Finite Group Databasebeta

← #27158509

Subgroups — #27158509

Order 12416.

456 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690898C211yesmaximal, characteristic, complemented
2D97⋊#131C211yesmaximal, characteristic, complemented
3#20261781C212yesmaximal, complemented
4#20261780C212yesmaximal, complemented
5#20261779C212yesmaximal, complemented
6#20261778C212yesmaximal, complemented
7#20261750C211yesmaximal, characteristic, complemented
8#20261734C211yesmaximal, characteristic, complemented
9#20261732C211yesmaximal, characteristic, complemented
10#20261701C211yesmaximal, characteristic, complemented
11#20261534C211yesmaximal, characteristic
12#15690898—21no—
13#15690898—21no—
14#15690898C2212yescomplemented
15#15690898C2211yescharacteristic, complemented
16#15690898C2212yescomplemented
17#15690898C2211yescharacteristic, complemented
18#15690897—21no—
19#15690897—21no—
20#15690897C2211yescharacteristic, complemented
21#15690897C2211yescharacteristic, complemented
22C2×#13095521—21no—
23C2×#13095521C2211yescharacteristic
24D97⋊M16—21no—
25D97⋊M16C2211yescharacteristic
26C2×#13095519—21no—
27C2×#13095519—21no—
28C2×#13095519C2211yescharacteristic
29#15690889C2211yescharacteristic
30#15690881C2212yescomplemented
31#15690880C2212yescomplemented
32#15690875C2212yescomplemented
33#15690875C2212yescomplemented
34#15690875C2212yescomplemented
35#15690874C2212yescomplemented
36#15690864C2212yescomplemented
37C8×#87604C2212yescomplemented
38C2×#13095512C2211yescharacteristic
39#41×D97C2211yescharacteristic
40#15690842C412yescomplemented
41#15690841C411yescharacteristic, complemented
42#15690838C411yescharacteristic, complemented
43#15690837C412yescomplemented
44#15690835C411yescharacteristic, complemented
45#15690833C2212yes—
46#15690831C2212yes—
47M16×D97C2211yescharacteristic
48#137×C97C411yescharacteristic, complemented
49#13095521—21no—
50#13095521—21no—
51#13095521—21no—
52#13095521—22no—
53#13095521—22no—
54#13095521C2312yes—
55#13095521C2311yescharacteristic
56#13095521C2311yescharacteristic
57#13095520—21no—
58#13095520—21no—
59#13095520—21no—
60#13095519—21no—
61#13095519—21no—
62#13095519—21no—
63#13095519—22no—
64#13095519—22no—
65#13095519D411yescharacteristic
66#13095519D411yescharacteristic
67#13095519C2311yescharacteristic
68#13095519C2311yescharacteristic
69C2×#11715897—21no—
70C2×#11715897—21no—
71C2×#11715897—21no—
72C2×#11715897D411yescharacteristic
73C2×#11715897D411yescharacteristic
74C4⋊#87604—21no—
75C22⋊#87604—21no—
76C4×#87604C2312yes—
77#13095512—21no—
78#13095512—21no—
79#13095512C2×C412yes—
80#13095512C2312yes—
81#13095512C2311yescharacteristic
82#13095511—21no—
83#13095511C2×C411yescharacteristic
84#13095510—21no—
85#13095510C2311yescharacteristic
86Q8×D97C2×C411yescharacteristic
87C2×D388—21no—
88C2×D388C2×C411yescharacteristic
89D4×D97—21no—
90D4×D97C2×C411yescharacteristic
91C2×C4×D97—21no—
92C2×C4×D97—21no—
93C2×C4×D97C2311yescharacteristic
94#13095501C2×C412yes—
95#13095500C2×C412yes—
96#13095498C2×C412yes—
97#13095498C2×C412yes—
98D776C2×C412yes—
