Finite Group Databasebeta

← C2×#26012530 #36370314

Subgroups — C2×#26012530 #36370314

Order 22144.

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×#10621924C211yesmaximal, characteristic, complemented
2C2×D4×#10621924C211yesmaximal, characteristic
3#26012530C218yesmaximal, direct product factor
4C2×#19149748C211yesmaximal, characteristic, complemented
5C2×#19149742C211yesmaximal, characteristic, complemented
6C2×#19149741C211yesmaximal, characteristic
7C2×#19149733C211yesmaximal, characteristic, complemented
8C2×SD16×D173C211yesmaximal, characteristic
9C23×#10621924—21no—
10C2×D173⋊C8C2211yescharacteristic
11Q8×#10621924—22no—
12Q8×#10621924C2214yescomplemented
13D4×#10621924—22no—
14D4×#10621924C2214yes—
15C2×C4⋊#10621924—21no—
16C2×C4⋊#10621924C2211yescharacteristic
17C2×C4⋊#10621924C2211yescharacteristic
18C2×C22⋊#10621924—21no—
19C2×C4×#10621924—21no—
20C2×C4×#10621924C2211yescharacteristic
21#19149748C2214yescomplemented
22#19149742C2214yescomplemented
23#19149741C2214yes—
24#19149733C2214yescomplemented
25C2×Q8×D173C2211yescharacteristic
26C2×D4×D173C2211yescharacteristic
27C2×#14583815C411yescharacteristic, complemented
28C2×#14583814C411yescharacteristic, complemented
29C2×#14583813C411yescharacteristic, complemented
30SD16×D173C414yescomplemented
31SD16×D173C2214yes—
32C2×C8×D173C2211yescharacteristic
33C2×SD16×C173C411yescharacteristic, complemented
34C22×#10621924—24no—
35C22×#10621924—21no—
36C22×#10621924—21no—
37C22×#10621924—44no—
38C22×#10621924D411yescharacteristic
39C22×#10621924D411yescharacteristic
40D173⋊C8—21no—
41D173⋊C8C2312yes—
42C2×#13063443—21no—
43C4⋊#10621924—24no—
44C4⋊#10621924—21no—
45C4⋊#10621924—21no—
46C4⋊#10621924C2312yes—
47C4⋊#10621924C2312yes—
48C22⋊#10621924—24no—
49C4×#10621924—24no—
50C4×#10621924D412yes—
51C4×#10621924C2312yes—
52C23×D173—21no—
53C2×Dic346—21no—
54C2×Dic346C2×C411yescharacteristic, complemented
55Q8×D173—21no—
56Q8×D173—21no—
57Q8×D173C2×C412yescomplemented
58Q8×D173C2312yes—
59C2×D692C2×C411yescharacteristic
60C2×C22⋊D173—21no—
61D4×D173—21no—
62D4×D173—21no—
63D4×D173C2×C412yes—
64D4×D173C2312yes—
65C2×C4×D173—21no—
66C2×C4×D173C2311yescharacteristic
67#14583815C2×C414yescomplemented
68#14583814C2×C414yescomplemented
69#14583813C2×C414yescomplemented
70C2×#13063436C2×C411yescharacteristic, complemented
71C8×D173C2×C412yescomplemented
72C8×D173C2312yes—
73C2×Q8×C173C2×C411yescharacteristic, complemented
74C2×D4×C173C2×C411yescharacteristic
75SD16×C173C2×C414yescomplemented
76C2×C8×C173C2×C411yescharacteristic, complemented
77C2×#10621924—22no—
78C2×#10621924—22no—
79C2×#10621924—22no—
80C2×#10621924—22no—
81C2×#10621924—48no—
82C2×#10621924—42no—
83C2×#10621924—42no—
84C2×#10621924C2×D412yes—
85C2×#10621924C2×D412yes—
86#13063443—22no—
87C22×D173—21no—
88C22×D173—22no—
89C22×D173—22no—
90C22×D173—42no—
91C22×D173—42no—
