Finite Group Databasebeta

← #15676682

Subgroups — #15676682

Order 2944.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#10625058C211yesmaximal, characteristic, complemented
2C2×(C22⋊D23)⋊C4C211yesmaximal, characteristic
3#13092358C212yesmaximal, complemented
4#13092343C211yesmaximal, characteristic, complemented
5#13092332C212yesmaximal
6#13092330C211yesmaximal, characteristic, complemented
7#13092329C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic23C212yesmaximal, complemented
9#13092264C212yesmaximal, complemented
10#13092262C211yesmaximal, characteristic, complemented
11C23×#397C211yesmaximal, characteristic, complemented
12C22×C22⋊D23C2211yescharacteristic
13C23×Dic23C2211yescharacteristic
14C22×C4×D23C2211yescharacteristic
15C22×Dic46C2211yescharacteristic
16C2×C4⋊C4×C23C2211yescharacteristic
17C2×C22⋊C4×C23C2211yescharacteristic
18C2×C22⋊C4×C23C2212yes—
19C2×C22⋊Dic23C2212yes—
20C2×(C2×D23)⋊C4C2211yescharacteristic
21C2×(C2×D23)⋊C4C2212yes—
22C2×(C2×D23)⋊C4C2212yes—
23C2×Dic23⋊C4C2211yescharacteristic
24C2×Dic23⋊C4C2212yes—
25C2×C4×Dic23C2212yes—
26C2×C4×Dic23C2212yes—
27C2×C4×Dic23C2211yescharacteristic
28#10625058—24no—
29(C22⋊D23)⋊C4—24no—
30C23×#95C2212yes—
31C23×#95C2211yescharacteristic
32C23⋊#95C2212yes—
33C23⋊#95C2212yes—
34C23⋊#95C2212yes—
35C23⋊#95C2212yes—
36C23⋊#95C2211yescharacteristic
37C24×C23C2311yescharacteristic
38C23×D23C2311yescharacteristic
39C22×C4×C23C2312yes—
40C22×C4×C23C2312yes—
41C22×C4×C23C2311yescharacteristic
42C22×C4×C23C2311yescharacteristic
43C2×C22⋊D23—22no—
44C2×C22⋊D23—22no—
45C2×C22⋊D23—22no—
46C22×Dic23—22no—
47C22×Dic23—24no—
48C22×Dic23C2312yes—
49C22×Dic23C2311yescharacteristic
50C22×Dic23C2311yescharacteristic
51C22×Dic23C2312yes—
52C22×Dic23C2311yescharacteristic
53C2×C4×D23—22no—
54C2×C4×D23—24no—
55C2×Dic46—22no—
56C2×Dic46—24no—
57C4⋊C4×C23—22no—
58C22⋊C4×C23—24no—
59C22⋊C4×C23—22no—
60C22⋊Dic23—24no—
61(C2×D23)⋊C4—24no—
62(C2×D23)⋊C4—22no—
63(C2×D23)⋊C4—24no—
64Dic23⋊C4—22no—
65Dic23⋊C4—24no—
66C4×Dic23—24no—
67C4×Dic23—24no—
68C4×Dic23D414yes—
69C23×C23—21no—
70C23×C23—22no—
71C23×C23—22no—
72C23×C23—21no—
73C23×C23#4112yes—
74C23×C23C2411yescharacteristic
75C22×D23—21no—
76C22×D23—22no—
77C22×D23—22no—
78C22×D23—21no—
79C22×D23#4112yes—
80C2×C4×C23—22no—
81C2×C4×C23—22no—
82C2×C4×C23—21no—
83C2×C4×C23—22no—
84C2×C4×C23—22no—
85C2×C4×C23—22no—
86C2×C4×C23—22no—
87C2×C4×C23—22no—
88C2×C4×C23—22no—
89C2×C4×C23C2×D412yes—
90C22⋊D23—28no—
91C2×Dic23—22no—
92C2×Dic23—22no—
93C2×Dic23—21no—
94C2×Dic23—21no—
95C2×Dic23—21no—
96C2×Dic23—22no—
97C2×Dic23—22no—
98C2×Dic23—22no—
99C2×Dic23—22no—
100C2×Dic23—44no—
101C2×Dic23C2×D412yes—
102C2×Dic23C2×D412yes—
103C2×Dic23#4114yes—
104C2×Dic23#4114yes—
105C4×D23—44no—
106Dic46—28no—
107#2398—231nomaximal, Sylow
108C22×C23—22no—
109C22×C23—24no—
110C22×C23—41no—
111C22×C23—41no—
112C22×C23—42no—
113C22×C23C22×D411yescharacteristic
114C22×C23C2×#4111yescharacteristic
115C22×C23C2×#4111yescharacteristic
116C22×C23C2×#4112yes—
117C22×C23C2×#4112yes—
118C2×D23—22no—
119C2×D23—24no—
120C2×D23—41no—
121C2×D23—41no—
122C2×D23—42no—
123C4×C23—22no—
124C4×C23—42no—
125C4×C23—41no—
126C4×C23—42no—
127Dic23—24no—
128Dic23—22no—
129Dic23—41no—
130Dic23—42no—
