Finite Group Databasebeta

← #15682668

Subgroups — #15682668

Order 2944.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10625160C211yesmaximal, characteristic, complemented
2C2×#10625120C211yesmaximal, characteristic, complemented
3C22×D184C211yesmaximal, characteristic, complemented
4#13092986C211yesmaximal, characteristic, complemented
5#13092565C214yesmaximal, complemented
6#13092552C214yesmaximal, complemented
7#13092421C211yesmaximal, characteristic, complemented
8#13092383C211yesmaximal, characteristic, complemented
9C23×#416C211yesmaximal, characteristic, complemented
10C2×#41×C23C2211yescharacteristic
11#10625160—24no—
12#10625160—24no—
13C2×#67292C2212yescomplemented
14C2×D4×D23—22no—
15C2×D4×D23C2212yescomplemented
16C2×#67288—21no—
17C2×#67288C2211yescharacteristic
18C22×D92—21no—
19C22×D92C2211yescharacteristic
20C22×D92C2211yescharacteristic, complemented
21C22×C8×C23C2211yescharacteristic, complemented
22C23×#117C2211yescharacteristic, complemented
23#10625120—24no—
24C2×#67274C2212yescomplemented
25C2×#67272C2212yescomplemented
26C2×#67267C2211yescharacteristic, complemented
27C2×D184—24no—
28C2×D184C2214yescomplemented
29#10625066C2214yescomplemented
30#10625053—22no—
31#10625046—21no—
32#10625045—21no—
33C23×#103C2212yescomplemented
34C23×#102C2212yescomplemented
35(C2×D23)⋊C8C2214yescomplemented
36#10624991C2214yescomplemented
37C23×D23—21no—
38C23×D23—22no—
39C23×D23—21no—
40#41×C23—22no—
41#41×C23D414yescomplemented
42C2×Q8×C23C2311yescharacteristic
43C2×D4×C23—21no—
44C2×D4×C23C2311yescharacteristic
45C22×C4×C23—21no—
46C22×C4×C23C2311yescharacteristic
47C2×C22⋊D23—22no—
48C2×C22⋊D23D412yes—
49#67292—24no—
50#67292—24no—
51D4×D23—216no—
52D4×D23—24no—
53D4×D23—24no—
54#67288—28no—
55#67288—22no—
56#67288—22no—
57C2×D92—28no—
58C2×D92—22no—
59C2×D92—21no—
60C2×D92—22no—
61C2×D92—22no—
62C2×D92—22no—
63C2×D92—21no—
64C2×D92—24no—
65C2×D92D411yescharacteristic
66C2×D92D414yescomplemented
67C2×D92C2312yes—
68C2×D92C2312yes—
69C2×C4×D23—22no—
70C2×C4×D23C2312yes—
71C2×Dic46D411yescharacteristic
72C2×C8×C23—22no—
73C2×C8×C23C2312yes—
74C4⋊C4×C23C2312yes—
75C22⋊C4×C23—21no—
76C42×C23—21no—
77#67274—24no—
78#67272—24no—
79(C2×D23)⋊C4—24no—
80#67267—22no—
81C2×C23⋊C8C2312yes—
82D184—28no—
83C23×C23—21no—
84C23×C23C2×D411yescharacteristic
85C22×D23—22no—
86C22×D23—28no—
87C22×D23—24no—
88C22×D23—22no—
89C22×D23—42no—
90C22×D23—84no—
91C22×D23—82no—
92C22×D23—82no—
93C22×D23C2×D412yes—
94Q8×C23—21no—
95Q8×C23C2×D412yes—
96D4×C23—24no—
97D4×C23—21no—
98D4×C23C2×D412yes—
99C2×C4×C23—21no—
100C2×C4×C23—24no—
101C2×C4×C23—22no—
102C2×C4×C23C2×D412yes—
103C2×C4×C23C2×D411yescharacteristic
104C2×C4×C23C2×D412yes—
105C2×C4×C23C2411yescharacteristic
106C22⋊D23—28no—
107C22⋊D23—24no—
108C2×Dic23C2×D412yes—
109D92—28no—
110D92—22no—
111D92—24no—
112D92—22no—
113D92—22no—
114D92—22no—
115D92—42no—
116D92—84no—
117D92C2×D414yes—
118C4×D23—28no—
119C4×D23—22no—
120C4×D23—22no—
121Dic46—22no—
122C8×C23—22no—
123C23⋊C8—22no—
124#3008—231nomaximal, Sylow
125C22×C23—22no—
126C22×C23—22no—
127C22×C23—42no—
128C22×C23#12012yes—
129C22×C23C22×D411yescharacteristic
130C2×D23—22no—
131C2×D23—24no—
132C2×D23—24no—
133C2×D23—44no—
134C2×D23—42no—
135C2×D23—44no—
136C2×D23—84no—
137C2×D23—88no—
138C2×D23—84no—
139C4×C23—22no—
140C4×C23—42no—
141C4×C23#12012yes—
142C4×C23C22×D411yescharacteristic
143C4×C23C22×D411yescharacteristic
144Dic23—22no—
145C2×H(5,𝔽2)—231no—
146C2×AGL(1,ℤ/8)—231no—
147C22×D8—231no—
148#535—231no—
149#449—234no—
150#447—234no—
151#418—231no—
152#416D2311yescharacteristic, complemented
153#408—231no—
154C2×C23—22no—
155C2×C23—42no—
156C2×C23C2×#12011yescharacteristic
157C2×C23#57612yes—
158D23—42no—
159D23—44no—
160D23—82no—
