Finite Group Databasebeta

← #19212320

Subgroups — #19212320

Order 5888.

492 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 111 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#15675940C211yesmaximal, characteristic
2#15673512C212yesmaximal, complemented
3#15671223C211yesmaximal, characteristic, complemented
4#15671217C212yesmaximal
5C23×#1272C211yesmaximal, characteristic, complemented
6C2×#10625118C411yescharacteristic
7C2×Q8×Dic23C411yescharacteristic
8C2×#10625064C411yescharacteristic
9C2×Dic23⋊C8—22no—
10#13092319C2211yescharacteristic
11#13092309—22no—
12#13092282C411yescharacteristic
13#13092264—22no—
14C42×Dic23C2211yescharacteristic
15#13092172C2212yes—
16#13092171—22no—
17C23×#391C2211yescharacteristic
18C23×#336C2212yes—
19C22×Q8×C23C2×C411yescharacteristic
20C22×D92C2×C411yescharacteristic
21C22×C4×D23—21no—
22C22×C8×C23—22no—
23C2×C4⋊C4×C23C2×C411yescharacteristic
24C2×C42×C23C2311yescharacteristic
25#10625118—22no—
26#10625118C814yescomplemented
27Q8×Dic23—42no—
28C2×(C2×D23)⋊C4—22no—
29C2×(C2×D23)⋊C4—22no—
30C2×(C2×D23)⋊C4—21no—
31C2×C4⋊Dic23—21no—
32C2×C4⋊Dic23C2×C411yescharacteristic
33C2×C4×Dic23—21no—
34C2×C4×Dic23C2×C411yescharacteristic
35C2×C4×Dic23C2×C411yescharacteristic
36C2×C4×Dic23D412yes—
37C2×C4×Dic23D412yes—
38C22×C23⋊C8—22no—
39#10625064—42no—
40C23×#95—22no—
41C23⋊#95—22no—
42Dic23⋊C8—44no—
43C23×D23—21no—
44C2×Q8×C23—21no—
45C2×Q8×C23—41no—
46C2×Q8×C23—41no—
47C2×Q8×C23C2×C812yes—
48C22×C4×C23—21no—
49C22×C4×C23—21no—
50C22×C4×C23C22×C411yescharacteristic
51C22×C4×C23C2×D412yes—
52C22×Dic23—21no—
53C22×Dic23C22⋊C411yescharacteristic
54C22×Dic23C22⋊C412yes—
55C22×Dic23C22⋊C411yescharacteristic
56C2×D92—21no—
57C2×D92—41no—
58C2×D92—41no—
59C2×D92C2×C812yes—
60C2×C4×D23—44no—
61C2×C4×D23—42no—
62C2×C8×C23—42no—
63C2×C8×C23—42no—
64C2×C8×C23—42no—
65C4⋊C4×C23—41no—
66C42×C23—41no—
67(C2×D23)⋊C4—28no—
68(C2×D23)⋊C4—42no—
69(C2×D23)⋊C4—42no—
70(C2×D23)⋊C4—42no—
71C4⋊Dic23—42no—
72C4⋊Dic23—41no—
73C4×Dic23—22no—
74C4×Dic23—22no—
75C4×Dic23—22no—
76C4×Dic23—42no—
77C4×Dic23—42no—
78C4×Dic23C2×C812yes—
79C4×Dic23M1612yes—
80C2×C23⋊C8—42no—
81C2×C23⋊C8—42no—
82C2×C23⋊C8—42no—
83#7601—231nomaximal, Sylow
84C23×C23C2×C22⋊C411yescharacteristic
85C22×D23—22no—
86C22×D23—41no—
87C22×D23—41no—
88C22×D23—42no—
89C22×D23—42no—
90Q8×C23—42no—
91Q8×C23—42no—
92C2×C4×C23—22no—
93C2×C4×C23—21no—
94C2×C4×C23—21no—
95C2×C4×C23—22no—
96C2×C4×C23—22no—
97C2×C4×C23—41no—
98C2×C4×C23—41no—
99C2×C4×C23—41no—
100C2×C4×C23—42no—
101C2×C4×C23C22×C811yescharacteristic
102C2×C4×C23C2×M1611yescharacteristic
103C2×C4×C23AGL(1,ℤ/8)12yes—
104C2×C4×C23#13712yes—
105C2×Dic23—22no—
106C2×Dic23—22no—
107C2×Dic23—22no—
108C2×Dic23—21no—
109C2×Dic23—22no—
110C2×Dic23—41no—
111C2×Dic23—42no—
112C2×Dic23C22⋊C814yes—
113C2×Dic23#10012yes—
114D92—42no—
115D92—42no—
116C4×D23—84no—
117C8×C23—82no—
118C23⋊C8—82no—
119C4×#124—231no—
120#1831—232no—
121#1272—231no—
122#1272D2311yescharacteristic, complemented
123#1270—232no—
124C22×C23C2×C22⋊C811yescharacteristic
125C22×C23#41011yescharacteristic
126C22×C23C2×#10011yescharacteristic
127C22×C23#41712yes—
128C22×C23#42112yes—
129C2×D23—42no—
130C2×D23—44no—
