Finite Group Databasebeta

← #27153527

Subgroups — #27153527

Order 12416.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261546C211yesmaximal, characteristic, complemented
2#20261543C211yesmaximal, characteristic, complemented
3#20261542C211yesmaximal, characteristic, complemented
4#20261106C211yesmaximal, characteristic, complemented
5#20261000C211yesmaximal, characteristic, complemented
6#20260998C211yesmaximal, characteristic, complemented
7#20260996C211yesmaximal, characteristic, complemented
8#20260995C211yesmaximal, characteristic, complemented
9#20260918C211yesmaximal, characteristic, complemented
10#20260916C211yesmaximal, characteristic, complemented
11#20260903C211yesmaximal, characteristic, complemented
12#20260901C211yesmaximal, characteristic, complemented
13#20260852C211yesmaximal, characteristic, complemented
14#20260850C211yesmaximal, characteristic, complemented
15#577×C97C211yesmaximal, characteristic, complemented
16#15690857C2211yescharacteristic
17#15690856C2211yescharacteristic, complemented
18#15690842—21no—
19#15690841—21no—
20#15690841C2211yescharacteristic, complemented
21#15690840—21no—
22#15690840C2211yescharacteristic, complemented
23#15690840C2211yescharacteristic, complemented
24#15690836—21no—
25#15690836C2211yescharacteristic
26#15690835—21no—
27#15690835C2211yescharacteristic
28#15690835C2211yescharacteristic
29#15690834—21no—
30#15690831—21no—
31#15690830—21no—
32#15690830—21no—
33#15690830C2211yescharacteristic
34C2×#13095500C2211yescharacteristic, complemented
35C2×#13095499—21no—
36C2×#13095499C2211yescharacteristic, complemented
37C2×#13095498—21no—
38C2×#13095498C2211yescharacteristic, complemented
39SD16×D97—21no—
40#15690817C2211yescharacteristic, complemented
41#15690817C2211yescharacteristic, complemented
42#15690816C2211yescharacteristic, complemented
43#15690808C2211yescharacteristic, complemented
44#15690806C2211yescharacteristic
45#15690746C2211yescharacteristic, complemented
46#15690745C2211yescharacteristic, complemented
47#15690737C2211yescharacteristic, complemented
48#15690737C2211yescharacteristic, complemented
49#15690737C2211yescharacteristic, complemented
50#15690736C2211yescharacteristic, complemented
51#15690729C2211yescharacteristic, complemented
52#15690728C2211yescharacteristic, complemented
53#15690727C2211yescharacteristic, complemented
54#15690726C2211yescharacteristic, complemented
55C8×Dic97C2211yescharacteristic, complemented
562−1+4×C97C2211yescharacteristic
57H(5,𝔽2)×C97C2211yescharacteristic, complemented
58#137×C97—21no—
59#137×C97C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C97—21no—
61AGL(1,ℤ/8)×C97C2211yescharacteristic
62#135×C97—21no—
63#135×C97C2211yescharacteristic
64C2×SD16×C97—21no—
65C2×SD16×C97C2211yescharacteristic, complemented
66#131×C97C2211yescharacteristic, complemented
67#13095512—21no—
68#13095511—21no—
69#13095511—21no—
70#13095511C2311yescharacteristic
71#13095510—21no—
72#13095510—41no—
73C2×Dic194—21no—
74C2×Dic194C2311yescharacteristic
75Q8×D97—21no—
76C2×D388—21no—
77C2×D388C2311yescharacteristic
