Finite Group Databasebeta

← #27153261

Subgroups — #27153261

Order 12416.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690834C211yesmaximal, characteristic, complemented
2#20261384C211yesmaximal, characteristic, complemented
3#20261373C211yesmaximal, characteristic, complemented
4#20261128C211yesmaximal, characteristic, complemented
5#20260990C211yesmaximal, characteristic, complemented
6#20260964C211yesmaximal, characteristic, complemented
7#20260952C211yesmaximal, characteristic, complemented
8#20260938C211yesmaximal, characteristic, complemented
9#20260897C211yesmaximal, characteristic, complemented
10#20260894C211yesmaximal, characteristic, complemented
11#20260884C211yesmaximal, characteristic, complemented
12#20260880C211yesmaximal, characteristic, complemented
13#20260847C211yesmaximal, characteristic, complemented
14#20260806C211yesmaximal, characteristic, complemented
15C2×#135×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18#15690834—24no—
19C2×#13095501C2211yescharacteristic, complemented
20C2×#13095500C2211yescharacteristic
21C2×#13095499C2211yescharacteristic
22C2×#13095498C2211yescharacteristic
23C2×#13095496C2211yescharacteristic
24C2×#13095494C2211yescharacteristic
25#15690808C2211yescharacteristic, complemented
26#15690799C2211yescharacteristic
27#15690793—21no—
28#15690793—21no—
29#15690791—21no—
30#15690789C2211yescharacteristic, complemented
31#15690786C2211yescharacteristic
32#15690786C2211yescharacteristic
33#15690784C2211yescharacteristic
34#15690782—21no—
35#15690782—21no—
36#15690778—21no—
37D4×Dic97C2211yescharacteristic
38D4×Dic97C2211yescharacteristic, complemented
39C4×C22⋊D97C2211yescharacteristic
40C2×#13095490—21no—
41C2×#13095488—21no—
42#15690742C2211yescharacteristic, complemented
43#15690739C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690734C2211yescharacteristic, complemented
46#15690734C2211yescharacteristic, complemented
47#15690732C2211yescharacteristic
48#15690731C2211yescharacteristic
49#15690731C2211yescharacteristic, complemented
50C22⋊#11715895C2211yescharacteristic, complemented
51#15690722C2211yescharacteristic
52#15690720C2211yescharacteristic, complemented
53C2×#41×C97C2211yescharacteristic
54C2×#41×C97C2211yescharacteristic, complemented
55#135×C97—24no—
56C2×Q16×C97C2211yescharacteristic, complemented
57C2×SD16×C97C2211yescharacteristic
58C2×SD16×C97C2211yescharacteristic, complemented
59C2×D8×C97C2211yescharacteristic
60C22×C8×C97C2211yescharacteristic, complemented
61C23×D97—21no—
62#13095510—22no—
63#13095510—41no—
64C2×Dic194C2311yescharacteristic
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C2×C22⋊D97D411yescharacteristic
70D4×D97—22no—
71D4×D97—41no—
72C22×Dic97—21no—
73C22×Dic97—21no—
74C22×Dic97—21no—
75C2×C4×D97C2311yescharacteristic
76#13095501—41no—
77#13095500—22no—
78#13095499—22no—
79#13095498—41no—
80#13095496—22no—
81#13095494—22no—
82C2×#11715895C2311yescharacteristic
83C2×#11715895C2311yescharacteristic
84C4⋊Dic97C2311yescharacteristic
85C4⋊Dic97C2311yescharacteristic
86#13095490—21no—
87#13095490—21no—
88#13095490—21no—
89#13095490—21no—
90#13095490—21no—
91#13095490D411yescharacteristic
92C22⋊Dic97—21no—
93C22⋊Dic97—21no—
94C22⋊Dic97—21no—
95C22⋊Dic97D411yescharacteristic
96#13095488—21no—
97#13095488—21no—
98#13095488—21no—
99#13095488—21no—
100#13095488—21no—
101#13095488D411yescharacteristic
102C4×Dic97C2311yescharacteristic
103#41×C97—22no—
104#41×C97—22no—
105#41×C97—41no—
106#41×C97—41no—
107C2×Q8×C97C2311yescharacteristic
108C2×Q8×C97C2311yescharacteristic