92C22×D173C2×D411yescharacteristic
93Dic346—21no—
94Dic346—22no—
95Dic346—22no—
96Dic346C22×C412yescomplemented
97D692—21no—
98D692C22×C412yes—
99C22⋊D173—22no—
100C22⋊D173—22no—
101C2×Dic173—21no—
102C2×Dic173#4111yescharacteristic
103C4×D173—22no—
104C4×D173—22no—
105C4×D173C22×C411yescharacteristic
106C4×D173C2×D411yescharacteristic
107C4×D173#4111yescharacteristic
108C4×D173C2411yescharacteristic
109#13063436C22×C412yescomplemented
110C23×C173—21no—
111Q8×C173—21no—
112Q8×C173C22×C412yescomplemented
113D4×C173—21no—
114D4×C173C22×C412yes—
115C2×C4×C173—21no—
116C2×C4×C173C22×C411yescharacteristic
117C8×C173C22×C412yescomplemented
118C22×C173—22no—
119C22×C173—42no—
120C22×C173C4×D411yescharacteristic
121C2×D173—22no—
122C2×D173—42no—
123C2×D173—42no—
124C2×D173—42no—
125C2×D173—44no—
126C2×D173C4×D411yescharacteristic
127C2×D173AGL(1,ℤ/8)12yes—
128C2×D173#13712yes—
129C2×D173C22×D411yescharacteristic
130#10621924—22no—
131#10621924—22no—
132#10621924—42no—
133#10621924—42no—
134C4×C173—22no—
135C4×C173C4×D411yescharacteristic
136C4×C173C23×C411yescharacteristic
137Dic173—22no—
138Dic173C4×D411yescharacteristic
139Dic173C2×#4111yescharacteristic
140C2×C173—42no—
141C2×C173#44012yescomplemented
142C2×C173C2×C4×D411yescharacteristic
143D173—42no—
144D173#44012yescomplemented
145D173C2×AGL(1,ℤ/8)11yescharacteristic
146D173C2×#13711yescharacteristic
147C173C2×#44011yescharacteristic, complemented, Sylow
148C2×#440—1731nomaximal, Sylow
149C22×SD16—1731no—
150C2×C4×Q8—1731no—
151C2×C4×D4—1731no—
152#440—1738no—
153C2×#107—1731no—
154C2×#103—1731no—
155C2×#102—1731no—
156C2×#97—1731no—
157C22×Q8—1731no—
158C22×D4—1731no—
159C23×C4—3461no—
160C2×SD16—1731no—
161C2×SD16—1731no—
162C2×SD16—1734no—
163C2×SD16—1734no—
164C2×SD16—1731no—
165C2×SD16#1062192411yescharacteristic, complemented
166C22×C8—1731no—
167C22×C8—1731no—
168C4×Q8—3462no—
169C4×Q8—1734no—
170C4×D4—3462no—
171C4×D4—1734no—
172C2×C4⋊C4—3461no—
173C2×C4⋊C4—1731no—
174C2×C4⋊C4—1731no—
175C2×C22⋊C4—3461no—
176C2×C42—3461no—
177C2×C42—1731no—
178#107—1734no—
179#103—1734no—
180#102—1734no—
181#97—1734no—
182C24—3461no—
183C2×Q8—3461no—
184C2×Q8—3461no—
185C2×Q8—3461no—
186C2×Q8—1732no—
187C2×Q8—1732no—
188C2×Q8—1731no—
189C2×Q8C2×#1062192411yescharacteristic, complemented
190C2×D4—3461no—
191C2×D4—3461no—
192C2×D4—3461no—
193C2×D4—1732no—
194C2×D4—1732no—
195C2×D4—1731no—
196C2×D4C2×#1062192411yescharacteristic
197C22×C4—3461no—
198C22×C4—3461no—
199C22×C4—3461no—
200C22×C4—3464no—
201C22×C4—1731no—
202C22×C4—1731no—
203C22×C4—1731no—
204C22×C4—6924no—
205SD16—1734no—
206SD16—1734no—
207SD16—1734no—
208SD16C2×#1062192414yescomplemented
209C2×C8—3461no—
210C2×C8—3461no—
211C2×C8—1732no—
212C2×C8—1731no—
213C2×C8—1732no—
214C2×C8—1732no—
215C2×C8C2×#1062192411yescharacteristic, complemented
216C4⋊C4—3464no—
217C4⋊C4—3461no—
218C4⋊C4—3461no—
219C4⋊C4—1732no—