78C2×C22⋊D97—41no—
79D4×D97—21no—
80D4×D97—41no—
81#13095501—21no—
82#13095501—21no—
83#13095501—21no—
84#13095501D411yescharacteristic, complemented
85#13095500—21no—
86#13095500—21no—
87#13095499—21no—
88#13095499—21no—
89#13095499—21no—
90#13095499—21no—
91#13095499—21no—
92#13095499D411yescharacteristic, complemented
93#13095498—21no—
94#13095498—21no—
95#13095498—21no—
96#13095498—21no—
97#13095498—21no—
98#13095498D411yescharacteristic, complemented
99#13095496—21no—
100#13095496—21no—
101#13095496—21no—
102#13095496D411yescharacteristic, complemented
103#13095495—21no—
104#13095495—21no—
105#13095495C2311yescharacteristic
106#13095495C2311yescharacteristic
107#13095495C2311yescharacteristic
108#13095494—21no—
109C2×#11715895—21no—
110C2×#11715895—21no—
111C2×#11715895C2311yescharacteristic
112C8×D97—21no—
113#13095490—41no—
114C4×Dic97C2311yescharacteristic
115#41×C97—21no—
116#41×C97—21no—
117#41×C97—21no—
118#41×C97—21no—
119#41×C97—41no—
120#41×C97C2311yescharacteristic
121#41×C97C2311yescharacteristic
122#41×C97C2311yescharacteristic
123C2×Q8×C97—21no—
124C2×Q8×C97—21no—
125C2×Q8×C97C2311yescharacteristic
126C2×D4×C97—21no—
127C2×D4×C97—21no—
128C2×D4×C97—41no—
129C2×D4×C97C2311yescharacteristic
130Q16×C97—21no—
131Q16×C97D411yescharacteristic, complemented
132SD16×C97—21no—
133SD16×C97—21no—
134SD16×C97—21no—
135SD16×C97—21no—
136SD16×C97D411yescharacteristic, complemented
137SD16×C97D411yescharacteristic, complemented
138D8×C97—21no—
139D8×C97D411yescharacteristic, complemented
140M16×C97—21no—
141M16×C97C2311yescharacteristic
142C2×C8×C97—21no—
143C2×C8×C97C2311yescharacteristic
144C23×C97—21no—
145C23×C97—41no—
146C22×D97—21no—
147C22×D97—41no—
148Q8×C97—21no—
149Q8×C97—21no—
150Q8×C97—41no—
151Q8×C97C2×D411yescharacteristic
152Q8×C97C2×D411yescharacteristic
153Q8×C97C2×D411yescharacteristic
154D4×C97—21no—
155D4×C97—21no—
156D4×C97—21no—
157D4×C97—21no—
158D4×C97—21no—
159D4×C97—41no—
160D4×C97—41no—
161D4×C97—41no—
162D4×C97C2×D411yescharacteristic
163D4×C97C2×D411yescharacteristic
164D4×C97C2×D411yescharacteristic
165C2×C4×C97—21no—
166C2×C4×C97—21no—
167C2×C4×C97—21no—
168C2×C4×C97—21no—
169C2×C4×C97—41no—
170C2×C4×C97—41no—
171C2×C4×C97—41no—
172C2×C4×C97C2411yescharacteristic
173C22⋊D97—21no—
174C22⋊D97—41no—
175C22⋊D97—41no—
176C22⋊D97—82no—
177C2×Dic97—21no—
178C2×Dic97—21no—
179C2×Dic97—41no—
180D388—21no—
181D388—21no—
182D388—41no—
183D388C2×D411yescharacteristic
184C4×D97—21no—
185C4×D97—41no—
186C4×D97—41no—
187Dic194—21no—
188Dic194—21no—
189Dic194—41no—
190Dic194C2×D411yescharacteristic
191C8×C97—21no—
192C8×C97C2×D411yescharacteristic
193C8×C97C2×D411yescharacteristic
194#11715895—21no—
195#11715895—21no—
196#11715895—21no—
197#11715895C2×D411yescharacteristic
198#11715895C2×D411yescharacteristic
199C22×C97—21no—
200C22×C97—21no—
201C22×C97—21no—
202C22×C97—41no—
203C22×C97—81no—
204C22×C97—161no—
205C22×C97H(5,𝔽2)11yescharacteristic
206C2×D97—21no—
207C2×D97—41no—
208C2×D97—81no—
209C2×D97—161no—
210C4×C97—21no—
211C4×C97—21no—
212C4×C97—21no—
213C4×C97—41no—
214C4×C97C22×D411yescharacteristic