109C2×D4×C97—21no—
110C2×D4×C97—21no—
111C2×D4×C97C2311yescharacteristic
112C2×D4×C97C2311yescharacteristic
113C22×C4×C97—21no—
114C22×C4×C97—21no—
115C22×C4×C97C2311yescharacteristic
116Q16×C97—22no—
117SD16×C97—22no—
118SD16×C97—22no—
119D8×C97D414yescomplemented
120C2×C8×C97—22no—
121C2×C8×C97C2311yescharacteristic
122C2×C8×C97C2311yescharacteristic
123C23×C97—21no—
124C23×C97—21no—
125C23×C97C2×D411yescharacteristic
126C22×D97—21no—
127C22×D97—42no—
128C22×D97—42no—
129C22×D97—81no—
130C22×D97C2×D411yescharacteristic
131Q8×C97—21no—
132Q8×C97—21no—
133Q8×C97—21no—
134Q8×C97—21no—
135D4×C97—21no—
136D4×C97—21no—
137D4×C97—42no—
138D4×C97—42no—
139D4×C97C2×D412yes—
140D4×C97C2×D412yes—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—21no—
145C2×C4×C97—42no—
146C2×C4×C97—42no—
147C2×C4×C97C2×D411yescharacteristic
148C2×C4×C97C2411yescharacteristic
149C22⋊D97—41no—
150C22⋊D97—42no—
151C22⋊D97—42no—
152C22⋊D97—81no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156C2×Dic97—42no—
157C2×Dic97—42no—
158C2×Dic97—81no—
159C2×Dic97C2×D411yescharacteristic
160C2×Dic97C2×D411yescharacteristic
161C2×Dic97C2×D411yescharacteristic
162D388—21no—
163D388—21no—
164C4×D97—21no—
165C4×D97—21no—
166Dic194—21no—
167Dic194—21no—
168C8×C97—21no—
169C8×C97C2×D412yes—
170#11715895—21no—
171#11715895—21no—
172C22×C97—22no—
173C22×C97—22no—
174C22×C97—21no—
175C22×C97—21no—
176C22×C97—21no—
177C22×C97—41no—
178C22×C97—41no—
179C22×C97—41no—
180C22×C97—41no—
181C22×C97C22×D411yescharacteristic
182C2×D97—21no—
183C2×D97—21no—
184C2×D97—21no—
185C2×D97—41no—
186C2×D97—41no—
187C2×D97—41no—
188C2×D97—84no—
189C4×C97—21no—
190C4×C97—41no—
191C4×C97—41no—
192C4×C97C22×D411yescharacteristic
193C4×C97H(5,𝔽2)11yescharacteristic
194Dic97—21no—
195Dic97—21no—
196Dic97—21no—
197Dic97—41no—
198Dic97—41no—
199Dic97—41no—
200C2×C97—42no—
201C2×C97—41no—
202C2×C97—42no—
203C2×C97D4211yescharacteristic
204C2×C97#57511yescharacteristic
205C2×C97#57711yescharacteristic
206D97—41no—
207D97—81no—
208#3267—971nomaximal, Sylow
209C97#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—971no—
211C2×#135D9711yescharacteristic, complemented
212#547—971no—
213#546—971no—
214#497—971no—
215#470—971no—
216#465—971no—
217#463—971no—
218#461—971no—
219#452—971no—
220#452—971no—
221#450—971no—
222#449—971no—
223#442—971no—
224#435—971no—
225C2×#41—971no—
226C2×#41C2×D9711yescharacteristic
227C2×#41C2×D9711yescharacteristic, complemented
228C22×D4—971no—
229AGL(1,ℤ/8)—1944no—
230#135—24no—
231C2×Q16—971no—
232C2×Q16C2×D9711yescharacteristic, complemented
233C2×SD16—971no—
234C2×SD16—971no—
235C2×SD16—971no—
236C2×SD16C2×D9711yescharacteristic
237C2×SD16C2×D9711yescharacteristic, complemented
238C2×D8—971no—
239C2×D8C2×D9711yescharacteristic
240C2×M16—971no—
241C22×C8C2×D9711yescharacteristic, complemented
242C4⋊Q8—971no—
243#124—971no—
244#123—1941no—
245#123—1941no—
246#123—1941no—
247C22⋊Q8—971no—
248C22⋊Q8—971no—
249C22⋊Q8—971no—
250#121—971no—
251#121—1941no—
252#121—1941no—
253#120—1941no—
254C4×D4—971no—
255C4×D4—971no—
256C4×D4—971no—
257C2×C4⋊C4—1941no—
258C2×C22⋊C4—1941no—
259#106—971no—
260C4⋊C8—971no—
261#103—971no—
262#103—971no—
263#103—971no—
264#102—971no—
265#102—971no—
266#102—971no—
267C22⋊C8—971no—
268C22⋊C8—971no—
269#97—971no—
270C24—1941no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—3881no—
276#41—1942no—
277C2×Q8—971no—
278C2×Q8C22×D9711yescharacteristic
279C2×Q8C22×D9711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—971no—
283C2×D4—971no—
284C2×D4—3881no—
