Finite Group Databasebeta

← #15676655

Subgroups — #15676655

Order 2944.

560 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C22⋊D23)⋊C4C211yesmaximal, characteristic
2C2×#10625048C211yesmaximal, characteristic, complemented
3#13092358C211yesmaximal, characteristic, complemented
4#13092352C211yesmaximal, characteristic, complemented
5#13092351C211yesmaximal, characteristic, complemented
6#13092346C211yesmaximal, characteristic, complemented
7#13092345C211yesmaximal, characteristic, complemented
8#13092333C211yesmaximal, characteristic
9#13092332C211yesmaximal, characteristic
10C4⋊C4×Dic23C211yesmaximal, characteristic, complemented
11#13092268C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic23C211yesmaximal, characteristic, complemented
13#13092264C211yesmaximal, characteristic, complemented
14#13092264C211yesmaximal, characteristic, complemented
15C23×#397C211yesmaximal, characteristic, complemented
16C22×C22⋊D23C2211yescharacteristic
17C23×Dic23C2211yescharacteristic
18C22×C4×D23C2211yescharacteristic
19C2×C4⋊C4×C23C2211yescharacteristic
20C2×C22⋊C4×C23C2211yescharacteristic
21C2×C22⋊C4×C23C2211yescharacteristic
22C2×C22⋊C4×C23C2211yescharacteristic
23C2×C22⋊Dic23C2211yescharacteristic
24C2×C22⋊Dic23C2211yescharacteristic
25C2×(C2×D23)⋊C4C2211yescharacteristic
26C2×(C2×D23)⋊C4C2211yescharacteristic
27C2×(C2×D23)⋊C4C2211yescharacteristic
28C2×(C2×D23)⋊C4C2211yescharacteristic
29C2×(C2×D23)⋊C4C2211yescharacteristic
30C2×C4⋊Dic23C2211yescharacteristic
31C2×Dic23⋊C4C2211yescharacteristic
32C2×Dic23⋊C4C2211yescharacteristic
33C2×Dic23⋊C4C2211yescharacteristic
34C2×Dic23⋊C4C2211yescharacteristic
35C2×Dic23⋊C4C2211yescharacteristic
36C2×C4×Dic23C2211yescharacteristic
37C2×C4×Dic23C2211yescharacteristic
38C2×C4×Dic23C2211yescharacteristic
39C2×C4×Dic23C2211yescharacteristic
40C2×C4×Dic23C2211yescharacteristic
41(C22⋊D23)⋊C4—24no—
42#10625048—24no—
43C23×#95C2211yescharacteristic
44C23×#95C2211yescharacteristic
45C23×#95C2211yescharacteristic
46C23⋊#95C2211yescharacteristic
47C23⋊#95C2211yescharacteristic
48C23⋊#95C2211yescharacteristic
49C23⋊#95C2211yescharacteristic
50C23⋊#95C2211yescharacteristic
51C23⋊#95C2211yescharacteristic
52C23⋊#95C2211yescharacteristic
53C24×C23C2311yescharacteristic
54C23×D23C2311yescharacteristic
55C22×C4×C23C2311yescharacteristic
56C22×C4×C23C2311yescharacteristic
57C22×C4×C23C2311yescharacteristic
58C22×C4×C23C2311yescharacteristic
59C22×C4×C23C2311yescharacteristic
60C22×C4×C23C2311yescharacteristic
61C2×C22⋊D23—22no—
62C2×C22⋊D23—22no—
63C2×C22⋊D23—22no—
64C22×Dic23—22no—
65C22×Dic23—22no—
66C22×Dic23D414yes—
67C22×Dic23C2311yescharacteristic
68C22×Dic23C2311yescharacteristic
69C22×Dic23C2311yescharacteristic
70C22×Dic23C2311yescharacteristic
71C22×Dic23C2311yescharacteristic
72C22×Dic23C2311yescharacteristic
73C22×Dic23C2311yescharacteristic
74C2×C4×D23—22no—
75C2×C4×D23—22no—
76C2×C4×D23—22no—
77C4⋊C4×C23—22no—
78C22⋊C4×C23—22no—
79C22⋊C4×C23—22no—
80C22⋊C4×C23—22no—
81C22⋊Dic23—22no—
82C22⋊Dic23—22no—
83(C2×D23)⋊C4—22no—
84(C2×D23)⋊C4—22no—
85(C2×D23)⋊C4—22no—
86(C2×D23)⋊C4—22no—
87(C2×D23)⋊C4—22no—
88C4⋊Dic23—22no—