220C4⋊C4—1732no—
221C22⋊C4—3464no—
222C42—3464no—
223C42—1732no—
224C42—1732no—
225C23—21no—
226C23—3462no—
227C23—3462no—
228C23—3461no—
229C23—1731no—
230C23—6922no—
231C23—6922no—
232Q8—21no—
233Q8—3462no—
234Q8—3462no—
235Q8—3461no—
236Q8—1732no—
237Q8C22×#1062192412yescomplemented
238D4—21no—
239D4—3462no—
240D4—3462no—
241D4—3461no—
242D4—1732no—
243D4C22×#1062192412yes—
244C2×C4—21no—
245C2×C4—3462no—
246C2×C4—3462no—
247C2×C4—3462no—
248C2×C4—3462no—
249C2×C4—3461no—
250C2×C4—3462no—
251C2×C4—3462no—
252C2×C4—1732no—
253C2×C4—1732no—
254C2×C4—1731no—
255C2×C4—1731no—
256C2×C4—1731no—
257C2×C4—1731no—
258C2×C4—1731no—
259C2×C4—6928no—
260C2×C4—6922no—
261C2×C4—6922no—
262C2×C4C22×#1062192411yescharacteristic
263C8—3462no—
264C8—1732no—
265C8C22×#1062192412yescomplemented
266C22—22no—
267C22—42no—
268C22—3462no—
269C22—1732no—
270C22—1732no—
271C22—1731no—
272C22—1731no—
273C22—6922no—
274C22—6922no—
275C22—6924no—
276C22—6922no—
277C22D4×#1062192411yescharacteristic
278C4—22no—
279C4—3462no—
280C4—3462no—
281C4—3462no—
282C4—1731no—
283C4—1731no—
284C4—6922no—
285C4—6922no—
286C4D4×#1062192411yescharacteristic
287C4C23×#1062192411yescharacteristic
288C2—42no—
289C2—1731no—
290C2—1731no—
291C2—1732no—
292C2—6922no—
293C2#2601253012yesdirect product factor
294C2C2×D4×#1062192411yescharacteristic
subgroupG/H
centerC22D4×#10621924
commutatorC4×C173C23×C4
FrattiniC4C23×#10621924
FittingC2×SD16×C173C4
socleC22×C173C4×D4
prime radicalC2×D4×D173C22
cosocleC4×D173C24
hypercenterC2×SD16#10621924
hypocenterC173C2×#440
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×Q8×#10621924C2semidirect
3. #26012530C2direct (N × H)
4. C2×#19149748C2semidirect
5. C2×#19149742C2semidirect
7. C2×#19149733C2semidirect
12. Q8×#10621924C22semidirect
21. #19149748C22semidirect
22. #19149742C22semidirect
24. #19149733C22semidirect
27. C2×#14583815C4semidirect
28. C2×#14583814C4semidirect
29. C2×#14583813C4semidirect
30. SD16×D173C4semidirect
33. C2×SD16×C173C4semidirect
54. C2×Dic346C2×C4semidirect
57. Q8×D173C2×C4semidirect
67. #14583815C2×C4semidirect
68. #14583814C2×C4semidirect
69. #14583813C2×C4semidirect
70. C2×#13063436C2×C4semidirect
71. C8×D173C2×C4semidirect
73. C2×Q8×C173C2×C4semidirect
75. SD16×C173C2×C4semidirect
76. C2×C8×C173C2×C4semidirect
96. Dic346C22×C4semidirect
109. #13063436C22×C4semidirect
112. Q8×C173C22×C4semidirect
117. C8×C173C22×C4semidirect
141. C2×C173#440semidirect
144. D173#440semidirect
147. C173C2×#440semidirect
165. C2×SD16#10621924semidirect
189. C2×Q8C2×#10621924semidirect
208. SD16C2×#10621924semidirect
215. C2×C8C2×#10621924semidirect
237. Q8C22×#10621924semidirect
265. C8C22×#10621924semidirect
293. C2#26012530direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×#106219242
C2×D4×#106219242
#260125302
C2×#191497482
C2×#191497422
C2×#191497412
C2×#191497332
C2×SD16×D1732
C2×#440173