Finite Group Databasebeta

← #15686987

Subgroups — #15686987

Order 2944.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13093149C213yesmaximal, complemented
2#13093143C211yesmaximal, characteristic, complemented
3#13093138C213yesmaximal, complemented
4#13093119C213yesmaximal, complemented
5#13093113C213yesmaximal, complemented
6#13093086C211yesmaximal, characteristic, complemented
7#13093078C211yesmaximal, characteristic, complemented
8#13092962C213yesmaximal, complemented
9#13092953C213yesmaximal, complemented
10#13092946C213yesmaximal, complemented
11#41×Dic23C211yesmaximal, characteristic, complemented
12C4×#67292C211yesmaximal, characteristic, complemented
13C4×#67290C213yesmaximal, complemented
14C2×Q8×Dic23C211yesmaximal, characteristic, complemented
15C23×#548C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C22×Q8×C23C2211yescharacteristic
18C2×#67292C2211yescharacteristic
19C2×#67290C2213yes—
20C23×#124C2213yes—
21C23×C22⋊Q8C2213yes—
22C23×#121C2213yes—
23C4×Q8×C23C2211yescharacteristic
24C4×D4×C23C2213yes—
25#10625118C2213yes—
26#10625117C2213yes—
27Q8×Dic23C2213yes—
28Q8×Dic23C2211yescharacteristic
29Q8×Dic23C2211yescharacteristic
30Q8×Dic23C2213yes—
31Q8×Dic23C2211yescharacteristic
32#10625111C2213yes—
33#10625110C2216yes—
34#10625109C2213yes—
35D4×Dic23C2213yes—
36D4×Dic23C2213yes—
37D4×Dic23C2213yes—
38#10625101C2213yes—
39#10625101C2213yes—
40C4×C22⋊D23C2213yes—
41C4×C22⋊D23C2213yes—
42#10625097C2213yes—
43C2×C4⋊Dic23C2213yes—
44C2×C4×Dic23C2213yes—
45C2×C4×Dic23C2213yes—
46#10625069C2216yes—
47#10625064C2213yes—
48#10625063C2213yes—
49#10625063C2213yes—
50#10625061C2213yes—
51#10625059C2213yes—
52#10625058C2213yes—
53#10625057C2216yes—
54#10625056C2216yes—
55#10625054C2216yes—
56(C22⋊D23)⋊C4C2213yes—
57(C22⋊D23)⋊C4C2213yes—
58(C22⋊D23)⋊C4C2216yes—
59#10625050C2216yes—
60#10625048C2211yescharacteristic
61#10625048C2213yes—
62C4×D92C2213yes—
63C42×D23C2213yes—
64#10625041C2213yes—
65C4×Dic46C2213yes—
66#41×C23—24no—
67C2×Q8×C23—24no—
68C2×Q8×C23C2313yes—
69C2×Q8×C23C2311yescharacteristic
70C2×D4×C23C2313yes—
71C2×D4×C23C2313yes—
72C22×C4×C23C2313yes—
73C22×C4×C23C2313yes—
74C2×C22⋊D23C2313yes—
75C2×C22⋊D23C2313yes—
76C2×C22⋊D23C2316yes—
77C22×Dic23C2313yes—
78C22×Dic23C2313yes—
79C22×Dic23C2313yes—
80C22×Dic23C2311yescharacteristic
81#67292—24no—
82#67290—212no—
83C2×D92C2313yes—
84C2×C4×D23C2313yes—
85C2×C4×D23C2313yes—
86C2×C4×D23C2313yes—
87C2×Dic46C2313yes—
88C4⋊C4×C23C2313yes—
89C4⋊C4×C23C2313yes—
90C22⋊C4×C23C2311yescharacteristic
91C22⋊C4×C23C2313yes—
92C22⋊C4×C23C2316yes—
93C42×C23C2313yes—
94C22⋊Dic23C2313yes—
95C22⋊Dic23C2311yescharacteristic
96C22⋊Dic23C2316yes—
97(C2×D23)⋊C4C2316yes—
98(C2×D23)⋊C4C2316yes—
99(C2×D23)⋊C4C2313yes—
100(C2×D23)⋊C4C2313yes—
101C4⋊Dic23C2313yes—
102C4⋊Dic23C2313yes—
103C4⋊Dic23C2313yes—
104C4⋊Dic23C2313yes—
105C4⋊Dic23C2316yes—
106Dic23⋊C4C2311yescharacteristic