89Dic23⋊C4—22no—
90Dic23⋊C4—22no—
91Dic23⋊C4—22no—
92Dic23⋊C4—22no—
93Dic23⋊C4—22no—
94C4×Dic23—22no—
95C4×Dic23—22no—
96C4×Dic23—22no—
97C4×Dic23—22no—
98C4×Dic23—22no—
99C23×C23—21no—
100C23×C23—21no—
101C23×C23—21no—
102C23×C23—21no—
103C23×C23—21no—
104C23×C23—21no—
105C23×C23C2×D412yes—
106C23×C23C2411yescharacteristic
107C22×D23—21no—
108C22×D23—21no—
109C22×D23—21no—
110C22×D23—21no—
111C22×D23—21no—
112C22×D23—21no—
113C22×D23#4112yes—
114C2×C4×C23—21no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—21no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23#4112yes—
132C22⋊D23—28no—
133C2×Dic23—21no—
134C2×Dic23—21no—
135C2×Dic23—21no—
136C2×Dic23—21no—
137C2×Dic23—21no—
138C2×Dic23—21no—
139C2×Dic23—28no—
140C2×Dic23—21no—
141C2×Dic23—21no—
142C2×Dic23—21no—
143C2×Dic23—21no—
144C2×Dic23—21no—
145C2×Dic23—21no—
146C2×Dic23—21no—
147C2×Dic23—21no—
148C2×Dic23—21no—
149C2×Dic23C2×D412yes—
150C2×Dic23C2×D412yes—
151C2×Dic23#4112yes—
152C2×Dic23#4112yes—
153C2×Dic23#4112yes—
154C2×Dic23#4112yes—
155C4×D23—44no—
156#2395—231nomaximal, Sylow
157C22×C23—22no—
158C22×C23—22no—
159C22×C23—22no—
160C22×C23—41no—
161C22×C23—41no—
162C22×C23—41no—
163C22×C23—41no—
164C22×C23C22×D411yescharacteristic
165C22×C23C2×#4111yescharacteristic
166C22×C23C2×#4111yescharacteristic
167C22×C23C2×#4111yescharacteristic
168C22×C23C2×#4111yescharacteristic
169C22×C23C2×#4111yescharacteristic
170C22×C23C2×#4111yescharacteristic
171C2×D23—22no—
172C2×D23—22no—
173C2×D23—22no—
174C2×D23—41no—
175C2×D23—41no—
176C2×D23—41no—
177C2×D23—41no—
178C4×C23—22no—
179C4×C23—41no—
180C4×C23—41no—
181C4×C23—41no—
182C4×C23—41no—
183C4×C23—41no—
184Dic23—22no—
185Dic23—22no—
186Dic23—22no—
187Dic23—41no—
188Dic23—41no—
189Dic23—41no—
190Dic23#12314yes—
191C2×C4×D4—231no—
192C2×#117—231no—
193#400—231no—
194#399—231no—
195#398—231no—
196#397—231no—
197#397—231no—
198#397D2311yescharacteristic, complemented
199#394—231no—
200#394—231no—
201C4×C4⋊C4—231no—
202C4×C4⋊C4—231no—
203C4×C22⋊C4—231no—
204C4×C22⋊C4—231no—
205C4×C22⋊C4—231no—
206C2×C23—42no—
207C2×C23C2×#12311yescharacteristic
208C2×C23#52511yescharacteristic
209C2×C23#52911yescharacteristic
210C2×C23#52911yescharacteristic
211C2×C23#52911yescharacteristic
212C2×C23#52911yescharacteristic
213C2×C23#53211yescharacteristic
214D23—42no—
215C22×D4—231no—
216C23×C4—231no—
217C23×C4—231no—
218C4×D4—464no—
219#117—464no—
220C2×C4⋊C4—231no—
221C2×C4⋊C4—231no—
222C2×C4⋊C4—231no—
223C2×C4⋊C4—231no—
224C2×C4⋊C4—231no—
225C2×C4⋊C4—231no—
226C2×C4⋊C4C2×D2311yescharacteristic
227C2×C22⋊C4—231no—
228C2×C22⋊C4—231no—
229C2×C22⋊C4—231no—
230C2×C22⋊C4—231no—
231C2×C22⋊C4—231no—
232C2×C22⋊C4—231no—
233C2×C22⋊C4—231no—
234C2×C22⋊C4C2×D2311yescharacteristic
235C2×C22⋊C4C2×D2311yescharacteristic
236C2×C22⋊C4C2×D2311yescharacteristic
237C2×C42—231no—
238C2×C42—231no—
239C2×C42—231no—
240C2×C42—231no—
241C2×C42—231no—
242#95—231no—
243#95—231no—
244#95—231no—
245#95—231no—
246#95—231no—
247#95—231no—
248#95—231no—
249#95C2×D2311yescharacteristic
250#95C2×D2311yescharacteristic
251#95C2×D2311yescharacteristic
252C23#239511yescharacteristic, complemented, Sylow
