Finite Group Databasebeta

← C2×#20261774 #27158487

Subgroups — C2×#20261774 #27158487

Order 12416.

544 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×#87604C211yesmaximal, characteristic, complemented
2C2×D4×#87604C211yesmaximal, characteristic
3#20261774C218yesmaximal, direct product factor
4C2×#15690879C211yesmaximal, characteristic, complemented
5C2×#15690873C211yesmaximal, characteristic, complemented
6C2×#15690872C211yesmaximal, characteristic
7C2×#15690864C211yesmaximal, characteristic, complemented
8C2×SD16×D97C211yesmaximal, characteristic
9C23×#87604—21no—
10C2×#13095519C2211yescharacteristic
11Q8×#87604—22no—
12Q8×#87604C2214yescomplemented
13D4×#87604—22no—
14D4×#87604C2214yes—
15C2×C4⋊#87604—21no—
16C2×C4⋊#87604C2211yescharacteristic
17C2×C4⋊#87604C2211yescharacteristic
18C2×C22⋊#87604—21no—
19C2×C4×#87604—21no—
20C2×C4×#87604C2211yescharacteristic
21#15690879C2214yescomplemented
22#15690873C2214yescomplemented
23#15690872C2214yes—
24#15690864C2214yescomplemented
25C2×Q8×D97C2211yescharacteristic
26C2×D4×D97C2211yescharacteristic
27C2×#13095500C411yescharacteristic, complemented
28C2×#13095499C411yescharacteristic, complemented
29C2×#13095498C411yescharacteristic, complemented
30SD16×D97C414yescomplemented
31SD16×D97C2214yes—
32C2×C8×D97C2211yescharacteristic
33C2×SD16×C97C411yescharacteristic, complemented
34C22×#87604—21no—
35C22×#87604—21no—
36C22×#87604—24no—
37C22×#87604—44no—
38C22×#87604D411yescharacteristic
39C22×#87604D411yescharacteristic
40#13095519—21no—
41#13095519C2312yes—
42C2×#11715897—21no—
43C4⋊#87604—24no—
44C4⋊#87604—21no—
45C4⋊#87604—21no—
46C4⋊#87604C2312yes—
47C4⋊#87604C2312yes—
48C22⋊#87604—24no—
49C4×#87604—24no—
50C4×#87604D412yes—
51C4×#87604C2312yes—
52C23×D97—21no—
53C2×Dic194—21no—
54C2×Dic194C2×C411yescharacteristic, complemented
55Q8×D97—21no—
56Q8×D97—21no—
57Q8×D97C2×C412yescomplemented
58Q8×D97C2312yes—
59C2×D388C2×C411yescharacteristic
60C2×C22⋊D97—21no—
61D4×D97—21no—
62D4×D97—21no—
63D4×D97C2×C412yes—
64D4×D97C2312yes—
65C2×C4×D97—21no—
66C2×C4×D97C2311yescharacteristic
67#13095500C2×C414yescomplemented
68#13095499C2×C414yescomplemented
69#13095498C2×C414yescomplemented
70C2×#11715895C2×C411yescharacteristic, complemented
71C8×D97C2×C412yescomplemented
72C8×D97C2312yes—
73C2×Q8×C97C2×C411yescharacteristic, complemented
74C2×D4×C97C2×C411yescharacteristic
75SD16×C97C2×C414yescomplemented
76C2×C8×C97C2×C411yescharacteristic, complemented
77C23×C97—21no—
78C22×D97—21no—
79C22×D97—22no—
80C22×D97—22no—
81C22×D97—42no—
82C22×D97—42no—
83C22×D97C2×D411yescharacteristic
84C2×#87604—22no—
85C2×#87604—22no—
86C2×#87604—22no—
87C2×#87604—22no—
88C2×#87604—48no—
89C2×#87604—42no—
90C2×#87604—42no—
91C2×#87604C2×D412yes—
92C2×#87604C2×D412yes—