131C2×D23—81no—
132C2×D23—82no—
133C2×D23—81no—
134C4×C23—21no—
135C4×C23—21no—
136C4×C23—22no—
137C4×C23—22no—
138C4×C23—42no—
139C4×C23—81no—
140Dic23—24no—
141Dic23—22no—
142Dic23—41no—
143Dic23—81no—
144C2×#124—231no—
145C2×C4×Q8—231no—
146C2×C4×D4—231no—
147C2×C4⋊C8—462no—
148#391C2×D2311yescharacteristic
149#390—231no—
150#388—462no—
151#382—231no—
152C4×C22⋊C4—462no—
153C43—231no—
154#336—232no—
155#336—462no—
156#336C2×D2312yes—
157C2×C23#146011yescharacteristic
158C2×C23#147712yes—
159C2×C23#151212yes—
160C2×C23#153311yescharacteristic
161C2×C23#156011yescharacteristic
162D23—82no—
163C22×Q8C4×D2311yescharacteristic
164C22×D4—231no—
165C23×C4—461no—
166C22×C8—22no—
167C22×C8—462no—
168#124—234no—
169#124—462no—
170C4×Q8—922no—
171C4×D4—922no—
172C2×C4⋊C4—231no—
173C2×C4⋊C4—461no—
174C2×C4⋊C4C4×D2311yescharacteristic
175C2×C22⋊C4—462no—
176C2×C22⋊C4—461no—
177C2×C22⋊C4—462no—
178C2×C42—231no—
179C2×C42—231no—
180C2×C42—232no—
181C2×C42—232no—
182C2×C42—461no—
183C2×C42C22×D2311yescharacteristic
184C4⋊C8—924no—
185#95—22no—
186#95—462no—
187C23#760111yescharacteristic, complemented, Sylow
188C24—461no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8C8×D2312yes—
193C2×D4—232no—
194C2×D4—461no—
195C2×D4—921no—
196C2×D4—921no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—232no—
200C22×C4—231no—
201C22×C4—231no—
202C22×C4—461no—
203C22×C4—922no—
204C22×C4—924no—
205C22×C4C2×C4×D2311yescharacteristic
206C22×C4D4×D2312yes—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—922no—
211C2×C8—922no—
212C2×C8—922no—
213C4⋊C4—41no—
214C4⋊C4—921no—
215C4⋊C4—922no—
216C22⋊C4—468no—
217C22⋊C4—922no—
218C22⋊C4—922no—
219C22⋊C4—922no—
220C42—41no—
221C42—232no—
222C42—232no—
223C42—462no—
224C42—462no—
225C42—462no—
226C42—922no—
227C42—922no—
228C23—462no—
229C23—921no—
230C23—921no—
231C23—922no—
232C23—922no—
233C23C22⋊C4×D2311yescharacteristic
234Q8—42no—
235Q8—42no—
236D4—922no—
237D4—922no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—234no—
248C2×C4—232no—
249C2×C4—462no—
250C2×C4—462no—
251C2×C4—462no—
252C2×C4—461no—
253C2×C4—462no—
254C2×C4—921no—
255C2×C4—922no—
256C2×C4—1844no—
257C2×C4C2×C8×D2311yescharacteristic
258C2×C4M16×D2311yescharacteristic
259C2×C4#1062508512yes—
260C2×C4#1062508912yes—
261C8—82no—
262C8—1842no—
263C22—922no—
264C22—924no—
265C22—1841no—
266C22—1842no—
267C22—1841no—
268C22C22⋊C8×D2311yescharacteristic
269C22#1309238911yescharacteristic
270C22#100×D2311yescharacteristic
271C22#1309241712yes—
272C22#1309243212yes—
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—81no—
279C4—462no—
280C4—464no—
281C4—921no—
282C4—1841no—
283C2—1842no—
284C2#1567171611yescharacteristic
285C2#1567178312yes—
286C2#1567193412yes—
287C2#1567201911yescharacteristic
288C2#1567213111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D23
commutatorC2×C4×C23C22×C8
FrattiniC2×C42C22×D23
FittingC23×#1272C2
socleC23×C23C2×C22⋊C4
hypercenter#1272D23
hypocenterC23#7601
LCSG ▷ C2×C4×C23 ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #15673512C2semidirect
3. #15671223C2semidirect
5. C23×#1272C2semidirect
26. #10625118C8semidirect
122. #1272D23semidirect
187. C23#7601semidirect