107Dic23⋊C4C2313yes—
108Dic23⋊C4C2316yes—
109Dic23⋊C4C2313yes—
110Dic23⋊C4C2311yescharacteristic
111C4×Dic23C2313yes—
112C4×Dic23C2313yes—
113C4×Dic23C2311yescharacteristic
114C4×Dic23C2313yes—
115C4×Dic23C2313yes—
116C4×Dic23C2313yes—
117C4×Dic23C2311yescharacteristic
118C4×Dic23C2313yes—
119C4×Dic23C2313yes—
120C23×C23C2413yes—
121C23×C23C2411yescharacteristic
122C22×D23C2413yes—
123Q8×C23—26no—
124Q8×C23#4114yescomplemented
125D4×C23—26no—
126D4×C23—26no—
127C2×C4×C23—26no—
128C2×C4×C23—26no—
129C2×C4×C23C2411yescharacteristic
130C2×C4×C23C2413yes—
131C2×C4×C23C2411yescharacteristic
132C2×C4×C23C2413yes—
133C2×C4×C23C2413yes—
134C22⋊D23—26no—
135C22⋊D23—26no—
136C22⋊D23—46no—
137C2×Dic23—26no—
138C2×Dic23—43no—
139C2×Dic23—43no—
140C2×Dic23#4114yescomplemented
141C2×Dic23C2411yescharacteristic
142C2×Dic23C2411yescharacteristic
143C2×Dic23C2413yes—
144C2×Dic23C2413yes—
145C2×Dic23C2413yes—
146C2×Dic23C2411yescharacteristic
147C2×Dic23C2411yescharacteristic
148D92—26no—
149C4×D23—26no—
150C4×D23—26no—
151C4×D23—43no—
152Dic46—26no—
153#3480—231nomaximal, Sylow
154C22×C23—21no—
155C22×C23—21no—
156C22×C23—23no—
157C22×C23—23no—
158C22×C23—23no—
159C22×C23C2×#4112yes—
160C22×C23C2511yescharacteristic
161C2×D23—23no—
162C2×D23—23no—
163C2×D23—23no—
164C4×C23—23no—
165C4×C23—21no—
166C4×C23—23no—
167C4×C23—21no—
168C4×C23C2×#4116yes—
169Dic23—23no—
170Dic23—21no—
171Dic23—23no—
172Dic23—21no—
173Dic23—23no—
174Dic23C2×#4112yes—
175Dic23C2×#4112yes—
176#559—233no—
177#557—231no—
178#556—233no—
179#553—233no—
180#553—233no—
181#550—231no—
182#548—231no—
183#548D2311yescharacteristic, complemented
184#532—233no—
185#529—233no—
186#529—233no—
187C4×#41—231no—
188C4×#41—231no—
189C4×#41—233no—
190C2×C4×Q8—231no—
191C2×C23—22no—
192C2×C23—43no—
193C2×C23C22×#4111yescharacteristic
194C2×C23C22×#4111yescharacteristic
195C2×C23#58511yescharacteristic
196D23—43no—
197C2×#41—231no—
198C2×#41—233no—
199C2×#41C2×D2311yescharacteristic
200C22×Q8C2×D2311yescharacteristic
201C4⋊Q8—233no—
202#127—233no—
203#126—236no—
204#126—236no—
205#125—233no—
206#125—233no—
207#124—233no—
208#124—236no—
209#124C2×D2313yes—
210#123—236no—
211#123—236no—
212C22⋊Q8—233no—
213C22⋊Q8C2×D2313yes—
214#121—233no—
215#121—236no—
216#121—233no—
217#121—233no—
218#121C2×D2313yes—
219C4×Q8—231no—
220C4×Q8—233no—
221C4×Q8—231no—
222C4×Q8—233no—
223C4×Q8—231no—
224C4×Q8—233no—
225C4×Q8—233no—
226C4×Q8C2×D2311yescharacteristic
227C4×D4—233no—
228C4×D4—233no—
229C4×D4—233no—
230C4×D4—233no—
231C4×D4—233no—
232C4×D4—233no—
233C4×D4—233no—
234C4×D4—233no—
235C4×D4—236no—
236C4×D4—233no—
237C4×D4C2×D2313yes—
238#117—233no—
239#117—233no—
240#117—233no—
241#117—231no—
242#117—233no—
243C2×C4⋊C4—233no—
244C2×C42—233no—
245C2×C42—233no—
246C2×C42—233no—
247C23#348011yescharacteristic, complemented, Sylow
248#41—24no—
249#41—4612no—
250#41—464no—
251C2×Q8—24no—