93Q8×C97—21no—
94Q8×C97C22×C412yescomplemented
95D4×C97—21no—
96D4×C97C22×C412yes—
97C2×C4×C97—21no—
98C2×C4×C97C22×C411yescharacteristic
99C22⋊D97—22no—
100C22⋊D97—22no—
101C2×Dic97—21no—
102C2×Dic97#4111yescharacteristic
103D388—21no—
104D388C22×C412yes—
105C4×D97—22no—
106C4×D97—22no—
107C4×D97C22×C411yescharacteristic
108C4×D97C2×D411yescharacteristic
109C4×D97#4111yescharacteristic
110C4×D97C2411yescharacteristic
111Dic194—21no—
112Dic194—22no—
113Dic194—22no—
114Dic194C22×C412yescomplemented
115#11715897—22no—
116C8×C97C22×C412yescomplemented
117#11715895C22×C412yescomplemented
118C22×C97—22no—
119C22×C97—42no—
120C22×C97C4×D411yescharacteristic
121C2×D97—22no—
122C2×D97—42no—
123C2×D97—42no—
124C2×D97—42no—
125C2×D97—44no—
126C2×D97C4×D411yescharacteristic
127C2×D97AGL(1,ℤ/8)12yes—
128C2×D97#13712yes—
129C2×D97C22×D411yescharacteristic
130#87604—22no—
131#87604—22no—
132#87604—42no—
133#87604—42no—
134C4×C97—22no—
135C4×C97C4×D411yescharacteristic
136C4×C97C23×C411yescharacteristic
137Dic97—22no—
138Dic97C4×D411yescharacteristic
139Dic97C2×#4111yescharacteristic
140C2×C97—42no—
141C2×C97#44012yescomplemented
142C2×C97C2×C4×D411yescharacteristic
143D97—42no—
144D97#44012yescomplemented
145D97C2×AGL(1,ℤ/8)11yescharacteristic
146D97C2×#13711yescharacteristic
147C2×#440—971nomaximal, Sylow
148C97C2×#44011yescharacteristic, complemented, Sylow
149C22×SD16—971no—
150C2×C4×Q8—971no—
151C2×C4×D4—971no—
152#440—978no—
153C2×#107—971no—
154C2×#103—971no—
155C2×#102—971no—
156C2×#97—971no—
157C22×Q8—971no—
158C22×D4—971no—
159C23×C4—1941no—
160C2×SD16—971no—
161C2×SD16—971no—
162C2×SD16—974no—
163C2×SD16—974no—
164C2×SD16—971no—
165C2×SD16#8760411yescharacteristic, complemented
166C22×C8—971no—
167C22×C8—971no—
168C4×Q8—974no—
169C4×Q8—1942no—
170C4×D4—974no—
171C4×D4—1942no—
172C2×C4⋊C4—971no—
173C2×C4⋊C4—971no—
174C2×C4⋊C4—1941no—
175C2×C22⋊C4—1941no—
176C2×C42—971no—
177C2×C42—1941no—
178#107—974no—
179#103—974no—
180#102—974no—
181#97—974no—
182C24—1941no—
183C2×Q8—972no—
184C2×Q8—972no—
185C2×Q8—971no—
186C2×Q8—1941no—
187C2×Q8—1941no—
188C2×Q8—1941no—
189C2×Q8C2×#8760411yescharacteristic, complemented
190C2×D4—972no—
191C2×D4—972no—
192C2×D4—971no—
193C2×D4—1941no—
194C2×D4—1941no—
195C2×D4—1941no—
196C2×D4C2×#8760411yescharacteristic
197C22×C4—971no—
198C22×C4—971no—
199C22×C4—971no—
200C22×C4—3884no—
201C22×C4—1941no—
202C22×C4—1944no—
203C22×C4—1941no—
204C22×C4—1941no—
205SD16—974no—
206SD16—974no—
207SD16—974no—
208SD16C2×#8760414yescomplemented
209C2×C8—971no—
210C2×C8—972no—
211C2×C8—972no—
212C2×C8—972no—
213C2×C8—1941no—
214C2×C8—1941no—
215C2×C8C2×#8760411yescharacteristic, complemented
216C4⋊C4—972no—
217C4⋊C4—972no—
218C4⋊C4—1941no—
219C4⋊C4—1941no—
220C4⋊C4—1944no—
