Finite Group Databasebeta

← #15676561

Subgroups — #15676561

Order 2944.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10625068C211yesmaximal, characteristic
2#13092321C211yesmaximal, characteristic, complemented
3#13092321C211yesmaximal, characteristic, complemented
4#13092315C211yesmaximal, characteristic, complemented
5#13092313C211yesmaximal, characteristic, complemented
6#13092309C211yesmaximal, characteristic
7#13092308C211yesmaximal, characteristic
8#13092307C211yesmaximal, characteristic
9#13092306C211yesmaximal, characteristic
10#13092303C211yesmaximal, characteristic, complemented
11#13092302C211yesmaximal, characteristic, complemented
12#13092300C211yesmaximal, characteristic, complemented
13#13092279C211yesmaximal, characteristic
14#13092276C211yesmaximal, characteristic
15C23×#387C211yesmaximal, characteristic, complemented
16C22×C4×D23C2211yescharacteristic
17C22×Dic46C2211yescharacteristic
18C2×C4⋊C4×C23C2211yescharacteristic
19C2×C4⋊C4×C23C2211yescharacteristic
20C2×C42×C23C2211yescharacteristic
21C2×(C2×D23)⋊C4C2211yescharacteristic
22C2×(C2×D23)⋊C4C2211yescharacteristic
23C2×(C2×D23)⋊C4C2211yescharacteristic
24C2×(C2×D23)⋊C4C2211yescharacteristic
25C2×(C2×D23)⋊C4C2211yescharacteristic
26C2×(C2×D23)⋊C4C2211yescharacteristic
27C2×C4⋊Dic23C2211yescharacteristic
28C2×C4⋊Dic23C2211yescharacteristic
29C2×C4⋊Dic23C2211yescharacteristic
30C2×Dic23⋊C4C2211yescharacteristic
31C2×Dic23⋊C4C2211yescharacteristic
32C2×Dic23⋊C4C2211yescharacteristic
33C2×Dic23⋊C4C2211yescharacteristic
34C2×C4×Dic23C2211yescharacteristic
35C2×C4×Dic23C2211yescharacteristic
36C2×C4×Dic23C2211yescharacteristic
37#10625068C418yescomplemented
38C23×#95C2211yescharacteristic
39C23×#95C2211yescharacteristic
40C23×#95C2211yescharacteristic
41C23×#95C2211yescharacteristic
42C23⋊#95C2211yescharacteristic
43C23⋊#95C2211yescharacteristic
44C23⋊#95C2211yescharacteristic
45C23⋊#95C2211yescharacteristic
46C23⋊#95C2211yescharacteristic
47C23⋊#95C2211yescharacteristic
48C23⋊#95C2211yescharacteristic
49C23⋊#95C2211yescharacteristic
50C23⋊#95C2211yescharacteristic
51C23⋊#95C2211yescharacteristic
52C23×D23C2311yescharacteristic
53C22×C4×C23C2311yescharacteristic
54C22×C4×C23C2311yescharacteristic
55C22×C4×C23C2311yescharacteristic
56C22×C4×C23C2311yescharacteristic
57C22×C4×C23C2311yescharacteristic
58C22×C4×C23C2311yescharacteristic
59C22×C4×C23C2311yescharacteristic
60C22×Dic23C2311yescharacteristic
61C22×Dic23C2311yescharacteristic
62C22×Dic23C2311yescharacteristic
63C22×Dic23C2311yescharacteristic
64C22×Dic23C2311yescharacteristic
65C22×Dic23C2311yescharacteristic
66C22×Dic23C2311yescharacteristic
67C2×C4×D23—41no—
68C2×C4×D23—41no—
69C2×C4×D23C2×C414yes—
70C2×Dic46—41no—
71C2×Dic46—41no—
72C2×Dic46C2×C414yes—
73C4⋊C4×C23—41no—
74C4⋊C4×C23C2×C414yes—
75C42×C23—41no—
76(C2×D23)⋊C4—41no—
77(C2×D23)⋊C4—41no—
78(C2×D23)⋊C4—41no—
79(C2×D23)⋊C4—41no—
