Finite Group Databasebeta

← #15683316

Subgroups — #15683316

Order 2944.

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

normal subgroups only

This group has 458 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
1#13092927C211yesmaximal, characteristic
2C2×C4⋊C4×D23C211yesmaximal, characteristic, complemented
3#13092754C212yesmaximal, complemented
4#13092748C211yesmaximal, characteristic, complemented
5#13092747C211yesmaximal, characteristic, complemented
6#13092734C212yesmaximal, complemented
7#13092728C211yesmaximal, characteristic, complemented
8#13092458C212yesmaximal, complemented
9#13092415C212yesmaximal, complemented
10#13092379C211yesmaximal, characteristic, complemented
11C23×#480C211yesmaximal, characteristic, complemented
12C2×#67290C2211yescharacteristic
13C22×C4×D23—21no—
14C22×C4×D23C2211yescharacteristic
15C23×#121C2211yescharacteristic
16C2×C4⋊C4×C23C2211yescharacteristic
17#10625109C2211yescharacteristic
18#10625108—21no—
19D4×Dic23—21no—
20C2×C4⋊Dic23C2211yescharacteristic
21#10625095C2211yescharacteristic
22C2×Dic23⋊C4—21no—
23C2×#67262C2212yes—
24#10625068C2211yescharacteristic
25#10625063C2211yescharacteristic
26C4⋊C4×D23—21no—
27C4⋊C4×D23—21no—
28C4⋊C4×D23C2212yes—
29C4⋊C4×D23C2212yes—
30#10625054—21no—
31(C22⋊D23)⋊C4—21no—
32C23×#107C2212yescomplemented
33C23×#102C2212yescomplemented
34C23×C22⋊C8C2211yescharacteristic
35#10625014C2212yescomplemented
36C22⋊(C23⋊C8)C2211yescharacteristic
37(C2×D23)⋊C8C2212yes—
38#10624999C2212yescomplemented
39#10624998C2212yescomplemented
40#10624992C2212yescomplemented
41#10624990C2212yescomplemented
42#10624989C2212yescomplemented
43C23×D23D411yescharacteristic
44C2×D4×C23—21no—
45C2×D4×C23C2311yescharacteristic
46C22×C4×C23—21no—
47C22×C4×C23C2311yescharacteristic
48C2×C22⋊D23—21no—
49C2×C22⋊D23—21no—
50C22×Dic23—21no—
51C22×Dic23—21no—
52C22×Dic23D411yescharacteristic
53#67290—24no—
54C2×C4×D23—24no—
55C2×C4×D23—22no—
56C2×C4×D23—22no—
57C2×C4×D23—44no—
58C2×C4×D23D411yescharacteristic
59C2×C4×D23D411yescharacteristic
60C2×C4×D23C2311yescharacteristic
61C2×C4×D23C2311yescharacteristic
62C2×Dic46C2311yescharacteristic
63C2×C8×C23C2312yes—
64C4⋊C4×C23—21no—
65C4⋊C4×C23C2312yes—
66C4⋊C4×C23C2311yescharacteristic
67C22⋊C4×C23—21no—
68C22⋊Dic23—21no—
69C22⋊Dic23—21no—
70(C2×D23)⋊C4—21no—
71(C2×D23)⋊C4—21no—
72C4⋊Dic23—21no—
73C4⋊Dic23—21no—
74C4⋊Dic23C2312yes—
75C4⋊Dic23C2311yescharacteristic
76Dic23⋊C4—21no—
77Dic23⋊C4—22no—
78Dic23⋊C4—22no—
79C4×Dic23—21no—
80C2×C23⋊C8C2312yes—
81#67262—24no—
82C23×C23—21no—
83C23×C23C2×D411yescharacteristic
84C22×D23—22no—
85C22×D23—22no—
86C22×D23—22no—
87C22×D23C2×D411yescharacteristic
88C22×D23C2×D411yescharacteristic
89D4×C23—22no—
90D4×C23—42no—
91C2×C4×C23—22no—
92C2×C4×C23—21no—
93C2×C4×C23—22no—
94C2×C4×C23—42no—
95C2×C4×C23C2×D411yescharacteristic
96C2×C4×C23C2411yescharacteristic
97C22⋊D23—42no—
98C22⋊D23—42no—
99C2×Dic23—22no—
100C2×Dic23—21no—
101C2×Dic23—22no—
102C2×Dic23—21no—
103C2×Dic23—42no—
104C2×Dic23—42no—
105C2×Dic23C2×D411yescharacteristic
106C2×Dic23C2×D411yescharacteristic
107C4×D23—22no—
108C4×D23—22no—
109C4×D23—22no—
110C4×D23—44no—
111C4×D23—44no—
112C4×D23#4114yes—
113Dic46—22no—
114C8×C23—22no—
115C23⋊C8—22no—
116#3092—231nomaximal, Sylow
117C22×C23—21no—
118C22×C23—21no—
119C22×C23—41no—
120C22×C23—41no—
121C22×C23—41no—
122C22×C23AGL(1,ℤ/8)12yes—
123C22×C23C22×D411yescharacteristic
124C2×D23—28no—
125C2×D23—21no—
126C2×D23—21no—
127C2×D23—21no—
128C2×D23—21no—
129C2×D23#12312yes—
130C2×D23#13712yes—
131C4×C23—21no—
132C4×C23—42no—
133C4×C23—41no—
134C4×C23C2×#4112yes—
135Dic23—21no—
136Dic23—41no—
137Dic23—42no—
138Dic23—41no—
139Dic23#12312yes—
140#525—231no—
