Finite Group Databasebeta

← #27158511

Subgroups — #27158511

Order 12416.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261815C211yesmaximal, characteristic, complemented
2#20261814C211yesmaximal, characteristic, complemented
3#20261781C212yesmaximal, complemented
4#20261780C212yesmaximal, complemented
5#20261779C212yesmaximal, complemented
6#20261778C212yesmaximal, complemented
7#20261750C211yesmaximal, characteristic, complemented
8C2×#15690875C211yesmaximal, characteristic, complemented
9D97⋊#104C211yesmaximal, characteristic, complemented
10M16×#87604C211yesmaximal, characteristic, complemented
11#20261534C211yesmaximal, characteristic
12#15690898—21no—
13#15690898—21no—
14#15690898C2212yescomplemented
15#15690898C2212yescomplemented
16#15690898C2212yescomplemented
17#15690897—22no—
18#15690897C2212yescomplemented
19C2×#13095521—21no—
20C2×#13095521C2211yescharacteristic
21D97⋊M16—21no—
22D97⋊M16—22no—
23D97⋊M16C2211yescharacteristic
24D97⋊M16C2211yescharacteristic
25C2×C4×#87604C2211yescharacteristic
26#15690881C2212yescomplemented
27#15690880C2212yescomplemented
28#15690875C2211yescharacteristic, complemented
29#15690875C2211yescharacteristic, complemented
30#15690875C2212yescomplemented
31#15690875C2212yescomplemented
32#15690874C2211yescharacteristic, complemented
33#15690874C2211yescharacteristic, complemented
34#15690864C2212yescomplemented
35C8×#87604C2212yescomplemented
36C2×#13095512C2211yescharacteristic
37#41×D97C2211yescharacteristic
38#15690842C412yescomplemented
39#15690841C411yescharacteristic, complemented
40#15690838C411yescharacteristic, complemented
41#15690837C412yescomplemented
42#15690835C411yescharacteristic, complemented
43#15690833C2212yes—
44#15690831C2212yes—
45M16×D97C2211yescharacteristic
46#137×C97C411yescharacteristic, complemented
47C22×#87604—21no—
48#13095521—21no—
49#13095521—22no—
50#13095521—22no—
51#13095521—22no—
52#13095521D411yescharacteristic
53#13095521D411yescharacteristic
54#13095521C2311yescharacteristic
55#13095521C2311yescharacteristic
56#13095521C2312yes—
57#13095520—21no—
58#13095520—22no—
59#13095520D412yes—
60#13095519—21no—
61#13095519—22no—
62#13095519—22no—
63#13095519—22no—
64#13095519C2312yes—
65C2×#11715897—21no—
66C2×#11715897—22no—
67C4×#87604—21no—
68C4×#87604C2311yescharacteristic
69C4×#87604C2311yescharacteristic
70#13095512—21no—
71#13095512—21no—
72#13095512C2×C412yes—
73#13095512C2312yes—
74#13095512C2311yescharacteristic
75#13095511—21no—
76#13095511C2×C411yescharacteristic
77#13095510—21no—
78#13095510C2311yescharacteristic
79Q8×D97C2×C411yescharacteristic
80C2×D388—21no—
81C2×D388C2×C411yescharacteristic
82D4×D97—21no—
83D4×D97C2×C411yescharacteristic
84C2×C4×D97—21no—
85C2×C4×D97—21no—
86C2×C4×D97C2311yescharacteristic
87#13095501C2×C412yescomplemented
88#13095500C2×C412yescomplemented
89#13095498C2×C412yescomplemented
90#13095498C2×C412yescomplemented
91D776C2×C412yescomplemented
92#13095496C2×C412yescomplemented
93#13095495C2×C411yescharacteristic
94#13095494C2×C412yes—
