Finite Group Databasebeta

← C2×#20266216 #27187883

Subgroups — C2×#20266216 #27187883

Order 12928.

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×#87938C211yesmaximal, characteristic, complemented
2C2×D4×#87938C211yesmaximal, characteristic
3#20266216C218yesmaximal, direct product factor
4C2×#15692324C211yesmaximal, characteristic, complemented
5C2×#15692318C211yesmaximal, characteristic, complemented
6C2×#15692317C211yesmaximal, characteristic
7C2×#15692309C211yesmaximal, characteristic, complemented
8C2×SD16×D101C211yesmaximal, characteristic
9C23×#87938—21no—
10C2×D101⋊C8C2211yescharacteristic
11Q8×#87938—22no—
12Q8×#87938C2214yescomplemented
13D4×#87938—22no—
14D4×#87938C2214yes—
15C2×C4⋊#87938—21no—
16C2×C4⋊#87938C2211yescharacteristic
17C2×C4⋊#87938C2211yescharacteristic
18C2×C22⋊#87938—21no—
19C2×C4×#87938—21no—
20C2×C4×#87938C2211yescharacteristic
21#15692324C2214yescomplemented
22#15692318C2214yescomplemented
23#15692317C2214yes—
24#15692309C2214yescomplemented
25C2×Q8×D101C2211yescharacteristic
26C2×D4×D101C2211yescharacteristic
27C2×#13108063C411yescharacteristic, complemented
28C2×#13108062C411yescharacteristic, complemented
29C2×#13108061C411yescharacteristic, complemented
30SD16×D101C414yescomplemented
31SD16×D101C2214yes—
32C2×C8×D101C2211yescharacteristic
33C2×SD16×C101C411yescharacteristic, complemented
34C22×#87938—21no—
35C22×#87938—21no—
36C22×#87938—24no—
37C22×#87938—44no—
38C22×#87938D411yescharacteristic
39C22×#87938D411yescharacteristic
40D101⋊C8—21no—
41D101⋊C8C2312yes—
42C2×#11717581—21no—
43C4⋊#87938—24no—
44C4⋊#87938—21no—
45C4⋊#87938—21no—
46C4⋊#87938C2312yes—
47C4⋊#87938C2312yes—
48C22⋊#87938—24no—
49C4×#87938—24no—
50C4×#87938D412yes—
51C4×#87938C2312yes—
52C23×D101—21no—
53C2×Dic202—21no—
54C2×Dic202C2×C411yescharacteristic, complemented
55Q8×D101—21no—
56Q8×D101—21no—
57Q8×D101C2×C412yescomplemented
58Q8×D101C2312yes—
59C2×D404C2×C411yescharacteristic
60C2×C22⋊D101—21no—
61D4×D101—21no—
62D4×D101—21no—
63D4×D101C2×C412yes—
64D4×D101C2312yes—
65C2×C4×D101—21no—
66C2×C4×D101C2311yescharacteristic
67#13108063C2×C414yescomplemented
68#13108062C2×C414yescomplemented
69#13108061C2×C414yescomplemented
70C2×#11717579C2×C411yescharacteristic, complemented
71C8×D101C2×C412yescomplemented
72C8×D101C2312yes—
73C2×Q8×C101C2×C411yescharacteristic, complemented
74C2×D4×C101C2×C411yescharacteristic
75SD16×C101C2×C414yescomplemented
76C2×C8×C101C2×C411yescharacteristic, complemented
77C23×C101—21no—
78C22×D101—21no—
79C22×D101—22no—
80C22×D101—22no—
81C22×D101—42no—
82C22×D101—42no—
83C22×D101C2×D411yescharacteristic
84C2×#87938—22no—
85C2×#87938—22no—
86C2×#87938—22no—
87C2×#87938—22no—
88C2×#87938—48no—
89C2×#87938—42no—
90C2×#87938—42no—
91C2×#87938C2×D412yes—
92C2×#87938C2×D412yes—
93Q8×C101—21no—