141C22×C4⋊C4—231no—
142#484—232no—
143#483—231no—
144#483—231no—
145#481—232no—
146#480—231no—
147#480D2311yescharacteristic, complemented
148#428—232no—
149#417—232no—
150#407—231no—
151C2×C23—22no—
152C2×C23—81no—
153C2×C23C2×#12311yescharacteristic
154C2×C23C2×AGL(1,ℤ/8)11yescharacteristic
155C2×C23C2×#13711yescharacteristic
156D23—22no—
157D23—22no—
158C2×#41—231no—
159C23×C4—231no—
160C23×C4—461no—
161C2×M16—232no—
162#123—461no—
163C22⋊Q8—231no—
164C22⋊Q8—231no—
165#121—231no—
166#121C2×D2311yescharacteristic
167#120—461no—
168C4×D4—461no—
169C4×D4—461no—
170#117—231no—
171C2×C4⋊C4—231no—
172C2×C4⋊C4—232no—
173C2×C4⋊C4—232no—
174C2×C4⋊C4—461no—
175C2×C4⋊C4—461no—
176C2×C4⋊C4—461no—
177C2×C4⋊C4C2×D2311yescharacteristic
178#107—232no—
179#107C2×D2312yescomplemented
180#106—232no—
181#106—232no—
182#103—232no—
183#103—232no—
184#102—232no—
185#102C2×D2312yescomplemented
186C22⋊C8—231no—
187C22⋊C8—232no—
188C22⋊C8C2×D2311yescharacteristic
189C23#309211yescharacteristic, complemented, Sylow
190C24—231no—
191#41—464no—
192C2×Q8—231no—
193C2×D4—21no—
194C2×D4—461no—
195C2×D4—461no—
196C2×D4C22×D2311yescharacteristic
197C22×C4—21no—
198C22×C4—231no—
199C22×C4—231no—
200C22×C4—231no—
201C22×C4—231no—
202C22×C4—231no—
203C22×C4—464no—
204C22×C4—462no—
205C22×C4—462no—
206C22×C4—461no—
207C22×C4—461no—
208C22×C4—924no—
209C22×C4C22×D2311yescharacteristic
210M16—464no—
211C2×C8—232no—
212C2×C8C22×D2312yes—
213C4⋊C4—21no—
214C4⋊C4—232no—
215C4⋊C4—231no—
216C4⋊C4—462no—
217C4⋊C4—462no—
218C4⋊C4—461no—
219C4⋊C4—461no—
220C4⋊C4—461no—
221C4⋊C4C22×D2312yes—
222C4⋊C4C22×D2311yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—461no—
225C22⋊C4—461no—
226C22⋊C4—461no—
227C22⋊C4—461no—
228C42—461no—
229C23—21no—
230C23—231no—
231C23—231no—
232C23—462no—
233C23—462no—
234C23—462no—
235C23D4×D2311yescharacteristic
236Q8—462no—
237D4—22no—
238D4—42no—
239D4—922no—
240D4—922no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—231no—
246C2×C4—234no—
247C2×C4—231no—
248C2×C4—462no—
249C2×C4—462no—
250C2×C4—462no—
251C2×C4—462no—
252C2×C4—461no—
253C2×C4—461no—
254C2×C4—462no—
255C2×C4—924no—
256C2×C4—924no—
257C2×C4—922no—
258C2×C4—922no—
259C2×C4D4×D2311yescharacteristic
260C2×C4C23×D2311yescharacteristic
261C8—22no—
262C8—462no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—232no—
269C22—232no—
270C22—468no—
271C22—461no—
272C22—461no—
273C22—461no—
274C22—461no—
275C22#1062508212yes—
276C22C2×D4×D2311yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—232no—
281C4—461no—
282C4—921no—
283C4—922no—
284C4—921no—
285C4#41×D2312yes—
286C2—22no—
287C2—81no—
288C2—462no—
289C2—462no—
290C2#123×D2311yescharacteristic
291C2C2×#1062508211yescharacteristic
292C2#137×D2311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#480C2
socleC22×C23C22×D4
prime radical#13092927C2
hypercenter#480D23
hypocenterC23#3092
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D23C2semidirect
3. #13092754C2semidirect
4. #13092748C2semidirect
5. #13092747C2semidirect
6. #13092734C2semidirect
7. #13092728C2semidirect
8. #13092458C2semidirect
9. #13092415C2semidirect
10. #13092379C2semidirect
11. C23×#480C2semidirect
32. C23×#107C22semidirect
33. C23×#102C22semidirect
35. #10625014C22semidirect
38. #10624999C22semidirect
39. #10624998C22semidirect
40. #10624992C22semidirect
41. #10624990C22semidirect
42. #10624989C22semidirect
147. #480D23semidirect
179. #107C2×D23semidirect
185. #102C2×D23semidirect
189. C23#3092semidirect

Maximal subgroups

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

subgroupindex
#130929272
C2×C4⋊C4×D232
#130927542
#130927482
#130927472
#130927342
#130927282
#130924582
#130924152
#130923792
C23×#4802
#309223