Finite Group Databasebeta

← #27153553

Subgroups — #27153553

Order 12416.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261443C211yesmaximal, characteristic, complemented
2D4×C22⋊D97C211yesmaximal, characteristic
3#20261080C211yesmaximal, characteristic, complemented
4#20261054C211yesmaximal, characteristic, complemented
5#20261052C211yesmaximal, characteristic, complemented
6#20261023C211yesmaximal, characteristic, complemented
7#20260966C211yesmaximal, characteristic, complemented
8#20260961C211yesmaximal, characteristic, complemented
9#20260933C211yesmaximal, characteristic, complemented
10#20260884C211yesmaximal, characteristic, complemented
11SD16×Dic97C211yesmaximal, characteristic, complemented
12#20260798C211yesmaximal, characteristic, complemented
13C2×#15690731C211yesmaximal, characteristic, complemented
14#102×D97C211yesmaximal, characteristic, complemented
15#450×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97—21no—
17C2×D4×D97C2211yescharacteristic
18C2×#13095498C2211yescharacteristic, complemented
19C22×#11715895C2211yescharacteristic, complemented
20C2×C8×D97C2211yescharacteristic, complemented
21#15690809C2211yescharacteristic, complemented
22#15690806C2211yescharacteristic
23#15690800—21no—
24#15690790—21no—
25#15690787—21no—
26#15690786C2211yescharacteristic, complemented
27#15690784C2211yescharacteristic
28#15690782—21no—
29#15690781C2211yescharacteristic
30#15690779—21no—
31#15690778—21no—
32Q8×Dic97C2211yescharacteristic, complemented
33D4×Dic97C2211yescharacteristic
34C4×C22⋊D97C2211yescharacteristic
35#15690770—21no—
36C2×C4⋊Dic97C2211yescharacteristic, complemented
37C4⋊C4×D97C2211yescharacteristic, complemented
38#15690742C2211yescharacteristic, complemented
39#15690742C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41#15690739C2211yescharacteristic, complemented
42#15690734C2211yescharacteristic, complemented
43#15690732C2211yescharacteristic, complemented
44#15690732C2211yescharacteristic, complemented
45#15690731C2211yescharacteristic
46#15690731C2211yescharacteristic
47#15690731C2211yescharacteristic, complemented
48#15690731C2211yescharacteristic, complemented
49#15690730C2211yescharacteristic, complemented
50#15690722C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C22×D4×C97C2211yescharacteristic
53C2×SD16×C97C2211yescharacteristic
54C2×SD16×C97C2211yescharacteristic, complemented
55C22⋊Q8×C97C2211yescharacteristic, complemented
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic, complemented
58C22⋊C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60C2×Dic194—21no—
61C2×D388C2311yescharacteristic
62C2×C22⋊D97—21no—
63C2×C22⋊D97—21no—
64C2×C22⋊D97—21no—
65C2×C22⋊D97—21no—
66C2×C22⋊D97—21no—
67C2×C22⋊D97—44no—
68C2×C22⋊D97D411yescharacteristic
69D4×D97—22no—
70D4×D97—41no—
71C22×Dic97—21no—
72C22×Dic97—21no—
73C2×C4×D97—21no—
74C2×C4×D97C2311yescharacteristic
75#13095498—41no—
76C2×#11715895—41no—
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C8×D97—41no—
80C4⋊Dic97—21no—
81C4⋊Dic97—21no—
82C4⋊Dic97C2311yescharacteristic
83C4⋊Dic97C2311yescharacteristic
84C4⋊Dic97C2311yescharacteristic
85#13095490—21no—
86#13095490—21no—
87#13095490—21no—
88#13095490D411yescharacteristic
89C22⋊Dic97—21no—
90C22⋊Dic97—21no—
91C22⋊Dic97D411yescharacteristic
92#13095488—21no—
93#13095488—21no—
94#13095488D411yescharacteristic
95C4×Dic97—21no—
96C4×Dic97C2311yescharacteristic
97C24×C97—21no—
98C2×Q8×C97C2311yescharacteristic
99C2×D4×C97—22no—
100C2×D4×C97—21no—
101C2×D4×C97—41no—