99#13095496C2×C412yes—
100#13095495C2×C411yescharacteristic
101#13095494C2×C412yes—
102C8×D97C2312yes—
103#41×C97C2×C411yescharacteristic
104C2×Q8×C97C2×C411yescharacteristic
105Q16×C97C2×C412yes—
106SD16×C97C2×C412yes—
107M16×C97C2×C411yescharacteristic
108C22×D97—21no—
109C22×D97—21no—
110C22×D97#4111yescharacteristic
111C2×#87604—22no—
112Q8×C97—21no—
113Q8×C97C22×C412yes—
114Q8×C97C22×C411yescharacteristic
115D4×C97—21no—
116D4×C97C22×C411yescharacteristic
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97C22×C411yescharacteristic
120C22⋊D97—21no—
121C22⋊D97—21no—
122C22⋊D97—21no—
123C2×Dic97—21no—
124C2×Dic97C2×D411yescharacteristic
125D388—22no—
126D388—21no—
127D388—22no—
128D388—21no—
129D388C22×C412yes—
130D388C22×C411yescharacteristic
131C4×D97—21no—
132C4×D97—21no—
133C4×D97—22no—
134C4×D97—21no—
135C4×D97—22no—
136C4×D97C22×C411yescharacteristic
137C4×D97C2×D411yescharacteristic
138C4×D97#4111yescharacteristic
139C4×D97C2411yescharacteristic
140Dic194—21no—
141Dic194C22×C411yescharacteristic
142#11715897—21no—
143#11715897—21no—
144#11715897—22no—
145#11715897C2×D411yescharacteristic
146#11715897C2×D411yescharacteristic
147#11715897C2×D411yescharacteristic
148#11715897C2×D411yescharacteristic
149C8×C97C22×C412yes—
150#11715895C22×C412yes—
151C22×C97—21no—
152C22×C97C4×D411yescharacteristic
153C2×D97—22no—
154C2×D97—21no—
155C2×D97—41no—
156C2×D97—42no—
157C2×D97—42no—
158C2×D97C4×D411yescharacteristic
159C2×D97C2×#4111yescharacteristic
160#87604—42no—
161C4×C97—21no—
162C4×C97—22no—
163C4×C97C4×D411yescharacteristic
164C4×C97C23×C411yescharacteristic
165Dic97—21no—
166Dic97C4×D411yescharacteristic
167Dic97C22×D411yescharacteristic
168C2×C97—21no—
169C2×C97—41no—
170C2×C97C2×C4×D411yescharacteristic
171D97—21no—
172D97—21no—
173D97—42no—
174D97—41no—
175#2955—971nomaximal, Sylow
176C97#295511yescharacteristic, complemented, Sylow
177#575—971no—
178C2×#131—971no—
179C2×#131—971no—
180#444—972no—
181#444—972no—
182#443—972no—
183#443—972no—
184#430—971no—
185#421—971no—
186#421—971no—
187#405—971no—
188C2×#41—971no—
189C2×#41—971no—
190#137—971no—
191#137—972no—
192#137#8760411yescharacteristic, complemented
193AGL(1,ℤ/8)—971no—
194AGL(1,ℤ/8)—972no—
195AGL(1,ℤ/8)—971no—
196#135—972no—
197#135—972no—
198#131—971no—
199#131—971no—
200#131—971no—
201#131—972no—
202#131—971no—
203#131—972no—
204#131—1941no—
205#131—1941no—
206#131—1941no—
207#131—1941no—
208C2×M16—971no—
209C2×M16—971no—
210C2×M16—971no—
211C2×M16—1941no—
212C2×M16—1941no—
213C22×C8—971no—
214C22×C8—1941no—
215C22×C8—1941no—
216#117—971no—
217#108—972no—
218#108—972no—
219#104—972no—
220#104—972no—
221#104—972no—
222#104—972no—
223#97—972no—
224C4×C8—972no—
225#41—972no—
226#41—971no—
227#41—972no—
228#41—971no—
229#41—971no—
230#41—1941no—
231#41—1941no—
232#41—1941no—
233#41—1941no—
234#41C2×#8760411yescharacteristic
235C2×Q8—971no—
236C2×Q8C2×#8760411yescharacteristic
237C2×D4—971no—
238C2×D4—971no—
239C2×D4—1941no—
240C2×D4—1941no—
241C22×C4—971no—
242C22×C4—1941no—
243C22×C4—1941no—