80(C2×D23)⋊C4C2×C414yes—
81(C2×D23)⋊C4C2×C414yes—
82C4⋊Dic23—41no—
83C4⋊Dic23C2×C414yes—
84C4⋊Dic23C2×C414yes—
85Dic23⋊C4—41no—
86Dic23⋊C4—41no—
87Dic23⋊C4—41no—
88Dic23⋊C4—41no—
89C4×Dic23—41no—
90C4×Dic23—41no—
91C4×Dic23—41no—
92C23×C23C2411yescharacteristic
93C22×D23—21no—
94C22×D23—21no—
95C22×D23—21no—
96C22×D23—21no—
97C22×D23—21no—
98C22×D23—21no—
99C22×D23C22×C412yes—
100C2×C4×C23—21no—
101C2×C4×C23—21no—
102C2×C4×C23—21no—
103C2×C4×C23—21no—
104C2×C4×C23—21no—
105C2×C4×C23—21no—
106C2×C4×C23—21no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23C22×C412yes—
119C2×C4×C23C22×C412yes—
120C2×C4×C23C22×C412yes—
121C2×Dic23—21no—
122C2×Dic23—21no—
123C2×Dic23—21no—
124C2×Dic23—21no—
125C2×Dic23—21no—
126C2×Dic23—21no—
127C2×Dic23—21no—
128C2×Dic23—21no—
129C2×Dic23—21no—
130C2×Dic23—21no—
131C2×Dic23—21no—
132C2×Dic23—21no—
133C2×Dic23—21no—
134C2×Dic23—21no—
135C2×Dic23—21no—
136C2×Dic23—21no—
137C2×Dic23—21no—
138C2×Dic23—21no—
139C2×Dic23C22×C412yes—
140C2×Dic23C22×C412yes—
141C2×Dic23C22×C412yes—
142C4×D23—44no—
143Dic46—44no—
144#2378—231nomaximal, Sylow
145C22×C23C23×C411yescharacteristic
146C22×C23H(5,𝔽2)11yescharacteristic
147C22×C23H(5,𝔽2)11yescharacteristic
148C22×C23H(5,𝔽2)11yescharacteristic
149C22×C232−1+411yescharacteristic
150C22×C232−1+411yescharacteristic
151C22×C232−1+411yescharacteristic
152C2×D23—22no—
153C2×D23—22no—
154C2×D23—22no—
155C2×D23—41no—
156C2×D23—41no—
157C2×D23—41no—
158C2×D23—41no—
159C4×C23—22no—
160C4×C23—22no—
161C4×C23—22no—
162C4×C23—41no—
163C4×C23—41no—
164C4×C23—41no—
165C4×C23—41no—
166Dic23—22no—
167Dic23—22no—
168Dic23—22no—
169Dic23—41no—
170Dic23—41no—
171Dic23—41no—
172Dic23—41no—
173C2×C22⋊Q8—231no—
174#391—231no—
175#391—231no—
176#389—231no—
177#389—231no—
178#388—231no—
179#388—231no—
180#388—231no—
181#388—231no—
182#387—231no—
183#387—231no—
184#387—231no—
185#387D2311yescharacteristic, complemented
186#381—231no—
187#380—231no—
188C2×C23#51811yescharacteristic
189C2×C23#51911yescharacteristic
190C2×C23#52011yescharacteristic
191C2×C23#56211yescharacteristic
192C2×C23#56211yescharacteristic
193C2×C23#56311yescharacteristic
194C2×C23#56311yescharacteristic
195D23—42no—
196C22×Q8—231no—
197C23×C4—231no—
198C22⋊Q8—238no—
199C2×C4⋊C4—231no—
200C2×C4⋊C4—231no—
201C2×C4⋊C4—231no—
202C2×C4⋊C4—231no—
203C2×C4⋊C4—231no—
204C2×C4⋊C4—231no—
205C2×C4⋊C4—231no—
206C2×C4⋊C4C2×D2311yescharacteristic
207C2×C4⋊C4C2×D2311yescharacteristic
208C2×C22⋊C4—231no—
209C2×C22⋊C4—231no—
210C2×C22⋊C4—231no—
211C2×C22⋊C4—231no—
212C2×C22⋊C4—231no—
213C2×C22⋊C4—231no—
214C2×C42—231no—
215C2×C42—231no—
216C2×C42—231no—
217C2×C42C2×D2311yescharacteristic
218#95—231no—
219#95—231no—
220#95—231no—
221#95—231no—
222#95—231no—
223#95—231no—
224#95—231no—
225#95—231no—
226#95—231no—
227#95—231no—
228#95C2×D2311yescharacteristic
229#95C2×D2311yescharacteristic
230#95C2×D2311yescharacteristic
231#95C2×D2311yescharacteristic