102C2×D4×C97C2311yescharacteristic
103C2×D4×C97C2311yescharacteristic
104C22×C4×C97C2311yescharacteristic
105SD16×C97—22no—
106SD16×C97—41no—
107C2×C8×C97C2311yescharacteristic
108C2×C8×C97C2311yescharacteristic
109C4⋊C4×C97—21no—
110C4⋊C4×C97C2311yescharacteristic
111C22⋊C4×C97—21no—
112C23×C97—21no—
113C23×C97—21no—
114C23×C97—42no—
115C23×C97—42no—
116C23×C97—81no—
117C23×C97C2×D411yescharacteristic
118C22×D97—21no—
119C22×D97—42no—
120C22×D97—42no—
121C22×D97—81no—
122C22×D97C2×D411yescharacteristic
123Q8×C97—21no—
124Q8×C97—21no—
125D4×C97—21no—
126D4×C97—21no—
127D4×C97—21no—
128D4×C97—42no—
129D4×C97#4112yes—
130C2×C4×C97—21no—
131C2×C4×C97—21no—
132C2×C4×C97—21no—
133C2×C4×C97—21no—
134C2×C4×C97C2×D411yescharacteristic
135C2×C4×C97C2411yescharacteristic
136C22⋊D97—42no—
137C22⋊D97—42no—
138C22⋊D97—42no—
139C22⋊D97—42no—
140C22⋊D97—81no—
141C22⋊D97SD1614yescomplemented
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145C2×Dic97—42no—
146C2×Dic97—81no—
147C2×Dic97C2×D411yescharacteristic
148C2×Dic97C2×D411yescharacteristic
149C2×Dic97C2×D411yescharacteristic
150D388—21no—
151D388—21no—
152C4×D97—21no—
153C4×D97—21no—
154C4×D97—81no—
155Dic194—81no—
156C8×C97—21no—
157C8×C97—21no—
158#11715895—21no—
159#11715895—21no—
160C22×C97—21no—
161C22×C97—21no—
162C22×C97—22no—
163C22×C97—41no—
164C22×C97—41no—
165C22×C97—41no—
166C22×C97—41no—
167C22×C97—41no—
168C22×C97—84no—
169C22×C97C2×SD1612yes—
170C22×C97C22×D411yescharacteristic
171C2×D97—21no—
172C2×D97—21no—
173C2×D97—41no—
174C2×D97—41no—
175C2×D97—41no—
176C2×D97—84no—
177C2×D97C2×SD1612yes—
178C4×C97—21no—
179C4×C97—41no—
180C4×C97—41no—
181C4×C97C2×#4111yescharacteristic
182C4×C97H(5,𝔽2)11yescharacteristic
183Dic97—21no—
184Dic97—21no—
185Dic97—41no—
186Dic97—41no—
187Dic97—41no—
188Dic97C2×SD1612yes—
189C2×C97—22no—
190C2×C97—42no—
191C2×C97—81no—
192C2×C97#54611yescharacteristic
193C2×C97C22×SD1611yescharacteristic
194C2×C97#57611yescharacteristic
195D97—22no—
196D97—81no—
197#3295—971nomaximal, Sylow
198C97#329511yescharacteristic, complemented, Sylow
199#554—971no—
200D42—971no—
201#486—971no—
202#481—971no—
203#481—971no—
204#476—971no—
205#465—971no—
206#465—971no—
207#460—971no—
208#450—971no—
209#450D9711yescharacteristic, complemented
210C4×SD16—971no—
211C8×D4—971no—
212C2×#102—971no—
213C2×#102—971no—
214C22×D4—971no—
215C22×D4—1941no—
216C22×D4C2×D9711yescharacteristic
217C2×SD16—971no—
218C2×SD16C2×D9711yescharacteristic
219C2×SD16C2×D9711yescharacteristic, complemented
220C22×C8—971no—
221C22×C8—971no—
222C4⋊Q8—971no—
223#127—971no—
224#125—1941no—
225C22⋊Q8—971no—
226C22⋊Q8—1941no—
227C22⋊Q8—1941no—
228C22⋊Q8C2×D9711yescharacteristic, complemented
229#121—971no—
230#121—971no—
231#121—1941no—
232#120—1941no—
233#120—1941no—
234C4×Q8—971no—
235C4×D4—971no—
236C4×D4—971no—
237C4×D4—1941no—
238C2×C4⋊C4—971no—
239C2×C4⋊C4—971no—
240#106—971no—
241#106—971no—
242#106—971no—
243C4⋊C8—971no—
244#103—971no—
245#102—971no—
246#102—971no—
247#102—971no—
248#102—971no—
249#102—971no—
250#102—971no—
251#102—971no—
252#102C2×D9711yescharacteristic
253#102C2×D9711yescharacteristic, complemented
254C22⋊C8—971no—
255C22⋊C8C2×D9711yescharacteristic, complemented
256C4×C8—971no—
257C24—21no—
258C24—1941no—
259C2×Q8—1941no—
260C2×Q8C22×D9711yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—971no—
265C2×D4—971no—
266C2×D4—3884no—
267C2×D4—3881no—
268C2×D4—1941no—
269C2×D4—1941no—
270C2×D4—1941no—
271C2×D4—1941no—
272C2×D4—1941no—
273C2×D4—1942no—
274C2×D4C22×D9711yescharacteristic
275C2×D4C22×D9711yescharacteristic
