Finite Group Databasebeta

← #15682651

Subgroups — #15682651

Order 2944.

806 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#67288C211yesmaximal, characteristic, complemented
2C2×#10625079C212yesmaximal, complemented
3C2×#10625101C211yesmaximal, characteristic, complemented
4#13092564C214yesmaximal, complemented
5#13092558C212yesmaximal, complemented
6#13092553C212yesmaximal, complemented
7#13092419C212yesmaximal, complemented
8C23×#407C211yesmaximal, characteristic, complemented
9C23×C4×C23C2211yescharacteristic
10C22×C22⋊D23—21no—
11C22×C22⋊D23—21no—
12C2×#67288—21no—
13C2×#67288—21no—
14C2×#67288—28no—
15C2×#67288C2212yes—
16C2×#67288C2212yescomplemented
17C22×D92C2211yescharacteristic, complemented
18C22×D92C2211yescharacteristic, complemented
19C22×C4×D23—21no—
20C22×Dic46C2211yescharacteristic, complemented
21C2×M16×C23C2212yescomplemented
22#10625101—21no—
23#10625101—21no—
24#10625101C2212yes—
25#10625101C2212yescomplemented
26C2×(C2×D23)⋊C4—21no—
27C2×C4⋊Dic23C2211yescharacteristic, complemented
28#10625079—28no—
29C2×D184C2214yescomplemented
30C2×#67263C2214yescomplemented
31C23×C22⋊C8C2212yescomplemented
32C23×C22⋊C8C2212yescomplemented
33#10625003C2214yescomplemented
34#10624998C2214yescomplemented
35C24×C23D411yescharacteristic
36C23×D23—21no—
37C23×D23—21no—
38C22×C4×C23—24no—
39C22×C4×C23D411yescharacteristic
40C22×C4×C23D411yescharacteristic
41C22×C4×C23D411yescharacteristic
42C22×C4×C23C2311yescharacteristic
43C22×C4×C23C2311yescharacteristic
44C22×C4×C23C2311yescharacteristic
45C2×C22⋊D23—28no—
46C2×C22⋊D23—22no—
47C2×C22⋊D23—22no—
48C2×C22⋊D23—22no—
49C2×C22⋊D23—22no—
50C2×C22⋊D23—44no—
51C22×Dic23—21no—
52C22×Dic23—21no—
53#67288—28no—
54#67288—28no—
55#67288—24no—
56#67288—24no—
57#67288—24no—
58#67288D418yescomplemented
59C2×D92—21no—
60C2×D92—24no—
61C2×D92—24no—
62C2×D92—21no—
63C2×D92C2312yes—
64C2×D92C2312yes—
65C2×C4×D23—28no—
66C2×C4×D23—22no—
67C2×C4×D23—22no—
68C2×Dic46—21no—
69C2×Dic46—24no—
70C2×Dic46C2312yes—
71M16×C23—24no—
72C2×C8×C23C2314yes—
73(C2×D23)⋊C4—22no—
74(C2×D23)⋊C4—22no—
75C4⋊Dic23—21no—
76C4⋊Dic23C2312yes—
77D184—28no—
78#67263—28no—
79C23×C23—22no—
80C23×C23—22no—
81C23×C23—41no—
82C23×C23C2×D411yescharacteristic
83C23×C23C2×D411yescharacteristic
84C23×C23C2×D411yescharacteristic
85C22×D23—24no—
86C22×D23—22no—
87C22×D23—22no—
88C22×D23—42no—
89C22×D23—42no—
90C22×D23—42no—
91C22×D23—42no—
92C22×D23—42no—
93C2×C4×C23—22no—
94C2×C4×C23—22no—
95C2×C4×C23—22no—
96C2×C4×C23—22no—
97C2×C4×C23—22no—
98C2×C4×C23C2×D411yescharacteristic
99C2×C4×C23C2×D411yescharacteristic
100C2×C4×C23C2×D411yescharacteristic
101C2×C4×C23C2×D414yes—
102C2×C4×C23C2411yescharacteristic
103C22⋊D23—28no—
104C22⋊D23—28no—
105C22⋊D23—44no—
106C22⋊D23—44no—
107C2×Dic23—24no—
108C2×Dic23—22no—
109C2×Dic23—22no—
110C2×Dic23—42no—
111D92—24no—
112D92—22no—
113D92—24no—
114D92—42no—
115D92C2×D414yescomplemented
116C4×D23—28no—
117C4×D23—28no—
118Dic46—24no—
119Dic46—22no—
120Dic46C2×D414yescomplemented
121C8×C23—24no—
122#3006—231nomaximal, Sylow
123C22×C23—21no—
124C22×C23—21no—
125C22×C23—21no—
126C22×C23—21no—
127C22×C23—21no—
128C22×C23—21no—
129C22×C23—21no—
130C22×C23—44no—
131C22×C23#12012yes—
132C22×C23AGL(1,ℤ/8)12yescomplemented
133C22×C23C22×D411yescharacteristic
134C2×D23—24no—
135C2×D23—48no—
136C2×D23—42no—
137C2×D23—42no—
138C2×D23—42no—
139C2×D23—42no—
140C2×D23—42no—
141C2×D23—84no—
142C4×C23—21no—
143C4×C23—21no—
144C4×C23#12012yes—
145C4×C23C22×D412yes—
146Dic23—24no—
147Dic23—42no—
148C22×#41—231no—
149C2×AGL(1,ℤ/8)—232no—
150C2×#121—231no—
151#449—234no—
152#448—232no—
153#447—232no—
154#417—232no—
155#407D2311yescharacteristic, complemented
156C2×C23—22no—
157C2×C23—22no—
158C2×C23—41no—
159C2×C23C2×#12011yescharacteristic
160C2×C23C2×AGL(1,ℤ/8)11yescharacteristic
161C2×C23#57511yescharacteristic
162D23—44no—
