescharacteristic
384C2×C4C2×C22⋊D9711yescharacteristic
385C2×C4C2×C22⋊D9711yescharacteristic
386C2×C4C23×D9711yescharacteristic
387C8—972no—
388C8—1941no—
389C8—1941no—
390C8—1941no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—7764no—
405C22—3881no—
406C22—3881no—
407C22—3881no—
408C22—3881no—
409C22—3881no—
410C22—1942no—
411C22C22×C22⋊D9711yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—3881no—
419C4—3881no—
420C4C2×D4×D9711yescharacteristic
421C4#1569085511yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—7761no—
427C2—3882no—
428C2D4×C22⋊D9711yescharacteristic
429C2#2026153311yescharacteristic
430C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#546×C97C2
socleC22×C97C22×D4
hypercenter#546D97
hypocenterC97#3273
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. C2×#15690835C2semidirect
3. #20261381C2semidirect
4. #20261117C2semidirect
5. #20260983C2semidirect
6. #20260979C2semidirect
7. #20260950C2semidirect
8. #20260947C2semidirect
9. #20260897C2semidirect
10. #20260895C2semidirect
11. #20260869C2semidirect
12. #20260868C2semidirect
13. C4×#13095496C2semidirect
14. #20260813C2semidirect
15. #546×C97C2semidirect
17. C22×D388C22semidirect
20. C2×#13095501C22semidirect
21. C2×#13095499C22semidirect
22. C2×#13095499C22semidirect
23. C2×#13095498C22semidirect
24. C2×#13095498C22semidirect
29. C2×#13095495C22semidirect
30. C2×#13095495C22semidirect
31. #15690798C22semidirect
33. #15690789C22semidirect
37. C4×D388C22semidirect
39. #15690743C22semidirect
40. C4⋊#11715895C22semidirect
41. #15690734C22semidirect
42. #15690733C22semidirect
43. #15690731C22semidirect
44. #15690731C22semidirect
45. #15690730C22semidirect
46. #15690730C22semidirect
47. C22⋊#11715895C22semidirect
48. C22⋊#11715895C22semidirect
49. C4×#11715895C22semidirect
51. C22×D4×C97C22semidirect
52. #124×C97C22semidirect
54. C22⋊Q8×C97C22semidirect
59. C4×D4×C97C22semidirect
82. #13095496D4semidirect
216. C97#3273semidirect
220. #546D97semidirect
235. C22×D4C2×D97semidirect
250. #124C2×D97semidirect
254. C22⋊Q8C2×D97semidirect
263. C4×D4C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
C2×#156908352
#202613812
#202611172
#202609832
#202609792
#202609502
#202609472
#202608972
#202608952
#202608692
#202608682
C4×#130954962
#202608132
#546×C972
#327397