285C2×D4—1941no—
286C2×D4—1941no—
287C2×D4—1941no—
288C2×D4—1942no—
289C2×D4C22×D9711yescharacteristic
290C2×D4C22×D9711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—971no—
294C22×C4—1941no—
295C22×C4—1941no—
296C22×C4—1941no—
297C22×C4C22×D9711yescharacteristic
298Q16—22no—
299Q16—3881no—
300SD16—22no—
301SD16—22no—
302SD16—3881no—
303SD16—1942no—
304SD16—1942no—
305D8—1942no—
306D8C22⋊D9714yescomplemented
307M16—1942no—
308C2×C8—22no—
309C2×C8—971no—
310C2×C8—971no—
311C2×C8C22×D9711yescharacteristic
312C2×C8C22×D9711yescharacteristic
313C4⋊C4—971no—
314C4⋊C4—971no—
315C4⋊C4—971no—
316C4⋊C4—1941no—
317C4⋊C4—1941no—
318C4⋊C4—1941no—
319C4⋊C4—1941no—
320C4⋊C4—1941no—
321C22⋊C4—971no—
322C22⋊C4—971no—
323C22⋊C4—1941no—
324C22⋊C4—1941no—
325C22⋊C4—1941no—
326C22⋊C4—1941no—
327C22⋊C4—1941no—
328C22⋊C4—1941no—
329C22⋊C4—1941no—
330C22⋊C4—1941no—
331C42—971no—
332C23—21no—
333C23—21no—
334C23—7761no—
335C23—971no—
336C23—3882no—
337C23—3882no—
338C23—1941no—
339C23D4×D9711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—1941no—
345Q8—1941no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—42no—
350D4—7761no—
351D4—3882no—
352D4—3882no—
353D4—3881no—
354D4—1941no—
355D4—1941no—
356D4C2×C22⋊D9712yes—
357D4C2×C22⋊D9712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—7761no—
365C2×C4—971no—
366C2×C4—971no—
367C2×C4—971no—
368C2×C4—3882no—
369C2×C4—3882no—
370C2×C4—1941no—
371C2×C4—1941no—
372C2×C4—1941no—
373C2×C4—1941no—
374C2×C4—1941no—
375C2×C4D4×D9711yescharacteristic
376C2×C4C23×D9711yescharacteristic
377C8—21no—
378C8—1941no—
379C8—1941no—
380C8C2×C22⋊D9712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—7764no—
391C22—3881no—
392C22—3881no—
393C22—3881no—
394C22—1941no—
395C22—1941no—
396C22—1941no—
397C22C2×D4×D9711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—3881no—
402C4—3881no—
403C4—3881no—
404C4—1941no—
405C4—1941no—
406C4—1941no—
407C4C22×C22⋊D9711yescharacteristic
408C4#1569085511yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—7761no—
413C2—3881no—
414C2D4×C22⋊D9711yescharacteristic
415C2#2026153211yescharacteristic
416C2#2026154211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#135×C97C2
socleC22×C97C22×D4
hypercenterC2×#135D97
hypocenterC97#3267
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#15690834C2semidirect
2. #20261384C2semidirect
3. #20261373C2semidirect
4. #20261128C2semidirect
5. #20260990C2semidirect
6. #20260964C2semidirect
7. #20260952C2semidirect
8. #20260938C2semidirect
9. #20260897C2semidirect
10. #20260894C2semidirect
11. #20260884C2semidirect
12. #20260880C2semidirect
13. #20260847C2semidirect
14. #20260806C2semidirect
15. C2×#135×C97C2semidirect
19. C2×#13095501C22semidirect
25. #15690808C22semidirect
30. #15690789C22semidirect
38. D4×Dic97C22semidirect
42. #15690742C22semidirect
43. #15690739C22semidirect
44. #15690735C22semidirect
45. #15690734C22semidirect
46. #15690734C22semidirect
49. #15690731C22semidirect
50. C22⋊#11715895C22semidirect
52. #15690720C22semidirect
54. C2×#41×C97C22semidirect
56. C2×Q16×C97C22semidirect
58. C2×SD16×C97C22semidirect
60. C22×C8×C97C22semidirect
119. D8×C97D4semidirect
209. C97#3267semidirect
211. C2×#135D97semidirect
227. C2×#41C2×D97semidirect
232. C2×Q16C2×D97semidirect
237. C2×SD16C2×D97semidirect
241. C22×C8C2×D97semidirect
306. D8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
#202613842
#202613732
#202611282
#202609902
#202609642
#202609522
#202609382
#202608972
#202608942
#202608842
#202608802
#202608472
#202608062
C2×#135×C972
#326797