Finite Group Databasebeta

← #15686111

Subgroups — #15686111

Order 2944.

372 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 123 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
1Q8×D92C211yesmaximal, characteristic, complemented
2#13092776C212yesmaximal, complemented
3#13092767C211yesmaximal, characteristic, complemented
4#13092712C212yesmaximal, complemented
5#13092697C211yesmaximal, characteristic, complemented
6#13092629C212yesmaximal
7#13092619C211yesmaximal, characteristic, complemented
8#13092548C211yesmaximal, characteristic, complemented
9#13092515C212yesmaximal, complemented
10C4×#67272C211yesmaximal, characteristic
11C23×#556C211yesmaximal, characteristic, complemented
12C2×Q8×D23—21no—
13C23×C4⋊Q8C2211yescharacteristic
14C23×#126—22no—
15C23×#124C2212yes—
16C23×C22⋊Q8—21no—
17C4×Q8×C23C2212yes—
18C4×Q8×C23C2211yescharacteristic
19C4×D4×C23C2211yescharacteristic
20C23×#117—21no—
21C2×#67274C2212yes—
22C2×#67272C2211yescharacteristic
23#10625068—21no—
24#10625068—22no—
25C4×D92C2211yescharacteristic
26C4×D92C2212yes—
27#10625040C2211yescharacteristic
28#10625040C2212yes—
29#10625017C2212yes—
30#10625017C2212yes—
31#10625017C2212yes—
32#10625014C2212yes—
33#10625014C2211yescharacteristic
34#10624991C2211yescharacteristic
35#10624989C2212yes—
36#10624989C2211yescharacteristic
37C4⋊(C23⋊C8)C2211yescharacteristic
38C4⋊(C23⋊C8)C2212yes—
39#10624985C2212yes—
40C4×C23⋊C8C2211yescharacteristic
41C2×Q8×C23D411yescharacteristic
42C2×Q8×C23C2312yes—
43C2×D4×C23C2311yescharacteristic
44C22×C4×C23—21no—
45Q8×D23—44no—
46C2×D92C2311yescharacteristic
47C2×C4×D23—22no—
48C2×C4×D23—21no—
49C2×Dic46—22no—
50C2×Dic46—21no—
51C4⋊C4×C23—21no—
52C4⋊C4×C23—22no—
53C4⋊C4×C23—21no—
54C4⋊C4×C23D411yescharacteristic
55C4⋊C4×C23D412yes—
56C4⋊C4×C23C2311yescharacteristic
57C22⋊C4×C23—22no—
58C22⋊C4×C23—21no—
59C22⋊C4×C23—22no—
60C42×C23—22no—
61C42×C23C2311yescharacteristic
62C42×C23C2312yes—
63#67274—24no—
64#67272—22no—
65(C2×D23)⋊C4—22no—
66(C2×D23)⋊C4—21no—
67C4⋊Dic23—22no—
68C4⋊Dic23—21no—
69C4⋊Dic23C2312yes—
70C4⋊Dic23C2311yescharacteristic
71C2×C23⋊C8C2311yescharacteristic
72C2×C23⋊C8C2312yes—
73C2×C23⋊C8C2311yescharacteristic
74C23×C23—21no—
75C22×D23—21no—
76Q8×C23—22no—
77Q8×C23—22no—
78Q8×C23—22no—
79D4×C23—21no—
80D4×C23—21no—
81C2×C4×C23—21no—
82C2×C4×C23—22no—
83C2×C4×C23—42no—
84C2×C4×C23C2×D412yes—
85C2×C4×C23C2×D411yescharacteristic
86C2×C4×C23C2×D412yes—
87C2×C4×C23C2×D411yescharacteristic
88C2×C4×C23C2411yescharacteristic
89C2×Dic23—21no—
90C2×Dic23—22no—
91D92—21no—
92D92#4112yes—
93C4×D23—44no—
94C4×D23—42no—
95Dic46—44no—
96Dic46—42no—
97C23⋊C8—21no—
98C23⋊C8—22no—
99C23⋊C8—21no—
100#3367—231nomaximal, Sylow
101C22×C23—41no—
102C22×C23—41no—
103C22×C23—41no—
104C22×C23C22×D411yescharacteristic
105C2×D23—22no—
106C2×D23—41no—
107C2×D23—41no—
108C4×C23—22no—
109C4×C23—22no—
110C4×C23—21no—
111C4×C23—42no—
112C4×C23—41no—
113C4×C23AGL(1,ℤ/8)12yes—
114C4×C23C2×#4111yescharacteristic
115C4×C232−1+411yescharacteristic
116Dic23—41no—
117Dic23—42no—
118#556D2311yescharacteristic, complemented