161H(5,𝔽2)—464no—
162H(5,𝔽2)—464no—
163C2×#41—231no—
164C2×#41—232no—
165C2×#41—461no—
166C2×#41C2×D2311yescharacteristic
167C22×D4—232no—
168C22×D4—231no—
169C22×D4—231no—
170C22×D4—461no—
171C22×D4—462no—
172AGL(1,ℤ/8)—464no—
173C2×SD16—232no—
174C2×D8—232no—
175C2×D8—234no—
176C2×D8—464no—
177C2×M16—231no—
178C22×C8C2×D2311yescharacteristic, complemented
179#127—461no—
180#124—461no—
181#121—234no—
182#120—462no—
183#117C2×D2311yescharacteristic, complemented
184#103C2×D2312yescomplemented
185#102—234no—
186#102C2×D2312yescomplemented
187C22⋊C8—234no—
188C23#300811yescharacteristic, complemented, Sylow
189C24—462no—
190C24—461no—
191C24—461no—
192#41—22no—
193#41—468no—
194#41—464no—
195#41—464no—
196#41—462no—
197#41—462no—
198#41D9214yescomplemented
199C2×Q8—231no—
200C2×Q8C22×D2311yescharacteristic
201C2×D4—21no—
202C2×D4—232no—
203C2×D4—234no—
204C2×D4—231no—
205C2×D4—232no—
206C2×D4—232no—
207C2×D4—462no—
208C2×D4—468no—
209C2×D4—464no—
210C2×D4—462no—
211C2×D4—464no—
212C2×D4—4616no—
213C2×D4—464no—
214C2×D4—462no—
215C2×D4—462no—
216C2×D4—461no—
217C2×D4—462no—
218C2×D4—461no—
219C2×D4C22×D2311yescharacteristic
220C22×C4—21no—
221C22×C4—232no—
222C22×C4—462no—
223C22×C4C22×D2311yescharacteristic
224SD16—464no—
225D8—468no—
226D8—464no—
227M16—462no—
228C2×C8—22no—
229C2×C8—232no—
230C2×C8C22×D2312yes—
231C4⋊C4C22×D2312yes—
232C22⋊C4—21no—
233C22⋊C4—464no—
234C42—21no—
235C23—21no—
236C23—232no—
237C23—468no—
238C23—462no—
239C23—464no—
240C23—462no—
241C23—922no—
242C23—1842no—
243C23—1842no—
244C23—1844no—
245C23D4×D2311yescharacteristic
246Q8—21no—
247Q8—462no—
248Q8C2×D9212yes—
249D4—24no—
250D4—21no—
251D4—234no—
252D4—468no—
253D4—468no—
254D4—464no—
255D4—464no—
256D4—462no—
257D4—462no—
258D4—462no—
259D4—462no—
260D4—922no—
261D4—1844no—
262D4C2×D9212yes—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—232no—
267C2×C4—468no—
268C2×C4—462no—
269C2×C4—462no—
270C2×C4C2×D9212yes—
271C2×C4D4×D2311yescharacteristic
272C2×C4D4×D2312yes—
273C2×C4C23×D2311yescharacteristic
274C8—22no—
275C8—462no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—464no—
280C22—462no—
281C22—464no—
282C22—924no—
283C22—922no—
284C22—924no—
285C22—1844no—
286C22—1844no—
287C22—1848no—
288C22#1062505312yes—
289C22C2×D4×D2311yescharacteristic
290C4—22no—
291C4—42no—
292C4—462no—
293C4#1062505312yes—
294C4C22×D9211yescharacteristic
295C4C2×D4×D2311yescharacteristic
296C2—22no—
297C2—42no—
298C2—922no—
299C2—924no—
300C2—1842no—
301C2C2×#1062505311yescharacteristic
302C2#1309321912yes—
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#416C2
socleC22×C23C22×D4
hypercenter#416D23
hypocenterC23#3008
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#10625160C2semidirect
2. C2×#10625120C2semidirect
3. C22×D184C2semidirect
4. #13092986C2semidirect
5. #13092565C2semidirect
6. #13092552C2semidirect
7. #13092421C2semidirect
8. #13092383C2semidirect
9. C23×#416C2semidirect
13. C2×#67292C22semidirect
15. C2×D4×D23C22semidirect
20. C22×D92C22semidirect
21. C22×C8×C23C22semidirect
22. C23×#117C22semidirect
24. C2×#67274C22semidirect
25. C2×#67272C22semidirect
26. C2×#67267C22semidirect
28. C2×D184C22semidirect
29. #10625066C22semidirect
33. C23×#103C22semidirect
34. C23×#102C22semidirect
35. (C2×D23)⋊C8C22semidirect
36. #10624991C22semidirect
41. #41×C23D4semidirect
66. C2×D92D4semidirect
152. #416D23semidirect
178. C22×C8C2×D23semidirect
183. #117C2×D23semidirect
184. #103C2×D23semidirect
186. #102C2×D23semidirect
188. C23#3008semidirect
198. #41D92semidirect

Maximal subgroups

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

subgroupindex
C2×#106251602
C2×#106251202
C22×D1842
#130929862
#130925652
#130925522
#130924212
#130923832
C23×#4162
#300823