221C22⋊C4—1944no—
222C42—972no—
223C42—972no—
224C42—1944no—
225C23—21no—
226C23—971no—
227C23—3882no—
228C23—3882no—
229C23—1942no—
230C23—1942no—
231C23—1941no—
232Q8—21no—
233Q8—972no—
234Q8—1942no—
235Q8—1942no—
236Q8—1941no—
237Q8C22×#8760412yescomplemented
238D4—21no—
239D4—972no—
240D4—1942no—
241D4—1942no—
242D4—1941no—
243D4C22×#8760412yes—
244C2×C4—21no—
245C2×C4—971no—
246C2×C4—971no—
247C2×C4—971no—
248C2×C4—971no—
249C2×C4—971no—
250C2×C4—972no—
251C2×C4—972no—
252C2×C4—3882no—
253C2×C4—3882no—
254C2×C4—3888no—
255C2×C4—1942no—
256C2×C4—1942no—
257C2×C4—1942no—
258C2×C4—1942no—
259C2×C4—1942no—
260C2×C4—1942no—
261C2×C4—1941no—
262C2×C4C22×#8760411yescharacteristic
263C8—972no—
264C8—1942no—
265C8C22×#8760412yescomplemented
266C22—22no—
267C22—42no—
268C22—972no—
269C22—972no—
270C22—971no—
271C22—971no—
272C22—3882no—
273C22—3882no—
274C22—3884no—
275C22—3882no—
276C22—1942no—
277C22D4×#8760411yescharacteristic
278C4—22no—
279C4—971no—
280C4—971no—
281C4—3882no—
282C4—3882no—
283C4—1942no—
284C4—1942no—
285C4—1942no—
286C4D4×#8760411yescharacteristic
287C4C23×#8760411yescharacteristic
288C2—42no—
289C2—971no—
290C2—971no—
291C2—972no—
292C2—3882no—
293C2#2026177412yesdirect product factor
294C2C2×D4×#8760411yescharacteristic
subgroupG/H
centerC22D4×#87604
commutatorC4×C97C23×C4
FrattiniC4C23×#87604
FittingC2×SD16×C97C4
socleC22×C97C4×D4
prime radicalC2×D4×D97C22
cosocleC4×D97C24
hypercenterC2×SD16#87604
hypocenterC97C2×#440
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×Q8×#87604C2semidirect
3. #20261774C2direct (N × H)
4. C2×#15690879C2semidirect
5. C2×#15690873C2semidirect
7. C2×#15690864C2semidirect
12. Q8×#87604C22semidirect
21. #15690879C22semidirect
22. #15690873C22semidirect
24. #15690864C22semidirect
27. C2×#13095500C4semidirect
28. C2×#13095499C4semidirect
29. C2×#13095498C4semidirect
30. SD16×D97C4semidirect
33. C2×SD16×C97C4semidirect
54. C2×Dic194C2×C4semidirect
57. Q8×D97C2×C4semidirect
67. #13095500C2×C4semidirect
68. #13095499C2×C4semidirect
69. #13095498C2×C4semidirect
70. C2×#11715895C2×C4semidirect
71. C8×D97C2×C4semidirect
73. C2×Q8×C97C2×C4semidirect
75. SD16×C97C2×C4semidirect
76. C2×C8×C97C2×C4semidirect
94. Q8×C97C22×C4semidirect
114. Dic194C22×C4semidirect
116. C8×C97C22×C4semidirect
117. #11715895C22×C4semidirect
141. C2×C97#440semidirect
144. D97#440semidirect
148. C97C2×#440semidirect
165. C2×SD16#87604semidirect
189. C2×Q8C2×#87604semidirect
208. SD16C2×#87604semidirect
215. C2×C8C2×#87604semidirect
237. Q8C22×#87604semidirect
265. C8C22×#87604semidirect
293. C2#20261774direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×#876042
C2×D4×#876042
#202617742
C2×#156908792
C2×#156908732
C2×#156908722
C2×#156908642
C2×SD16×D972
C2×#44097