252C2×Q8—233no—
253C2×Q8C22×D2313yes—
254C2×Q8C22×D2311yescharacteristic
255C2×D4—233no—
256C2×D4—233no—
257C2×D4—236no—
258C2×D4—233no—
259C2×D4C22×D2313yes—
260C2×D4C22×D2313yes—
261C22×C4—231no—
262C22×C4—233no—
263C22×C4—233no—
264C22×C4—233no—
265C22×C4—233no—
266C22×C4—233no—
267C22×C4—233no—
268C22×C4C22×D2313yes—
269C22×C4C22×D2313yes—
270C4⋊C4—233no—
271C4⋊C4—233no—
272C4⋊C4—236no—
273C4⋊C4—231no—
274C4⋊C4—231no—
275C4⋊C4—233no—
276C4⋊C4—233no—
277C4⋊C4—233no—
278C4⋊C4—233no—
279C4⋊C4—236no—
280C4⋊C4C22×D2313yes—
281C4⋊C4C22×D2313yes—
282C22⋊C4—231no—
283C22⋊C4—233no—
284C22⋊C4—236no—
285C22⋊C4—233no—
286C22⋊C4—236no—
287C22⋊C4—233no—
288C22⋊C4—236no—
289C22⋊C4C22×D2311yescharacteristic
290C22⋊C4C22×D2313yes—
291C22⋊C4C22×D2316yes—
292C42—231no—
293C42—231no—
294C42—233no—
295C42—233no—
296C42—233no—
297C42—233no—
298C42—233no—
299C42—233no—
300C42—233no—
301C42C22×D2313yes—
302C23—233no—
303C23C23×D2313yes—
304C23C23×D2311yescharacteristic
305Q8—26no—
306Q8—466no—
307Q8#6729014yescomplemented
308D4—26no—
309D4—26no—
310D4—466no—
311D4—466no—
312D4—466no—
313D4—926no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—234no—
317C2×C4—231no—
318C2×C4—231no—
319C2×C4—231no—
320C2×C4—231no—
321C2×C4—233no—
322C2×C4—233no—
323C2×C4—233no—
324C2×C4—466no—
325C2×C4—466no—
326C2×C4—466no—
327C2×C4—923no—
328C2×C4—923no—
329C2×C4—923no—
330C2×C4C23×D2311yescharacteristic
331C2×C4C23×D2313yes—
332C2×C4C23×D2311yescharacteristic
333C2×C4C23×D2313yes—
334C2×C4C23×D2313yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—463no—
341C22—463no—
342C22—463no—
343C22#41×D2312yes—
344C22C24×D2311yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—232no—
350C4—232no—
351C4—463no—
352C4—461no—
353C4—463no—
354C4—461no—
355C4—463no—
356C4C2×#6729016yes—
357C2—22no—
358C2—43no—
359C2—923no—
360C2C2×#41×D2311yescharacteristic
361C2C22×#6729011yescharacteristic
362C2#1309325411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D23
commutator, socle, cosocleC22×C23C25
FittingC23×#548C2
hypercenter#548D23
hypocenterC23#3480
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #13093149C2semidirect
2. #13093143C2semidirect
3. #13093138C2semidirect
4. #13093119C2semidirect
5. #13093113C2semidirect
6. #13093086C2semidirect
7. #13093078C2semidirect
8. #13092962C2semidirect
9. #13092953C2semidirect
10. #13092946C2semidirect
11. #41×Dic23C2semidirect
12. C4×#67292C2semidirect
13. C4×#67290C2semidirect
14. C2×Q8×Dic23C2semidirect
15. C23×#548C2semidirect
124. Q8×C23#41semidirect
140. C2×Dic23#41semidirect
183. #548D23semidirect
247. C23#3480semidirect
307. Q8#67290semidirect

Maximal subgroups

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

subgroupindex
#130931492
#130931432
#130931382
#130931192
#130931132
#130930862
#130930782
#130929622
#130929532
#130929462
#41×Dic232
C4×#672922
C4×#672902
C2×Q8×Dic232
C23×#5482
#348023