94Q8×C101C22×C412yescomplemented
95D4×C101—21no—
96D4×C101C22×C412yes—
97C2×C4×C101—21no—
98C2×C4×C101C22×C411yescharacteristic
99C22⋊D101—22no—
100C22⋊D101—22no—
101C2×Dic101—21no—
102C2×Dic101#4111yescharacteristic
103D404—21no—
104D404C22×C412yes—
105C4×D101—22no—
106C4×D101—22no—
107C4×D101C22×C411yescharacteristic
108C4×D101C2×D411yescharacteristic
109C4×D101#4111yescharacteristic
110C4×D101C2411yescharacteristic
111Dic202—21no—
112Dic202—22no—
113Dic202—22no—
114Dic202C22×C412yescomplemented
115#11717581—22no—
116C8×C101C22×C412yescomplemented
117#11717579C22×C412yescomplemented
118C22×C101—22no—
119C22×C101—42no—
120C22×C101C4×D411yescharacteristic
121C2×D101—22no—
122C2×D101—42no—
123C2×D101—42no—
124C2×D101—42no—
125C2×D101—44no—
126C2×D101C4×D411yescharacteristic
127C2×D101AGL(1,ℤ/8)12yes—
128C2×D101#13712yes—
129C2×D101C22×D411yescharacteristic
130#87938—22no—
131#87938—22no—
132#87938—42no—
133#87938—42no—
134C4×C101—22no—
135C4×C101C4×D411yescharacteristic
136C4×C101C23×C411yescharacteristic
137Dic101—22no—
138Dic101C4×D411yescharacteristic
139Dic101C2×#4111yescharacteristic
140C2×C101—42no—
141C2×C101#44012yescomplemented
142C2×C101C2×C4×D411yescharacteristic
143D101—42no—
144D101#44012yescomplemented
145D101C2×AGL(1,ℤ/8)11yescharacteristic
146D101C2×#13711yescharacteristic
147C2×#440—1011nomaximal, Sylow
148C101C2×#44011yescharacteristic, complemented, Sylow
149C22×SD16—1011no—
150C2×C4×Q8—1011no—
151C2×C4×D4—1011no—
152#440—1018no—
153C2×#107—1011no—
154C2×#103—1011no—
155C2×#102—1011no—
156C2×#97—1011no—
157C22×Q8—1011no—
158C22×D4—1011no—
159C23×C4—2021no—
160C2×SD16—1011no—
161C2×SD16—1011no—
162C2×SD16—1014no—
163C2×SD16—1014no—
164C2×SD16—1011no—
165C2×SD16#8793811yescharacteristic, complemented
166C22×C8—1011no—
167C22×C8—1011no—
168C4×Q8—1014no—
169C4×Q8—2022no—
170C4×D4—1014no—
171C4×D4—2022no—
172C2×C4⋊C4—1011no—
173C2×C4⋊C4—1011no—
174C2×C4⋊C4—2021no—
175C2×C22⋊C4—2021no—
176C2×C42—1011no—
177C2×C42—2021no—
178#107—1014no—
179#103—1014no—
180#102—1014no—
181#97—1014no—
182C24—2021no—
183C2×Q8—1012no—
184C2×Q8—1012no—
185C2×Q8—1011no—
186C2×Q8—2021no—
187C2×Q8—2021no—
188C2×Q8—2021no—
189C2×Q8C2×#8793811yescharacteristic, complemented
190C2×D4—1012no—
191C2×D4—1012no—
192C2×D4—1011no—
193C2×D4—2021no—
194C2×D4—2021no—
195C2×D4—2021no—
196C2×D4C2×#8793811yescharacteristic
197C22×C4—1011no—
198C22×C4—1011no—
199C22×C4—1011no—
200C22×C4—4044no—
201C22×C4—2021no—
202C22×C4—2024no—
203C22×C4—2021no—
204C22×C4—2021no—
205SD16—1014no—
206SD16—1014no—
207SD16—1014no—
208SD16C2×#8793814yescomplemented
209C2×C8—1011no—
210C2×C8—1012no—
211C2×C8—1012no—
212C2×C8—1012no—
213C2×C8—2021no—
214C2×C8—2021no—
215C2×C8C2×#8793811yescharacteristic, complemented
216C4⋊C4—1012no—
217C4⋊C4—1012no—
218C4⋊C4—2021no—
219C4⋊C4—2021no—