276C22×C4—971no—
277C22×C4—1941no—
278C22×C4—1941no—
279C22×C4—1941no—
280C22×C4C22×D9711yescharacteristic
281SD16—22no—
282SD16—41no—
283SD16—3881no—
284C2×C8—971no—
285C2×C8—971no—
286C2×C8—3881no—
287C2×C8—3881no—
288C2×C8C22×D9711yescharacteristic
289C2×C8C22×D9711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—971no—
292C4⋊C4—971no—
293C4⋊C4—971no—
294C4⋊C4—971no—
295C4⋊C4—1941no—
296C4⋊C4—1941no—
297C4⋊C4—1941no—
298C4⋊C4—1941no—
299C4⋊C4—1941no—
300C4⋊C4C22×D9711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—971no—
303C22⋊C4—971no—
304C22⋊C4—1941no—
305C22⋊C4—1941no—
306C22⋊C4—1941no—
307C22⋊C4—1941no—
308C42—971no—
309C42—1941no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—7761no—
316C23—971no—
317C23—3882no—
318C23—3882no—
319C23—1941no—
320C23D4×D9711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—7761no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—7761no—
329D4—974no—
330D4—3882no—
331D4—3882no—
332D4—3882no—
333D4—3882no—
334D4—1941no—
335D4—1941no—
336D4#1309551012yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—7761no—
342C2×C4—7761no—
343C2×C4—971no—
344C2×C4—971no—
345C2×C4—971no—
346C2×C4—3882no—
347C2×C4—1941no—
348C2×C4—1941no—
349C2×C4—1941no—
350C2×C4—1941no—
351C2×C4—1941no—
352C2×C4D4×D9711yescharacteristic
353C2×C4C23×D9711yescharacteristic
354C8—21no—
355C8—21no—
356C8—1941no—
357C8—1941no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—7764no—
368C22—972no—
369C22—3881no—
370C22—3881no—
371C22—3881no—
372C22—1941no—
373C22—1941no—
374C22SD16×D9712yes—
375C22C2×D4×D9711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—972no—
380C4—3881no—
381C4—3881no—
382C4—3881no—
383C4—1941no—
384C4—1941no—
385C4C2×#1309551011yescharacteristic
386C4#1569085511yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—7761no—
391C2—1942no—
392C2#2026137211yescharacteristic
393C2C2×SD16×D9711yescharacteristic
394C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#450×C97C2
socleC22×C97C22×D4
prime radicalD4×C22⋊D97C2
hypercenter#450D97
hypocenterC97#3295
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20261443C2semidirect
3. #20261080C2semidirect
4. #20261054C2semidirect
5. #20261052C2semidirect
6. #20261023C2semidirect
7. #20260966C2semidirect
8. #20260961C2semidirect
9. #20260933C2semidirect
10. #20260884C2semidirect
11. SD16×Dic97C2semidirect
12. #20260798C2semidirect
13. C2×#15690731C2semidirect
14. #102×D97C2semidirect
15. #450×C97C2semidirect
18. C2×#13095498C22semidirect
19. C22×#11715895C22semidirect
20. C2×C8×D97C22semidirect
21. #15690809C22semidirect
26. #15690786C22semidirect
32. Q8×Dic97C22semidirect
36. C2×C4⋊Dic97C22semidirect
37. C4⋊C4×D97C22semidirect
38. #15690742C22semidirect
39. #15690742C22semidirect
40. #15690741C22semidirect
41. #15690739C22semidirect
42. #15690734C22semidirect
43. #15690732C22semidirect
44. #15690732C22semidirect
47. #15690731C22semidirect
48. #15690731C22semidirect
49. #15690730C22semidirect
50. #15690722C22semidirect
51. C8×Dic97C22semidirect
54. C2×SD16×C97C22semidirect
55. C22⋊Q8×C97C22semidirect
57. #102×C97C22semidirect
58. C22⋊C8×C97C22semidirect
141. C22⋊D97SD16semidirect
198. C97#3295semidirect
209. #450D97semidirect
219. C2×SD16C2×D97semidirect
228. C22⋊Q8C2×D97semidirect
253. #102C2×D97semidirect
255. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614432
D4×C22⋊D972
#202610802
#202610542
#202610522
#202610232
#202609662
#202609612
#202609332
#202608842
SD16×Dic972
#202607982
C2×#156907312
#102×D972
#450×C972
#329597