232C23#237811yescharacteristic, complemented, Sylow
233C24—231no—
234C2×Q8—234no—
235C2×Q8—921no—
236C2×Q8—921no—
237C22×C4—231no—
238C22×C4—231no—
239C22×C4—231no—
240C22×C4—234no—
241C22×C4—231no—
242C22×C4—231no—
243C22×C4—231no—
244C22×C4—231no—
245C22×C4—921no—
246C22×C4—921no—
247C22×C4C22×D2311yescharacteristic
248C22×C4C22×D2311yescharacteristic
249C22×C4C22×D2311yescharacteristic
250C22×C4C22×D2311yescharacteristic
251C22×C4C22×D2311yescharacteristic
252C22×C4C22×D2311yescharacteristic
253C22×C4C22×D2311yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—234no—
256C4⋊C4—234no—
257C4⋊C4—921no—
258C4⋊C4—921no—
259C4⋊C4—921no—
260C4⋊C4—921no—
261C4⋊C4—921no—
262C4⋊C4C4×D2314yes—
263C22⋊C4—234no—
264C22⋊C4—234no—
265C22⋊C4—921no—
266C22⋊C4—921no—
267C22⋊C4—921no—
268C22⋊C4—921no—
269C42—41no—
270C42—921no—
271C42—921no—
272C42—921no—
273C23—232no—
274C23—461no—
275C23—461no—
276C23—461no—
277C23—461no—
278C23—461no—
279C23—461no—
280C23C23×D2311yescharacteristic
281Q8—924no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—232no—
301C2×C4—232no—
302C2×C4—232no—
303C2×C4—461no—
304C2×C4—461no—
305C2×C4—461no—
306C2×C4—461no—
307C2×C4—461no—
308C2×C4—461no—
309C2×C4—461no—
310C2×C4—461no—
311C2×C4—461no—
312C2×C4—461no—
313C2×C4—461no—
314C2×C4—461no—
315C2×C4—461no—
316C2×C4—461no—
317C2×C4—461no—
318C2×C4—461no—
319C2×C4—461no—
320C2×C4—461no—
321C2×C4—924no—
322C2×C4C2×C4×D2312yes—
323C2×C4C2×C4×D2312yes—
324C2×C4C2×C4×D2312yes—
325C22—462no—
326C22—462no—
327C22—462no—
328C22—921no—
329C22—921no—
330C22—921no—
331C22—921no—
332C22C22×C4×D2311yescharacteristic
333C22#1062515511yescharacteristic
334C22#1062515811yescharacteristic
335C22#1062516011yescharacteristic
336C22#1062516011yescharacteristic
337C22#1062516111yescharacteristic
338C22#1062516111yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—462no—
347C4—462no—
348C4—462no—
349C4—921no—
350C4—921no—
351C4—921no—
352C4—921no—
353C2—922no—
354C2#1309289111yescharacteristic
355C2#1309289311yescharacteristic
356C2#1309289611yescharacteristic
357C2#1309316311yescharacteristic
358C2#1309316411yescharacteristic
359C2#1309316911yescharacteristic
360C2#1309317111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutatorC22×C23C23×C4
FittingC23×#387C2
socle, cosocleC23×C23C24
prime radicalC23×D23C23
hypercenter#387D23
hypocenterC23#2378
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #13092321C2semidirect
3. #13092321C2semidirect
4. #13092315C2semidirect
5. #13092313C2semidirect
10. #13092303C2semidirect
11. #13092302C2semidirect
12. #13092300C2semidirect
15. C23×#387C2semidirect
37. #10625068C4semidirect
185. #387D23semidirect
232. C23#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#106250682
#130923212
#130923152
#130923132
#130923092
#130923082
#130923072
#130923062
#130923032
#130923022
#130923002
#130922792
#130922762
C23×#3872
#237823