Finite Group Databasebeta

← #27150563

Subgroups — #27150563

Order 12416.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690837C211yesmaximal, characteristic
2#20261217C211yesmaximal, characteristic, complemented
3#20261207C211yesmaximal, characteristic, complemented
4#20260846C211yesmaximal, characteristic, complemented
5#20260845C211yesmaximal, characteristic, complemented
6#20260836C211yesmaximal, characteristic, complemented
7#20260833C211yesmaximal, characteristic, complemented
8#20260820C211yesmaximal, characteristic, complemented
9C4×D776C211yesmaximal, characteristic, complemented
10#20260815C211yesmaximal, characteristic, complemented
11#20260774C211yesmaximal, characteristic, complemented
12#20260730C211yesmaximal, characteristic, complemented
13#102×D97C211yesmaximal, characteristic, complemented
14#20260689C211yesmaximal, characteristic, complemented
15C4×SD16×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18#15690837C418yescomplemented
19C2×#13095498C2211yescharacteristic
20C2×D776C2211yescharacteristic
21C2×#13095496C2211yescharacteristic
22C2×#13095494C2211yescharacteristic
23C2×C8×D97C2211yescharacteristic
24Q8×Dic97C2211yescharacteristic, complemented
25C4×D388—21no—
26C4×D388C2211yescharacteristic
27C4×D388C2211yescharacteristic, complemented
28D4×Dic97C2211yescharacteristic
29#15690772—21no—
30#15690772C2211yescharacteristic
31#15690772C2211yescharacteristic, complemented
32C4×C22⋊D97—21no—
33#15690770—21no—
34#15690768C2211yescharacteristic
35#15690767—21no—
36#15690767C2211yescharacteristic
37C4⋊C4×D97—21no—
38C4⋊C4×D97C2211yescharacteristic
39C2×#13095488—21no—
40C22⋊C4×D97—21no—
41#15690744C2211yescharacteristic, complemented
42#15690743C2211yescharacteristic, complemented
43#15690741C2211yescharacteristic, complemented
44#15690734C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic
46#15690732C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic
48#15690730C2211yescharacteristic
49#15690730C2211yescharacteristic, complemented
50#15690721C2211yescharacteristic, complemented
51#15690720C2211yescharacteristic, complemented
52C8×Dic97C2211yescharacteristic, complemented
53C2×SD16×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic, complemented
55C4×D4×C97C2211yescharacteristic
56#106×C97C2211yescharacteristic, complemented
57#103×C97C2211yescharacteristic, complemented
58#102×C97C2211yescharacteristic
59C4×C8×C97C2211yescharacteristic, complemented
60C23×D97—21no—
61#13095512—22no—
62#13095512C2×C414yescomplemented
63C2×D388—21no—
64C2×D388C2311yescharacteristic
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67D4×D97—22no—
68D4×D97C2×C414yescomplemented
69C22×Dic97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74C2×C4×D97—21no—
75C2×C4×D97—21no—
76C2×C4×D97C2311yescharacteristic
77#13095498C2×C414yescomplemented
78D776C2×C414yescomplemented
79#13095496C2×C414yescomplemented
80#13095494C2×C414yescomplemented
81C2×#11715895C2311yescharacteristic
82C2×#11715895C2311yescharacteristic
83C8×D97—22no—
84C4⋊Dic97—21no—
85C4⋊Dic97C2311yescharacteristic
86C4⋊Dic97C2311yescharacteristic
87#13095490—21no—
88#13095490D411yescharacteristic
89#13095490D411yescharacteristic
90C22⋊Dic97—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96#13095488D411yescharacteristic
97#13095488D411yescharacteristic
98C4×Dic97—21no—
99C4×Dic97—21no—
100C4×Dic97C2311yescharacteristic
101C2×Q8×C97C2311yescharacteristic
102C2×D4×C97C2311yescharacteristic
103C22×C4×C97—21no—
