Finite Group Databasebeta

← #44408048

Subgroups — #44408048

Order 29056.

582 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20300281C211yesmaximal, characteristic, complemented
2C2×C42×D227C211yesmaximal, characteristic, complemented
3#28408926C211yesmaximal, characteristic, complemented
4#28408925C211yesmaximal, characteristic, complemented
5#28408918C211yesmaximal, characteristic, complemented
6#28408916C211yesmaximal, characteristic, complemented
7#28408915C211yesmaximal, characteristic, complemented
8#28408908C211yesmaximal, characteristic
9#28408908C211yesmaximal, characteristic
10#28408908C211yesmaximal, characteristic
11#28408906C211yesmaximal, characteristic, complemented
12#28408906C211yesmaximal, characteristic, complemented
13#28408903C211yesmaximal, characteristic, complemented
14#28408899C211yesmaximal, characteristic, complemented
15#388×C227C211yesmaximal, characteristic, complemented
16C22×C22⋊D227C2211yescharacteristic
17C22×C4×D227C2211yescharacteristic
18C22×C4×D227C2211yescharacteristic
19C22×C4×D227C2211yescharacteristic
20#20300281—24no—
21C2×C4⋊Dic227C2211yescharacteristic
22C2×C4⋊Dic227C2211yescharacteristic
23C2×Dic227⋊C4C2211yescharacteristic
24C2×Dic227⋊C4C2211yescharacteristic
25C2×Dic227⋊C4C2211yescharacteristic
26C2×C22⋊Dic227C2211yescharacteristic
27C2×C22⋊Dic227C2211yescharacteristic
28C2×C22⋊Dic227C2211yescharacteristic
29C2×(C2×D227)⋊C4C2211yescharacteristic
30C2×(C2×D227)⋊C4C2211yescharacteristic
31C2×(C2×D227)⋊C4C2211yescharacteristic
32C2×C4×Dic227C2211yescharacteristic
33C2×C4×Dic227C2211yescharacteristic
34C2×C4×Dic227C2211yescharacteristic
35C42×D227—24no—
36C227⋊#95C2211yescharacteristic
37C227⋊#95C2211yescharacteristic
38C227⋊#95C2211yescharacteristic
39C227⋊#95C2211yescharacteristic
40C227⋊#95C2211yescharacteristic
41C227⋊#95C2211yescharacteristic
42C227⋊#95C2211yescharacteristic
43C227⋊#95C2211yescharacteristic
44C227⋊#95C2211yescharacteristic
45C227⋊#95C2211yescharacteristic
46C2×C4⋊C4×C227C2211yescharacteristic
47C2×C22⋊C4×C227C2211yescharacteristic
48C2×C22⋊C4×C227C2211yescharacteristic
49C2×C22⋊C4×C227C2211yescharacteristic
50C2×C42×C227C2211yescharacteristic
51#95×C227C2211yescharacteristic
52#95×C227C2211yescharacteristic
53C23×D227C2311yescharacteristic
54C2×C22⋊D227—22no—
55C2×C22⋊D227—22no—
56C2×C22⋊D227—22no—
57C22×Dic227C2311yescharacteristic
58C22×Dic227C2311yescharacteristic
59C22×Dic227C2311yescharacteristic
60C22×Dic227C2311yescharacteristic
61C22×Dic227C2311yescharacteristic
62C22×Dic227C2311yescharacteristic
63C22×Dic227C2311yescharacteristic
64C2×C4×D227—22no—
65C2×C4×D227—22no—
66C2×C4×D227—22no—
67C2×C4×D227—22no—
68C2×C4×D227—22no—
69C2×C4×D227—22no—
70C2×C4×D227—22no—
71C2×C4×D227—22no—
72C2×C4×D227—22no—
73C4⋊Dic227—22no—
74C4⋊Dic227—22no—
75Dic227⋊C4—22no—
76Dic227⋊C4—22no—
77Dic227⋊C4—22no—
78C22⋊Dic227—22no—
79C22⋊Dic227—22no—
80C22⋊Dic227—22no—
81(C2×D227)⋊C4—22no—
82(C2×D227)⋊C4—22no—
83(C2×D227)⋊C4—22no—
84C4×Dic227—22no—
85C4×Dic227—22no—
86C4×Dic227D414yescomplemented
87C24×C227C2311yescharacteristic
88C22×C4×C227C2311yescharacteristic
89C22×C4×C227C2311yescharacteristic
90C22×C4×C227C2311yescharacteristic
91C22×C4×C227C2311yescharacteristic
92C22×C4×C227C2311yescharacteristic
93C22×C4×C227C2311yescharacteristic
94C4⋊C4×C227—22no—
95C22⋊C4×C227—22no—