244Q16—972no—
245Q16C2×#8760412yes—
246SD16—972no—
247SD16—972no—
248SD16—972no—
249SD16C2×#8760412yes—
250D8—972no—
251D8—972no—
252M16—971no—
253M16—972no—
254M16—972no—
255M16—971no—
256M16—971no—
257M16—1941no—
258M16—1941no—
259M16—1941no—
260M16—1942no—
261M16—1941no—
262M16—1942no—
263M16—1941no—
264M16—1941no—
265M16C2×#8760411yescharacteristic
266C2×C8—971no—
267C2×C8—971no—
268C2×C8—971no—
269C2×C8—971no—
270C2×C8—971no—
271C2×C8—971no—
272C2×C8—972no—
273C2×C8—1941no—
274C2×C8—1941no—
275C2×C8—1941no—
276C2×C8—1942no—
277C2×C8—1941no—
278C2×C8—1942no—
279C2×C8—1941no—
280C2×C8—1941no—
281C4⋊C4—1941no—
282C22⋊C4—1941no—
283C42—972no—
284C23—971no—
285C23—1941no—
286C23—1941no—
287Q8—21no—
288Q8—971no—
289Q8—1941no—
290Q8C22×#8760411yescharacteristic
291Q8C22×#8760412yes—
292D4—21no—
293D4—972no—
294D4—971no—
295D4—1941no—
296D4—1942no—
297D4—1942no—
298D4—1941no—
299D4—1941no—
300D4—1941no—
301D4—1941no—
302D4C22×#8760411yescharacteristic
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—971no—
306C2×C4—971no—
307C2×C4—971no—
308C2×C4—971no—
309C2×C4—971no—
310C2×C4—1941no—
311C2×C4—1942no—
312C2×C4—1942no—
313C2×C4—1941no—
314C2×C4—1941no—
315C2×C4—1942no—
316C2×C4—1941no—
317C2×C4C22×#8760411yescharacteristic
318C8—971no—
319C8—971no—
320C8—971no—
321C8—971no—
322C8—972no—
323C8—1941no—
324C8—1941no—
325C8—1942no—
326C8C22×#8760412yes—
327C22—21no—
328C22—971no—
329C22—971no—
330C22—3882no—
331C22—3881no—
332C22—3882no—
333C22—1942no—
334C22—1941no—
335C22D4×#8760411yescharacteristic
336C4—21no—
337C4—22no—
338C4—971no—
339C4—971no—
340C4—3882no—
341C4—1941no—
342C4D4×#8760411yescharacteristic
343C4C23×#8760411yescharacteristic
344C2—21no—
345C2—41no—
346C2—3882no—
347C2—3881no—
348C2—1941no—
349C2—1941no—
350C2C2×D4×#8760411yescharacteristic
subgroupG/H
centerC2C2×D4×#87604
commutatorC4×C97C23×C4
FrattiniC4C23×#87604
Fitting#137×C97C4
socleC2×C97C2×C4×D4
prime radical#20261534C2
cosocleC4×D97C24
hypercenter#137#87604
hypocenterC97#2955
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#15690898C2semidirect
2. D97⋊#131C2semidirect
3. #20261781C2semidirect
4. #20261780C2semidirect
5. #20261779C2semidirect
6. #20261778C2semidirect
7. #20261750C2semidirect
8. #20261734C2semidirect
9. #20261732C2semidirect
10. #20261701C2semidirect
14. #15690898C22semidirect
15. #15690898C22semidirect
16. #15690898C22semidirect
17. #15690898C22semidirect
20. #15690897C22semidirect
21. #15690897C22semidirect
30. #15690881C22semidirect
31. #15690880C22semidirect
32. #15690875C22semidirect
33. #15690875C22semidirect
34. #15690875C22semidirect
35. #15690874C22semidirect
36. #15690864C22semidirect
37. C8×#87604C22semidirect
40. #15690842C4semidirect
41. #15690841C4semidirect
42. #15690838C4semidirect
43. #15690837C4semidirect
44. #15690835C4semidirect
48. #137×C97C4semidirect
176. C97#2955semidirect
192. #137#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908982
D97⋊#1312
#202617812
#202617802
#202617792
#202617782
#202617502
#202617342
#202617322
#202617012
#202615342
#295597