131Dic23#12414yes—
132C2×C4×Q8—231no—
133C2×C4×D4—231no—
134#400—232no—
135#397—231no—
136#397D2311yescharacteristic, complemented
137#394—232no—
138#393—231no—
139#393—231no—
140C4×C22⋊C4—232no—
141C4×C22⋊C4—232no—
142#376—231no—
143C2×C23—42no—
144C2×C23#52511yescharacteristic
145C2×C23#52511yescharacteristic
146C2×C23C2×#12411yescharacteristic
147C2×C23#52912yes—
148C2×C23#52912yes—
149D23—42no—
150C22×Q8—231no—
151C22×D4—231no—
152C23×C4—231no—
153C23×C4—231no—
154C4×Q8—464no—
155C4×D4—464no—
156C2×C4⋊C4—231no—
157C2×C4⋊C4—232no—
158C2×C4⋊C4C2×D2311yescharacteristic
159C2×C22⋊C4—232no—
160C2×C22⋊C4—231no—
161C2×C22⋊C4—232no—
162C2×C22⋊C4—232no—
163C2×C22⋊C4C2×D2311yescharacteristic
164C2×C22⋊C4C2×D2312yes—
165C2×C42—231no—
166C2×C42—232no—
167C2×C42—232no—
168#95—231no—
169#95—232no—
170#95—232no—
171#95—232no—
172#95—232no—
173#95C2×D2311yescharacteristic
174#95C2×D2312yes—
175C23#239811yescharacteristic, complemented, Sylow
176C24—231no—
177C24C22×D2311yescharacteristic
178C2×Q8—464no—
179C2×Q8—462no—
180C2×D4—462no—
181C2×D4—462no—
182C2×D4—462no—
183C22×C4—231no—
184C22×C4—232no—
185C22×C4—231no—
186C22×C4—232no—
187C22×C4—231no—
188C22×C4—462no—
189C22×C4—464no—
190C22×C4—462no—
191C22×C4—464no—
192C22×C4C22×D2312yes—
193C22×C4C22×D2311yescharacteristic
194C22×C4C22×D2312yes—
195C22×C4C22×D2311yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—462no—
198C4⋊C4—464no—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—464no—
202C22⋊C4—464no—
203C22⋊C4—464no—
204C22⋊C4—462no—
205C42—234no—
206C42—464no—
207C42—464no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—232no—
213C23—461no—
214C23—461no—
215C23—462no—
216C23—462no—
217C23#6729012yes—
218C23C23×D2311yescharacteristic
219Q8—468no—
220D4—468no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—232no—
231C2×C4—234no—
232C2×C4—232no—
233C2×C4—234no—
234C2×C4—462no—
235C2×C4—461no—
236C2×C4—461no—
237C2×C4—462no—
238C2×C4—461no—
239C2×C4—462no—
240C2×C4—462no—
241C2×C4—462no—
242C2×C4—462no—
243C2×C4—924no—
244C2×C4—924no—
245C2×C4D4×D2312yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—462no—
252C22—464no—
253C22—921no—
254C22—921no—
255C22—922no—
256C22C2×D4×D2311yescharacteristic
257C22C2×#6729011yescharacteristic
258C22#41×D2312yes—
259C22#41×D2311yescharacteristic
260C22#41×D2312yes—
261C4—22no—
262C4—42no—
263C4—41no—
264C4—42no—
265C4—234no—
266C4—464no—
267C4—462no—
268C4—921no—
269C4—922no—
270C2—42no—
271C2—922no—
272C2#1309292511yescharacteristic
273C2#1309292711yescharacteristic
274C2#124×D2311yescharacteristic
275C2#1309294412yes—
276C2#1309295012yes—
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#397C2
prime radicalC22×C22⋊D23C22
hypercenter#397D23
hypocenterC23#2398
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#10625058C2semidirect
3. #13092358C2semidirect
4. #13092343C2semidirect
6. #13092330C2semidirect
7. #13092329C2semidirect
8. C22⋊C4×Dic23C2semidirect
9. #13092264C2semidirect
10. #13092262C2semidirect
11. C23×#397C2semidirect
136. #397D23semidirect
175. C23#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#106250582
C2×(C22⋊D23)⋊C42
#130923582
#130923432
#130923322
#130923302
#130923292
C22⋊C4×Dic232
#130922642
#130922622
C23×#3972
#239823