95C8×D97C2312yes—
96#41×C97C2×C411yescharacteristic
97C2×Q8×C97C2×C411yescharacteristic
98Q16×C97C2×C412yescomplemented
99SD16×C97C2×C412yescomplemented
100M16×C97C2×C411yescharacteristic
101C22×D97—21no—
102C22×D97—21no—
103C22×D97#4111yescharacteristic
104C2×#87604—21no—
105C2×#87604—21no—
106C2×#87604—42no—
107Q8×C97—21no—
108Q8×C97C22×C412yes—
109Q8×C97C22×C411yescharacteristic
110D4×C97—21no—
111D4×C97C22×C411yescharacteristic
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97C22×C411yescharacteristic
115C22⋊D97—21no—
116C22⋊D97—21no—
117C22⋊D97—21no—
118C2×Dic97—21no—
119C2×Dic97C2×D411yescharacteristic
120D388—22no—
121D388—21no—
122D388—22no—
123D388—21no—
124D388C22×C412yes—
125D388C22×C411yescharacteristic
126C4×D97—21no—
127C4×D97—21no—
128C4×D97—22no—
129C4×D97—21no—
130C4×D97—22no—
131C4×D97C22×C411yescharacteristic
132C4×D97C2×D411yescharacteristic
133C4×D97#4111yescharacteristic
134C4×D97C2411yescharacteristic
135Dic194—21no—
136Dic194C22×C411yescharacteristic
137#11715897—22no—
138#11715897—21no—
139#11715897—21no—
140#11715897C2×D412yes—
141#11715897C2×D412yes—
142C8×C97C22×C412yes—
143#11715895C22×C412yes—
144C22×C97—21no—
145C22×C97C4×D411yescharacteristic
146C2×D97—22no—
147C2×D97—21no—
148C2×D97—41no—
149C2×D97—81no—
150C2×D97—81no—
151C2×D97C4×D411yescharacteristic
152C2×D97C2×#4111yescharacteristic
153#87604—22no—
154#87604—41no—
155C4×C97—21no—
156C4×C97—22no—
157C4×C97C4×D411yescharacteristic
158C4×C97C23×C411yescharacteristic
159Dic97—21no—
160Dic97C4×D411yescharacteristic
161Dic97C22×D411yescharacteristic
162C2×C97—21no—
163C2×C97—41no—
164C2×C97C2×C4×D411yescharacteristic
165D97—21no—
166D97—21no—
167D97—42no—
168D97—41no—
169#2956—971nomaximal, Sylow
170C97#295611yescharacteristic, complemented, Sylow
171#575—971no—
172#568—971no—
173#568—971no—
174#444—972no—
175#444—972no—
176#443—972no—
177#443—972no—
178#430—971no—
179C2×#104—971no—
180C2×#104—971no—
181C4×M16—971no—
182C2×#41—971no—
183C2×#41—971no—
184#137—971no—
185#137—972no—
186#137#8760411yescharacteristic, complemented
187AGL(1,ℤ/8)—971no—
188AGL(1,ℤ/8)—972no—
189AGL(1,ℤ/8)—971no—
190#135—972no—
191#135—972no—
192#131—972no—
193#131—972no—
194#131—972no—
195#131—972no—
196#131—1942no—
197#131—1941no—
198#131—1941no—
199C2×M16—971no—
200C2×M16—971no—
201C2×M16—971no—
202C2×M16—971no—
203C2×M16—1942no—
204C2×M16—1941no—
205C2×M16—1941no—
206C2×C42—971no—
207#108—972no—
208#108—972no—
209#104—971no—
210#104—971no—
211#104—971no—
212#104—972no—
213#104—971no—
214#104—972no—
215#97—972no—
216C4×C8—972no—
217#41—972no—
218#41—971no—
219#41—972no—
220#41—971no—
221#41—971no—
222#41—1941no—
223#41—1941no—
224#41—1941no—
225#41—1941no—
226#41C2×#8760411yescharacteristic
227C2×Q8—971no—
228C2×Q8C2×#8760411yescharacteristic
229C2×D4—971no—
230C2×D4—971no—
231C2×D4—1941no—
232C2×D4—1941no—
233C22×C4—971no—
234C22×C4—1941no—
235C22×C4—1941no—
236C22×C4—1941no—
237Q16—972no—
238Q16C2×#8760412yescomplemented
239SD16—972no—
240SD16—972no—
241SD16—972no—