104SD16×C97C2×C414yescomplemented
105C2×C8×C97C2311yescharacteristic
106C2×C8×C97C2311yescharacteristic
107C4⋊C4×C97—21no—
108C4⋊C4×C97C2311yescharacteristic
109C4⋊C4×C97C2311yescharacteristic
110C22⋊C4×C97—21no—
111C42×C97—21no—
112C42×C97C2311yescharacteristic
113C23×C97—21no—
114C22×D97—21no—
115C22×D97—21no—
116C22×D97—24no—
117C22×D97—81no—
118C22×D97—81no—
119C22×D97C2×D411yescharacteristic
120Q8×C97—21no—
121Q8×C97C22×C412yes—
122D4×C97—21no—
123D4×C97C22×C412yes—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—81no—
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2411yescharacteristic
131C22⋊D97—24no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—81no—
135C2×Dic97C2×D411yescharacteristic
136C2×Dic97C2×D411yescharacteristic
137C2×Dic97C2×D411yescharacteristic
138D388—24no—
139D388—21no—
140D388—21no—
141D388C22×C412yes—
142D388C22×C412yes—
143C4×D97—24no—
144C4×D97—81no—
145C4×D97—81no—
146C4×D97—81no—
147C4×D97—81no—
148C4×D97—81no—
149C4×D97C22×C412yes—
150C4×D97#4112yes—
151C8×C97—21no—
152C8×C97C22×C412yes—
153#11715895—21no—
154#11715895C22×C412yes—
155C22×C97—22no—
156C22×C97—41no—
157C22×C97—41no—
158C22×C97C22×D411yescharacteristic
159C2×D97—21no—
160C2×D97—21no—
161C2×D97—22no—
162C2×D97—22no—
163C2×D97—41no—
164C2×D97—41no—
165C2×D97—41no—
166C2×D97—41no—
167C2×D97—84no—
168C2×D97C4×D412yes—
169C4×C97—22no—
170C4×C97—21no—
171C4×C97—21no—
172C4×C97—41no—
173C4×C97—41no—
174C4×C97C23×C411yescharacteristic
175C4×C97C2×#4111yescharacteristic
176Dic97—21no—
177Dic97—21no—
178Dic97—41no—
179Dic97—41no—
180Dic97C4×D412yes—
181C2×C97—42no—
182C2×C97C2×C4×D411yescharacteristic
183C2×C97#57611yescharacteristic
184C2×C97#57711yescharacteristic
185D97—22no—
186D97—42no—
187D97—42no—
188#2946—971nomaximal, Sylow
189C97#294611yescharacteristic, complemented, Sylow
190C2×AGL(1,ℤ/8)—971no—
191#520—971no—
192#518—971no—
193#442—971no—
194#442—971no—
195#440—971no—
196#440—971no—
197C4×SD16—971no—
198C4×SD16D9711yescharacteristic, complemented
199C4×D8—971no—
200C4×D8—971no—
201#428—971no—
202#416—971no—
203C2×#102—971no—
204#405—971no—
205C2×#41—971no—
206C22×D4—971no—
207AGL(1,ℤ/8)—978no—
208C2×SD16—971no—
209C2×SD16C2×D9711yescharacteristic
210C2×D8—971no—
211C2×D8—971no—
212C2×M16—971no—
213C22×C8—971no—
214C4×Q8—971no—
215C4×Q8C2×D9711yescharacteristic, complemented
216C4×D4—971no—
217C4×D4—971no—
218C4×D4—971no—
219C4×D4—971no—
220C4×D4—971no—
221C4×D4—1941no—
222C4×D4—1941no—
223C4×D4—1941no—
224C4×D4—1941no—
225C4×D4C2×D9711yescharacteristic
226#117—971no—
227#117—971no—
228#117—1941no—
229C2×C4⋊C4—971no—
230C2×C4⋊C4—1941no—
231C2×C22⋊C4—1941no—
232C2×C22⋊C4—1941no—
233#107—971no—
234#107—971no—
235#106—971no—
236#106C2×D9711yescharacteristic, complemented
237#103—971no—
238#103C2×D9711yescharacteristic, complemented
239#102—971no—
240#102—971no—
241#102—971no—
242#102—971no—
243#102—971no—
244#102C2×D9711yescharacteristic
245#97—971no—
246#97—971no—
247C4×C8—971no—
248C4×C8C2×D9711yescharacteristic, complemented
249C24—1941no—
250#41—974no—
251#41—1942no—
252C2×Q8C22×D9711yescharacteristic
253C2×D4—974no—
254C2×D4—971no—
255C2×D4—971no—
256C2×D4—1941no—
257C2×D4—1942no—
258C2×D4—1941no—
259C2×D4C22×D9711yescharacteristic
260C22×C4—21no—
261C22×C4—971no—
262C22×C4—1941no—
263C22×C4—1941no—
264C22×C4—1941no—
265C22×C4—1941no—
266C22×C4—1941no—
267C22×C4—1941no—
268C22×C4—1941no—
269SD16—974no—
270SD16C4×D9714yescomplemented
271D8—974no—
272D8—974no—
273M16—974no—