96C22⋊C4×C227—22no—
97C22⋊C4×C227—22no—
98C42×C227—22no—
99C22×D227—21no—
100C22×D227—21no—
101C22×D227—21no—
102C22×D227—21no—
103C22×D227#4112yes—
104C22×D227#4112yes—
105C22×D227#4112yes—
106C22⋊D227—44no—
107C2×Dic227—21no—
108C2×Dic227—21no—
109C2×Dic227—21no—
110C2×Dic227—21no—
111C2×Dic227—21no—
112C2×Dic227—21no—
113C2×Dic227—21no—
114C2×Dic227—21no—
115C2×Dic227—21no—
116C2×Dic227—21no—
117C2×Dic227—21no—
118C2×Dic227—21no—
119C2×Dic227—21no—
120C2×Dic227—21no—
121C2×Dic227—21no—
122C2×Dic227—21no—
123C2×Dic227—21no—
124C2×Dic227—21no—
125C2×Dic227C2×D412yes—
126C2×Dic227C2×D412yes—
127C2×Dic227#4112yes—
128C4×D227—28no—
129C4×D227—28no—
130C4×D227—28no—
131C23×C227—21no—
132C23×C227—21no—
133C23×C227—21no—
134C23×C227—21no—
135C23×C227—21no—
136C23×C227—21no—
137C23×C227—21no—
138C23×C227C2411yescharacteristic
139C2×C4×C227—21no—
140C2×C4×C227—21no—
141C2×C4×C227—21no—
142C2×C4×C227—21no—
143C2×C4×C227—21no—
144C2×C4×C227—21no—
145C2×C4×C227—21no—
146C2×C4×C227—21no—
147C2×C4×C227—21no—
148C2×C4×C227—21no—
149C2×C4×C227—21no—
150C2×C4×C227—21no—
151C2×C4×C227—21no—
152C2×C4×C227—21no—
153C2×C4×C227—21no—
154C2×C4×C227C2×D412yes—
155C2×C4×C227#4112yes—
156C2×C4×C227#4112yes—
157C22×C227—41no—
158C22×C227—41no—
159C22×C227—41no—
160C22×C227—41no—
161C22×C227—41no—
162C22×C227—41no—
163C22×C227—41no—
164C22×C227C22×D411yescharacteristic
165C22×C227C2×#4111yescharacteristic
166C22×C227C2×#4111yescharacteristic
167C22×C227C2×#4111yescharacteristic
168C22×C227C2×#4111yescharacteristic
169C22×C227C2×#4111yescharacteristic
170C22×C227C2×#4111yescharacteristic
171C2×D227—22no—
172C2×D227—22no—
173C2×D227—22no—
174C2×D227—22no—
175C2×D227—22no—
176C2×D227—22no—
177C2×D227#12614yescomplemented
178C4×C227—22no—
179C4×C227—22no—
180C4×C227—22no—
181C4×C227—41no—
182C4×C227—41no—
183C4×C227—41no—
184Dic227—22no—
185Dic227—22no—
186Dic227—22no—
187Dic227—41no—
188Dic227—41no—
189Dic227—41no—
190Dic227—41no—
191C2×C227—81no—
192C2×C227#52511yescharacteristic
193C2×C227#52511yescharacteristic
194C2×C227#52511yescharacteristic
195C2×C227C2×#12611yescharacteristic
196C2×C227#52911yescharacteristic
197C2×C227#52911yescharacteristic
198C2×C227#52911yescharacteristic
199D227—24no—
200C227#285211yescharacteristic, complemented, Sylow
201#2852—2271nomaximal, Sylow
202C2×#126—2271no—
203C22×C42—2271no—
204#388—2271no—
205#388—2271no—
206#388D22711yescharacteristic, complemented
207#387—2271no—
208#387—2271no—
209#387—2271no—
210#386—2271no—
211#386—2271no—
212#386—2271no—
213#385—2271no—
214#385—2271no—
215#385—2271no—
216#383—2271no—
217C22×D4—2271no—
218C23×C4—2271no—
219C23×C4—2271no—
220C23×C4—2271no—
221#126—4544no—
222C2×C4⋊C4—2271no—
223C2×C4⋊C4—2271no—
224C2×C4⋊C4—2271no—
225C2×C4⋊C4—2271no—
226C2×C4⋊C4—2271no—
227C2×C4⋊C4C2×D22711yescharacteristic
228C2×C22⋊C4—2271no—
229C2×C22⋊C4—2271no—
230C2×C22⋊C4—2271no—
231C2×C22⋊C4—2271no—
232C2×C22⋊C4—2271no—
233C2×C22⋊C4—2271no—
234C2×C22⋊C4C2×D22711yescharacteristic
235C2×C22⋊C4C2×D22711yescharacteristic
236C2×C22⋊C4C2×D22711yescharacteristic
237C2×C42—4544no—
238C2×C42—2271no—
239C2×C42—2271no—
240C2×C42—2271no—
241C2×C42C2×D22711yescharacteristic
242#95—2271no—
243#95—2271no—
244#95—2271no—
245#95—2271no—
246#95—2271no—
247#95—2271no—
248#95—2271no—
249#95—2271no—
250#95—2271no—