220C4⋊C4—2024no—
221C22⋊C4—2024no—
222C42—1012no—
223C42—1012no—
224C42—2024no—
225C23—21no—
226C23—1011no—
227C23—4042no—
228C23—4042no—
229C23—2022no—
230C23—2022no—
231C23—2021no—
232Q8—21no—
233Q8—1012no—
234Q8—2022no—
235Q8—2022no—
236Q8—2021no—
237Q8C22×#8793812yescomplemented
238D4—21no—
239D4—1012no—
240D4—2022no—
241D4—2022no—
242D4—2021no—
243D4C22×#8793812yes—
244C2×C4—21no—
245C2×C4—1011no—
246C2×C4—1011no—
247C2×C4—1011no—
248C2×C4—1011no—
249C2×C4—1011no—
250C2×C4—1012no—
251C2×C4—1012no—
252C2×C4—4042no—
253C2×C4—4042no—
254C2×C4—4048no—
255C2×C4—2022no—
256C2×C4—2022no—
257C2×C4—2022no—
258C2×C4—2022no—
259C2×C4—2022no—
260C2×C4—2022no—
261C2×C4—2021no—
262C2×C4C22×#8793811yescharacteristic
263C8—1012no—
264C8—2022no—
265C8C22×#8793812yescomplemented
266C22—22no—
267C22—42no—
268C22—1011no—
269C22—1011no—
270C22—1012no—
271C22—1012no—
272C22—4042no—
273C22—4042no—
274C22—4044no—
275C22—4042no—
276C22—2022no—
277C22D4×#8793811yescharacteristic
278C4—22no—
279C4—1011no—
280C4—1011no—
281C4—4042no—
282C4—4042no—
283C4—2022no—
284C4—2022no—
285C4—2022no—
286C4D4×#8793811yescharacteristic
287C4C23×#8793811yescharacteristic
288C2—42no—
289C2—1011no—
290C2—1011no—
291C2—1012no—
292C2—4042no—
293C2#2026621612yesdirect product factor
294C2C2×D4×#8793811yescharacteristic
subgroupG/H
centerC22D4×#87938
commutatorC4×C101C23×C4
FrattiniC4C23×#87938
FittingC2×SD16×C101C4
socleC22×C101C4×D4
prime radicalC2×D4×D101C22
cosocleC4×D101C24
hypercenterC2×SD16#87938
hypocenterC101C2×#440
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×Q8×#87938C2semidirect
3. #20266216C2direct (N × H)
4. C2×#15692324C2semidirect
5. C2×#15692318C2semidirect
7. C2×#15692309C2semidirect
12. Q8×#87938C22semidirect
21. #15692324C22semidirect
22. #15692318C22semidirect
24. #15692309C22semidirect
27. C2×#13108063C4semidirect
28. C2×#13108062C4semidirect
29. C2×#13108061C4semidirect
30. SD16×D101C4semidirect
33. C2×SD16×C101C4semidirect
54. C2×Dic202C2×C4semidirect
57. Q8×D101C2×C4semidirect
67. #13108063C2×C4semidirect
68. #13108062C2×C4semidirect
69. #13108061C2×C4semidirect
70. C2×#11717579C2×C4semidirect
71. C8×D101C2×C4semidirect
73. C2×Q8×C101C2×C4semidirect
75. SD16×C101C2×C4semidirect
76. C2×C8×C101C2×C4semidirect
94. Q8×C101C22×C4semidirect
114. Dic202C22×C4semidirect
116. C8×C101C22×C4semidirect
117. #11717579C22×C4semidirect
141. C2×C101#440semidirect
144. D101#440semidirect
148. C101C2×#440semidirect
165. C2×SD16#87938semidirect
189. C2×Q8C2×#87938semidirect
208. SD16C2×#87938semidirect
215. C2×C8C2×#87938semidirect
237. Q8C22×#87938semidirect
265. C8C22×#87938semidirect
293. C2#20266216direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×#879382
C2×D4×#879382
#202662162
C2×#156923242
C2×#156923182
C2×#156923172
C2×#156923092
C2×SD16×D1012
C2×#440101