119D4×Q8—231no—
120#489—232no—
121#487—231no—
122#477—232no—
123#474—231no—
124#463—232no—
125#461—231no—
126#446—231no—
127#440—232no—
128C4×D8—231no—
129C2×C23—81no—
130C2×C23#54811yescharacteristic
131C2×C23C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C23#57811yescharacteristic
133D23—42no—
134C22×Q8—461no—
135C2×SD16—232no—
136C2×D8—231no—
137C4⋊Q8—231no—
138C4⋊Q8—232no—
139C4⋊Q8C2×D2311yescharacteristic
140#126—22no—
141#124C2×D2312yes—
142C22⋊Q8—21no—
143C22⋊Q8—461no—
144C22⋊Q8—462no—
145C4×Q8C2×D2312yes—
146C4×Q8C2×D2311yescharacteristic
147C4×D4—231no—
148C4×D4—232no—
149C4×D4C2×D2311yescharacteristic
150#117—21no—
151#107—231no—
152#107—232no—
153C4⋊C8—231no—
154C4⋊C8—232no—
155#103—232no—
156#103—232no—
157#103—232no—
158#102—231no—
159#102—232no—
160#102—231no—
161#97—232no—
162C4×C8—231no—
163C23#336711yescharacteristic, complemented, Sylow
164C2×Q8—462no—
165C2×Q8—461no—
166C2×Q8—924no—
167C2×Q8C22⋊D2311yescharacteristic
168C2×Q8C22×D2312yes—
169C2×D4—231no—
170C2×D4C22×D2311yescharacteristic
171C22×C4—21no—
172C22×C4—462no—
173C22×C4—461no—
174SD16—464no—
175D8—462no—
176C2×C8—231no—
177C2×C8—231no—
178C2×C8—232no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—231no—
183C4⋊C4—232no—
184C4⋊C4—462no—
185C4⋊C4—461no—
186C4⋊C4C22⋊D2311yescharacteristic
187C4⋊C4C22⋊D2312yes—
188C4⋊C4C22×D2311yescharacteristic
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C22⋊C4—462no—
193C22⋊C4—461no—
194C42—22no—
195C42C22×D2311yescharacteristic
196C42C22×D2312yes—
197C23—21no—
198C23—461no—
199Q8—22no—
200Q8—22no—
201Q8—22no—
202Q8—924no—
203Q8—922no—
204D4—21no—
205D4—21no—
206D4—232no—
207D4—461no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—462no—
212C2×C4—461no—
213C2×C4—924no—
214C2×C4—922no—
215C2×C4C2×C22⋊D2312yes—
216C2×C4C2×C22⋊D2311yescharacteristic
217C2×C4C2×C22⋊D2312yes—
218C2×C4C2×C22⋊D2311yescharacteristic
219C2×C4C23×D2311yescharacteristic
220C8—461no—
221C8—462no—
222C8—461no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—462no—
227C22—921no—
228C22—921no—
229C22C22×C22⋊D2311yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—41no—
234C4—42no—
235C4—922no—
236C4—921no—
237C4#1062512012yes—
238C4#41×D2311yescharacteristic
239C4#1062516111yescharacteristic
240C2—81no—
241C2—922no—
242C2#1309307411yescharacteristic
243C2C2×#1062512011yescharacteristic
244C2#1309323211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#556C2
socleC22×C23C22×D4
prime radicalC2×#67272C22
hypercenter#556D23
hypocenterC23#3367
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. Q8×D92C2semidirect
2. #13092776C2semidirect
3. #13092767C2semidirect
4. #13092712C2semidirect
5. #13092697C2semidirect
7. #13092619C2semidirect
8. #13092548C2semidirect
9. #13092515C2semidirect
11. C23×#556C2semidirect
118. #556D23semidirect
163. C23#3367semidirect

Maximal subgroups

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

subgroupindex
Q8×D922
#130927762
#130927672
#130927122
#130926972
#130926292
#130926192
#130925482
#130925152
C4×#672722
C23×#5562
#336723