215C4×C97C22×D411yescharacteristic
216Dic97—22no—
217Dic97—21no—
218Dic97—41no—
219C2×C97—21no—
220C2×C97—41no—
221C2×C97—41no—
222C2×C97—41no—
223C2×C97—81no—
224C2×C97D4211yescharacteristic
225D97—41no—
226D97—81no—
227#3291—971nomaximal, Sylow
228C97#329111yescharacteristic, complemented, Sylow
229#577—971no—
230#577—971no—
231#577—971no—
232#577D9711yescharacteristic, complemented
233#492—971no—
234#471—971no—
235#471—971no—
236#471—971no—
237#471—971no—
238#456—971no—
239#456—971no—
240#453—971no—
241#453—971no—
242#443—971no—
243#443—971no—
2442−1+4—971no—
2452−1+4C2×D9711yescharacteristic
246H(5,𝔽2)—971no—
247H(5,𝔽2)C2×D9711yescharacteristic, complemented
248#137—21no—
249#137—971no—
250#137—971no—
251#137—971no—
252#137—1941no—
253#137—1941no—
254#137—1941no—
255#137C2×D9711yescharacteristic, complemented
256AGL(1,ℤ/8)—21no—
257AGL(1,ℤ/8)—971no—
258AGL(1,ℤ/8)—971no—
259AGL(1,ℤ/8)—971no—
260AGL(1,ℤ/8)—1941no—
261AGL(1,ℤ/8)—1941no—
262AGL(1,ℤ/8)—1941no—
263AGL(1,ℤ/8)C2×D9711yescharacteristic
264#135—21no—
265#135—971no—
266#135—1941no—
267#135—1941no—
268#135—1941no—
269#135C2×D9711yescharacteristic
270C2×SD16—21no—
271C2×SD16—971no—
272C2×SD16—971no—
273C2×SD16—971no—
274C2×SD16—1941no—
275C2×SD16—1941no—
276C2×SD16—1941no—
277C2×SD16C2×D9711yescharacteristic, complemented
278#131—971no—
279#131—971no—
280#131—971no—
281#131C2×D9711yescharacteristic, complemented
282C4⋊Q8—971no—
283#127—971no—
284#108—971no—
285#108—971no—
286#104—971no—
287#104—971no—
288#104—971no—
289#104—971no—
290#101—971no—
291#101—971no—
292#100—971no—
293#100—971no—
294C4×C8—971no—
295#41—21no—
296#41—21no—
297#41—21no—
298#41—21no—
299#41—41no—
300#41—971no—
301#41—3881no—
302#41—1941no—
303#41—1941no—
304#41—1941no—
305#41—1941no—
306#41C22×D9711yescharacteristic
307#41C22×D9711yescharacteristic
308#41C22×D9711yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—971no—
312C2×Q8—1941no—
313C2×Q8—1941no—
314C2×Q8C22×D9711yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—971no—
319C2×D4—3881no—
320C2×D4—3881no—
321C2×D4—1941no—
322C2×D4—1941no—
323C2×D4C22×D9711yescharacteristic
324Q16—21no—
325Q16—971no—
326Q16—1941no—
327Q16—1941no—
328Q16—1941no—
329Q16C22⋊D9711yescharacteristic, complemented
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—971no—
335SD16—971no—
336SD16—1941no—
337SD16—1941no—
338SD16—1941no—
339SD16—1941no—
340SD16—1941no—
341SD16—1941no—
342SD16—1941no—
343SD16—1941no—
344SD16—1941no—
345SD16—1941no—
346SD16—1941no—
347SD16—1941no—
348SD16C22⋊D9711yescharacteristic, complemented
349SD16C22⋊D9711yescharacteristic, complemented
350D8—21no—
351D8—971no—
352D8—1941no—
353D8—1941no—
354D8—1941no—
355D8C22⋊D9711yescharacteristic, complemented
356M16—21no—
357M16—971no—
358M16—971no—
359M16—971no—
360M16—1941no—
361M16—1941no—
362M16—1941no—
363M16C22×D9711yescharacteristic
364C2×C8—21no—
365C2×C8—971no—
366C2×C8—1941no—
367C2×C8—1941no—
368C2×C8—1941no—
369C2×C8C22×D9711yescharacteristic
370C4⋊C4—3881no—
371C42—971no—
372C23—21no—
373C23—41no—
374C23—3881no—
375C23—1941no—
376Q8—21no—