274C2×C8—971no—
275C2×C8—971no—
276C2×C8—1942no—
277C2×C8C22×D9711yescharacteristic
278C2×C8C22×D9711yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—971no—
281C4⋊C4—971no—
282C4⋊C4—971no—
283C4⋊C4—971no—
284C4⋊C4—1941no—
285C4⋊C4—1941no—
286C4⋊C4C22×D9711yescharacteristic
287C4⋊C4C22×D9711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—971no—
290C22⋊C4—971no—
291C22⋊C4—1941no—
292C22⋊C4—1941no—
293C22⋊C4—1941no—
294C22⋊C4—1941no—
295C22⋊C4—1941no—
296C22⋊C4—1941no—
297C42—21no—
298C42—971no—
299C42—1941no—
300C42—1941no—
301C42C22×D9711yescharacteristic
302C23—21no—
303C23—7761no—
304C23—7761no—
305C23—971no—
306C23—1944no—
307C23—1941no—
308C23—1941no—
309Q8—21no—
310Q8C2×C4×D9712yes—
311D4—21no—
312D4—972no—
313D4—972no—
314D4—1944no—
315D4—1944no—
316D4—1941no—
317D4—1941no—
318D4C2×C4×D9712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—7761no—
324C2×C4—7761no—
325C2×C4—7761no—
326C2×C4—7761no—
327C2×C4—7761no—
328C2×C4—7761no—
329C2×C4—971no—
330C2×C4—972no—
331C2×C4—972no—
332C2×C4—971no—
333C2×C4—971no—
334C2×C4—1944no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4D4×D9711yescharacteristic
338C2×C4D4×D9711yescharacteristic
339C2×C4C23×D9711yescharacteristic
340C8—21no—
341C8—972no—
342C8—1941no—
343C8C2×C4×D9712yes—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—7764no—
348C22—972no—
349C22—3881no—
350C22—3881no—
351C22—3881no—
352C22—3881no—
353C22—1942no—
354C22—1942no—
355C22—1941no—
356C22—1941no—
357C22C2×D4×D9711yescharacteristic
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—972no—
364C4—3881no—
365C4—3881no—
366C4—1941no—
367C4—1941no—
368C4C22×C4×D9711yescharacteristic
369C4#41×D9711yescharacteristic
370C2—42no—
371C2—3882no—
372C2—3882no—
373C2—1942no—
374C2C4×D4×D9711yescharacteristic
375C2#2026153711yescharacteristic
376C2#2026154111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
FittingC4×SD16×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690837C2
cosocleC2×C4×C97C24
hypercenterC4×SD16D97
hypocenterC97#2946
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #20261217C2semidirect
3. #20261207C2semidirect
4. #20260846C2semidirect
5. #20260845C2semidirect
6. #20260836C2semidirect
7. #20260833C2semidirect
8. #20260820C2semidirect
9. C4×D776C2semidirect
10. #20260815C2semidirect
11. #20260774C2semidirect
12. #20260730C2semidirect
13. #102×D97C2semidirect
14. #20260689C2semidirect
15. C4×SD16×C97C2semidirect
18. #15690837C4semidirect
24. Q8×Dic97C22semidirect
27. C4×D388C22semidirect
31. #15690772C22semidirect
41. #15690744C22semidirect
42. #15690743C22semidirect
43. #15690741C22semidirect
44. #15690734C22semidirect
46. #15690732C22semidirect
49. #15690730C22semidirect
50. #15690721C22semidirect
51. #15690720C22semidirect
52. C8×Dic97C22semidirect
54. C4×Q8×C97C22semidirect
56. #106×C97C22semidirect
57. #103×C97C22semidirect
59. C4×C8×C97C22semidirect
62. #13095512C2×C4semidirect
68. D4×D97C2×C4semidirect
77. #13095498C2×C4semidirect
78. D776C2×C4semidirect
79. #13095496C2×C4semidirect
80. #13095494C2×C4semidirect
104. SD16×C97C2×C4semidirect
189. C97#2946semidirect
198. C4×SD16D97semidirect
215. C4×Q8C2×D97semidirect
236. #106C2×D97semidirect
238. #103C2×D97semidirect
248. C4×C8C2×D97semidirect
270. SD16C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908372
#202612172
#202612072
#202608462
#202608452
#202608362
#202608332
#202608202
C4×D7762
#202608152
#202607742
#202607302
#102×D972
#202606892
C4×SD16×C972
#294697