253C24—231no—
254C24C22×D2311yescharacteristic
255C2×D4—462no—
256C2×D4—462no—
257C2×D4—462no—
258C22×C4—231no—
259C22×C4—234no—
260C22×C4—231no—
261C22×C4—231no—
262C22×C4—231no—
263C22×C4—231no—
264C22×C4—231no—
265C22×C4—231no—
266C22×C4—462no—
267C22×C4—462no—
268C22×C4—462no—
269C22×C4—462no—
270C22×C4—462no—
271C22×C4C22×D2311yescharacteristic
272C22×C4C22×D2311yescharacteristic
273C22×C4C22×D2311yescharacteristic
274C22×C4C22×D2311yescharacteristic
275C22×C4C22×D2311yescharacteristic
276C22×C4C22×D2311yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—462no—
279C4⋊C4—462no—
280C4⋊C4—462no—
281C4⋊C4—462no—
282C4⋊C4—462no—
283C4⋊C4—462no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—462no—
288C22⋊C4—462no—
289C22⋊C4—462no—
290C22⋊C4—462no—
291C22⋊C4—462no—
292C22⋊C4—462no—
293C22⋊C4—462no—
294C42—462no—
295C42—462no—
296C42—462no—
297C42—462no—
298C42—462no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—232no—
306C23—461no—
307C23—461no—
308C23—461no—
309C23—461no—
310C23—461no—
311C23—461no—
312C23D4×D2312yes—
313C23C23×D2311yescharacteristic
314D4—468no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—232no—
333C2×C4—232no—
334C2×C4—232no—
335C2×C4—232no—
336C2×C4—232no—
337C2×C4—232no—
338C2×C4—468no—
339C2×C4—461no—
340C2×C4—461no—
341C2×C4—461no—
342C2×C4—461no—
343C2×C4—461no—
344C2×C4—461no—
345C2×C4—461no—
346C2×C4—461no—
347C2×C4—461no—
348C2×C4—461no—
349C2×C4—461no—
350C2×C4—461no—
351C2×C4—461no—
352C2×C4—461no—
353C2×C4—461no—
354C2×C4—924no—
355C2×C4#6729012yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—462no—
364C22—462no—
365C22—462no—
366C22—921no—
367C22—921no—
368C22—921no—
369C22—921no—
370C22C2×D4×D2311yescharacteristic
371C22C2×#6729011yescharacteristic
372C22#41×D2311yescharacteristic
373C22#41×D2311yescharacteristic
374C22#41×D2311yescharacteristic
375C22#41×D2311yescharacteristic
376C22#41×D2311yescharacteristic
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—234no—
384C4—462no—
385C4—462no—
386C4—462no—
387C4—921no—
388C4—921no—
389C4—921no—
390C2—42no—
391C2—922no—
392C2#123×D2311yescharacteristic
393C2#1309292511yescharacteristic
394C2#1309294411yescharacteristic
395C2#1309294611yescharacteristic
396C2#1309295011yescharacteristic
397C2#1309295011yescharacteristic
398C2#1309296211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D23
commutator, socle, cosocleC23×C23C24
FittingC23×#397C2
prime radicalC22×C22⋊D23C22
hypercenter#397D23
hypocenterC23#2395
LCSG ▷ C23×C23 ▷ C23
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#10625048C2semidirect
3. #13092358C2semidirect
4. #13092352C2semidirect
5. #13092351C2semidirect
6. #13092346C2semidirect
7. #13092345C2semidirect
10. C4⋊C4×Dic23C2semidirect
11. #13092268C2semidirect
12. C22⋊C4×Dic23C2semidirect
13. #13092264C2semidirect
14. #13092264C2semidirect
15. C23×#397C2semidirect
198. #397D23semidirect
252. C23#2395semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D23)⋊C42
C2×#106250482
#130923582
#130923522
#130923512
#130923462
#130923452
#130923332
#130923322
C4⋊C4×Dic232
#130922682
C22⋊C4×Dic232
#130922642
C23×#3972
#239523