377Q8—21no—
378Q8—41no—
379Q8—971no—
380Q8—3881no—
381Q8—1941no—
382Q8—1941no—
383Q8D4×D9711yescharacteristic
384Q8C2×C22⋊D9711yescharacteristic
385Q8C2×C22⋊D9711yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—7762no—
395D4—971no—
396D4—3881no—
397D4—3881no—
398D4—3881no—
399D4—1941no—
400D4—1941no—
401D4—1941no—
402D4D4×D9711yescharacteristic
403D4C2×C22⋊D9711yescharacteristic
404D4C2×C22⋊D9711yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—3881no—
413C2×C4—3881no—
414C2×C4—3881no—
415C2×C4—1941no—
416C2×C4—1941no—
417C2×C4—1941no—
418C2×C4C23×D9711yescharacteristic
419C8—21no—
420C8—971no—
421C8—971no—
422C8—1941no—
423C8—1941no—
424C8—1941no—
425C8C2×C22⋊D9711yescharacteristic
426C8C2×C22⋊D9711yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—81no—
432C22—7761no—
433C22—161no—
434C22—15521no—
435C22—3881no—
436C22—1941no—
437C22#1569085511yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—3881no—
443C4—1942no—
444C4—1941no—
445C4C2×D4×D9711yescharacteristic
446C4C22×C22⋊D9711yescharacteristic
447C2—21no—
448C2—41no—
449C2—41no—
450C2—41no—
451C2—81no—
452C2—7761no—
453C2—3881no—
454C2D4×C22⋊D9711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#577×C97C2
socleC2×C97D42
hypercenter#577D97
hypocenterC97#3291
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #20261546C2semidirect
2. #20261543C2semidirect
3. #20261542C2semidirect
4. #20261106C2semidirect
5. #20261000C2semidirect
6. #20260998C2semidirect
7. #20260996C2semidirect
8. #20260995C2semidirect
9. #20260918C2semidirect
10. #20260916C2semidirect
11. #20260903C2semidirect
12. #20260901C2semidirect
13. #20260852C2semidirect
14. #20260850C2semidirect
15. #577×C97C2semidirect
17. #15690856C22semidirect
20. #15690841C22semidirect
22. #15690840C22semidirect
23. #15690840C22semidirect
34. C2×#13095500C22semidirect
36. C2×#13095499C22semidirect
38. C2×#13095498C22semidirect
40. #15690817C22semidirect
41. #15690817C22semidirect
42. #15690816C22semidirect
43. #15690808C22semidirect
45. #15690746C22semidirect
46. #15690745C22semidirect
47. #15690737C22semidirect
48. #15690737C22semidirect
49. #15690737C22semidirect
50. #15690736C22semidirect
51. #15690729C22semidirect
52. #15690728C22semidirect
53. #15690727C22semidirect
54. #15690726C22semidirect
55. C8×Dic97C22semidirect
57. H(5,𝔽2)×C97C22semidirect
59. #137×C97C22semidirect
65. C2×SD16×C97C22semidirect
66. #131×C97C22semidirect
84. #13095501D4semidirect
92. #13095499D4semidirect
98. #13095498D4semidirect
102. #13095496D4semidirect
131. Q16×C97D4semidirect
136. SD16×C97D4semidirect
137. SD16×C97D4semidirect
139. D8×C97D4semidirect
228. C97#3291semidirect
232. #577D97semidirect
247. H(5,𝔽2)C2×D97semidirect
255. #137C2×D97semidirect
277. C2×SD16C2×D97semidirect
281. #131C2×D97semidirect
329. Q16C22⋊D97semidirect
348. SD16C22⋊D97semidirect
349. SD16C22⋊D97semidirect
355. D8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202615462
#202615432
#202615422
#202611062
#202610002
#202609982
#202609962
#202609952
#202609182
#202609162
#202609032
#202609012
#202608522
#202608502
#577×C972
#329197