163D23—82no—
164C2×#41—232no—
165C2×#41—232no—
166C2×#41—468no—
167C2×#41—461no—
168C2×#41—461no—
169C22×Q8—231no—
170C22×D4—231no—
171C22×D4—231no—
172C22×D4—461no—
173C22×D4—461no—
174C23×C4—461no—
175C23×C4C2×D2311yescharacteristic
176AGL(1,ℤ/8)—468no—
177C2×SD16—234no—
178C2×D8—234no—
179C2×M16C2×D2312yescomplemented
180#121—232no—
181#121—232no—
182#121—461no—
183#121—461no—
184C2×C4⋊C4—231no—
185C2×C22⋊C4—461no—
186#103—234no—
187#102—234no—
188C22⋊C8C2×D2312yescomplemented
189C22⋊C8C2×D2312yescomplemented
190C23#300611yescharacteristic, complemented, Sylow
191C24—461no—
192C24—461no—
193C24D9211yescharacteristic
194#41—238no—
195#41—464no—
196#41—464no—
197#41—468no—
198#41—468no—
199#41—464no—
200C2×Q8—232no—
201C2×Q8—464no—
202C2×Q8—461no—
203C2×D4—232no—
204C2×D4—232no—
205C2×D4—464no—
206C2×D4—468no—
207C2×D4—462no—
208C2×D4—462no—
209C2×D4—462no—
210C2×D4—462no—
211C2×D4—461no—
212C2×D4—461no—
213C2×D4—464no—
214C2×D4—924no—
215C22×C4—24no—
216C22×C4—468no—
217C22×C4—461no—
218C22×C4—462no—
219C22×C4—462no—
220C22×C4—461no—
221C22×C4D9211yescharacteristic
222C22×C4D9211yescharacteristic
223C22×C4D9211yescharacteristic
224C22×C4C22×D2311yescharacteristic
225C22×C4C22×D2311yescharacteristic
226C22×C4C22×D2311yescharacteristic
227SD16—468no—
228D8—468no—
229M16—24no—
230C2×C8C22×D2314yes—
231C4⋊C4—232no—
232C4⋊C4—461no—
233C22⋊C4—462no—
234C22⋊C4—462no—
235C23—22no—
236C23—22no—
237C23—41no—
238C23—464no—
239C23—462no—
240C23—462no—
241C23—922no—
242C23—922no—
243C23—922no—
244C23—922no—
245C23—922no—
246C23C2×D9211yescharacteristic
247C23C2×D9211yescharacteristic
248C23C2×D9211yescharacteristic
249Q8—234no—
250Q8—464no—
251Q8—462no—
252D4—234no—
253D4—464no—
254D4—468no—
255D4—468no—
256D4—462no—
257D4—464no—
258D4—924no—
259D4—924no—
260D4—922no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—468no—
267C2×C4—464no—
268C2×C4—462no—
269C2×C4—468no—
270C2×C4—462no—
271C2×C4—922no—
272C2×C4C2×D9211yescharacteristic
273C2×C4C2×D9211yescharacteristic
274C2×C4C2×D9211yescharacteristic
275C2×C4D4×D2314yes—
276C2×C4C23×D2311yescharacteristic
277C8—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—464no—
287C22—928no—
288C22—922no—
289C22—922no—
290C22—922no—
291C22—922no—
292C22—922no—
293C22—1844no—
294C22#1062505312yes—
295C22#1062507912yescomplemented
296C22C22×D9211yescharacteristic
297C4—21no—
298C4—21no—
299C4—464no—
300C4—922no—
301C4#1062505312yes—
302C4C2×D4×D2312yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—924no—
307C2—1842no—
308C2C2×#1062505311yescharacteristic
309C2C2×#1062507911yescharacteristic
310C2#1309321711yescharacteristic
subgroupG/H
centerC22C22×D92
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#407C2
socleC22×C23C22×D4
hypercenter#407D23
hypocenterC23#3006
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#67288C2semidirect
2. C2×#10625079C2semidirect
3. C2×#10625101C2semidirect
4. #13092564C2semidirect
5. #13092558C2semidirect
6. #13092553C2semidirect
7. #13092419C2semidirect
8. C23×#407C2semidirect
16. C2×#67288C22semidirect
17. C22×D92C22semidirect
18. C22×D92C22semidirect
20. C22×Dic46C22semidirect
21. C2×M16×C23C22semidirect
25. #10625101C22semidirect
27. C2×C4⋊Dic23C22semidirect
29. C2×D184C22semidirect
30. C2×#67263C22semidirect
31. C23×C22⋊C8C22semidirect
32. C23×C22⋊C8C22semidirect
33. #10625003C22semidirect
34. #10624998C22semidirect
58. #67288D4semidirect
115. D92C2×D4semidirect
120. Dic46C2×D4semidirect
132. C22×C23AGL(1,ℤ/8)semidirect
155. #407D23semidirect
179. C2×M16C2×D23semidirect
188. C22⋊C8C2×D23semidirect
189. C22⋊C8C2×D23semidirect
190. C23#3006semidirect
295. C22#10625079semidirect

Maximal subgroups

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

subgroupindex
C22×#672882
C2×#106250792
C2×#106251012
#130925642
#130925582
#130925532
#130924192
C23×#4072
#300623