242SD16C2×#8760412yescomplemented
243D8—972no—
244D8—972no—
245M16—972no—
246M16—971no—
247M16—971no—
248M16—971no—
249M16—972no—
250M16—971no—
251M16—971no—
252M16—972no—
253M16—1942no—
254M16—1942no—
255M16—1942no—
256M16—1942no—
257M16—1941no—
258M16—1941no—
259M16C2×#8760411yescharacteristic
260C2×C8—972no—
261C2×C8—972no—
262C2×C8—1942no—
263C2×C8—1942no—
264C2×C8—1942no—
265C2×C8—1942no—
266C2×C8—1941no—
267C2×C8—1941no—
268C42—971no—
269C42—971no—
270C42—1941no—
271C23—971no—
272C23—1941no—
273C23—1941no—
274Q8—21no—
275Q8—971no—
276Q8—1941no—
277Q8C22×#8760411yescharacteristic
278Q8C22×#8760412yes—
279D4—21no—
280D4—972no—
281D4—971no—
282D4—1941no—
283D4—1942no—
284D4—1942no—
285D4—1941no—
286D4—1941no—
287D4—1941no—
288D4—1941no—
289D4C22×#8760411yescharacteristic
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—971no—
293C2×C4—971no—
294C2×C4—971no—
295C2×C4—971no—
296C2×C4—971no—
297C2×C4—3882no—
298C2×C4—1941no—
299C2×C4—1942no—
300C2×C4—1942no—
301C2×C4—1941no—
302C2×C4—1941no—
303C2×C4—1941no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4C22×#8760411yescharacteristic
307C8—972no—
308C8—972no—
309C8—972no—
310C8—1942no—
311C8—1941no—
312C8—1941no—
313C8C22×#8760412yes—
314C22—21no—
315C22—7761no—
316C22—7761no—
317C22—971no—
318C22—971no—
319C22—3881no—
320C22—1942no—
321C22—1941no—
322C22D4×#8760411yescharacteristic
323C4—21no—
324C4—22no—
325C4—971no—
326C4—971no—
327C4—3881no—
328C4—1942no—
329C4—1941no—
330C4D4×#8760411yescharacteristic
331C4C23×#8760411yescharacteristic
332C2—21no—
333C2—41no—
334C2—3882no—
335C2—3881no—
336C2—1941no—
337C2—1941no—
338C2C2×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#2956
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. #20261815C2semidirect
2. #20261814C2semidirect
3. #20261781C2semidirect
4. #20261780C2semidirect
5. #20261779C2semidirect
6. #20261778C2semidirect
7. #20261750C2semidirect
8. C2×#15690875C2semidirect
9. D97⋊#104C2semidirect
10. M16×#87604C2semidirect
14. #15690898C22semidirect
15. #15690898C22semidirect
16. #15690898C22semidirect
18. #15690897C22semidirect
26. #15690881C22semidirect
27. #15690880C22semidirect
28. #15690875C22semidirect
29. #15690875C22semidirect
30. #15690875C22semidirect
31. #15690875C22semidirect
32. #15690874C22semidirect
33. #15690874C22semidirect
34. #15690864C22semidirect
35. C8×#87604C22semidirect
38. #15690842C4semidirect
39. #15690841C4semidirect
40. #15690838C4semidirect
41. #15690837C4semidirect
42. #15690835C4semidirect
46. #137×C97C4semidirect
87. #13095501C2×C4semidirect
88. #13095500C2×C4semidirect
89. #13095498C2×C4semidirect
90. #13095498C2×C4semidirect
91. D776C2×C4semidirect
92. #13095496C2×C4semidirect
98. Q16×C97C2×C4semidirect
99. SD16×C97C2×C4semidirect
170. C97#2956semidirect
186. #137#87604semidirect
238. Q16C2×#87604semidirect
242. SD16C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202618152
#202618142
#202617812
#202617802
#202617792
#202617782
#202617502
C2×#156908752
D97⋊#1042
M16×#876042
#202615342
#295697