251#95—2271no—
252#95C2×D22711yescharacteristic
253#95C2×D22711yescharacteristic
254C24—2271no—
255C24C22×D22711yescharacteristic
256C2×D4—4542no—
257C2×D4—4542no—
258C2×D4—4542no—
259C22×C4—4542no—
260C22×C4—4542no—
261C22×C4—4542no—
262C22×C4—4542no—
263C22×C4—4542no—
264C22×C4—4542no—
265C22×C4—4542no—
266C22×C4—4542no—
267C22×C4—4542no—
268C22×C4—2271no—
269C22×C4—2271no—
270C22×C4—2271no—
271C22×C4—2271no—
272C22×C4—2271no—
273C22×C4—2271no—
274C22×C4—2271no—
275C22×C4C22×D22711yescharacteristic
276C22×C4C22×D22711yescharacteristic
277C22×C4C22×D22711yescharacteristic
278C22×C4C22×D22711yescharacteristic
279C22×C4C22×D22711yescharacteristic
280C22×C4C22×D22711yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—4542no—
283C4⋊C4—4542no—
284C4⋊C4—4542no—
285C4⋊C4—4542no—
286C4⋊C4—4542no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—4542no—
291C22⋊C4—4542no—
292C22⋊C4—4542no—
293C22⋊C4—4542no—
294C22⋊C4—4542no—
295C22⋊C4—4542no—
296C42—22no—
297C42—4542no—
298C42—4542no—
299C42—2274no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—4541no—
308C23—4541no—
309C23—4541no—
310C23—4541no—
311C23—2272no—
312C23—2272no—
313C23—2272no—
314C23C23×D22711yescharacteristic
315D4—9084no—
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—4548no—
332C2×C4—4548no—
333C2×C4—4548no—
334C2×C4—4541no—
335C2×C4—4541no—
336C2×C4—4541no—
337C2×C4—4541no—
338C2×C4—4541no—
339C2×C4—4541no—
340C2×C4—4541no—
341C2×C4—4541no—
342C2×C4—4541no—
343C2×C4—4541no—
344C2×C4—4541no—
345C2×C4—4541no—
346C2×C4—4541no—
347C2×C4—4541no—
348C2×C4—4541no—
349C2×C4—4541no—
350C2×C4—4541no—
351C2×C4—4541no—
352C2×C4—2272no—
353C2×C4—2272no—
354C2×C4—2272no—
355C2×C4D4×D22712yes—
356C2×C4#1681388312yes—
357C2×C4#1681388412yes—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—4542no—
366C22—4542no—
367C22—4542no—
368C22—4542no—
369C22—4542no—
370C22—4542no—
371C22—2274no—
372C22C2×D4×D22711yescharacteristic
373C22#41×D22711yescharacteristic
374C22#41×D22711yescharacteristic
375C22#41×D22711yescharacteristic
376C22#41×D22711yescharacteristic
377C22C2×#1681388311yescharacteristic
378C22C2×#1681388411yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—9081no—
386C4—9081no—
387C4—9081no—
388C4—9081no—
389C4—4542no—
390C4—4542no—
391C4—4542no—
392C2—81no—
393C2—4544no—
394C2#2840954011yescharacteristic
395C2#2840954111yescharacteristic
396C2#2840954211yescharacteristic
397C2#126×D22711yescharacteristic
398C2#2840956011yescharacteristic
399C2#2840956111yescharacteristic
400C2#2840956411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting#388×C227C2
prime radicalC22×C22⋊D227C22
hypercenter#388D227
hypocenterC227#2852
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#20300281C2semidirect
2. C2×C42×D227C2semidirect
3. #28408926C2semidirect
4. #28408925C2semidirect
5. #28408918C2semidirect
6. #28408916C2semidirect
7. #28408915C2semidirect
11. #28408906C2semidirect
12. #28408906C2semidirect
13. #28408903C2semidirect
14. #28408899C2semidirect
15. #388×C227C2semidirect
86. C4×Dic227D4semidirect
177. C2×D227#126semidirect
200. C227#2852semidirect
206. #388D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203002812
C2×C42×D2272
#284089262
#284089252
#284089182
#284089162
#284089152
#284089082
#284089062
#284089032
#284088992
#388×C2272
#2852227