Finite Group Databasebeta

← #44402592

Subgroups — #44402592

Order 29056.

532 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
1C2×#20300244C211yesmaximal, characteristic
2#28408919C211yesmaximal, characteristic, complemented
3#28408919C211yesmaximal, characteristic, complemented
4#28408913C211yesmaximal, characteristic, complemented
5#28408913C211yesmaximal, characteristic, complemented
6#28408907C211yesmaximal, characteristic
7#28408899C211yesmaximal, characteristic, complemented
8#28408897C211yesmaximal, characteristic, complemented
9#28408893C211yesmaximal, characteristic
10C4⋊C4×Dic227C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic227C211yesmaximal, characteristic
12C4×C22⋊Dic227C211yesmaximal, characteristic
13#28408877C211yesmaximal, characteristic, complemented
14C2×C227⋊#95C211yesmaximal, characteristic
15#385×C227C211yesmaximal, characteristic
16C23×Dic227C2211yescharacteristic
17C23×Dic227C2211yescharacteristic
18#20300244C418yescomplemented
19C2×C4⋊Dic227C2211yescharacteristic
20C2×C4⋊Dic227C2211yescharacteristic
21C2×Dic227⋊C4C2211yescharacteristic
22C2×Dic227⋊C4C2211yescharacteristic
23C2×C22⋊Dic227C2211yescharacteristic
24C2×C22⋊Dic227C2211yescharacteristic
25C2×C4×Dic227C2211yescharacteristic
26C2×C4×Dic227C2211yescharacteristic
27C2×C4×Dic227C2211yescharacteristic
28C2×C4×Dic227C2211yescharacteristic
29C2×C4×Dic227C2211yescharacteristic
30C2×C4×Dic227C2211yescharacteristic
31C227⋊#95C2211yescharacteristic
32C227⋊#95C2211yescharacteristic
33C227⋊#95C2211yescharacteristic
34C227⋊#95C2211yescharacteristic
35C227⋊#95C2211yescharacteristic
36C227⋊#95C2211yescharacteristic
37C227⋊#95C2211yescharacteristic
38C227⋊#95C2211yescharacteristic
39C227⋊#95C2211yescharacteristic
40C227⋊#95C2211yescharacteristic
41C227⋊#95C2211yescharacteristic
42C227⋊#95C2211yescharacteristic
43C227⋊#95C2211yescharacteristic
44C227⋊#95C2211yescharacteristic
45C23×C4×C227C2211yescharacteristic
46C2×C4⋊C4×C227C2211yescharacteristic
47C2×C4⋊C4×C227C2211yescharacteristic
48C2×C22⋊C4×C227C2211yescharacteristic
49C2×C22⋊C4×C227C2211yescharacteristic
50#95×C227C2211yescharacteristic
51#95×C227C2211yescharacteristic
52C22×Dic227—22no—
53C22×Dic227—22no—
54C22×Dic227—22no—
55C22×Dic227—22no—
56C22×Dic227—41no—
57C22×Dic227C2×C414yes—
58C22×Dic227C2311yescharacteristic
59C22×Dic227C2311yescharacteristic
60C22×Dic227C2311yescharacteristic
61C22×Dic227C2311yescharacteristic
62C22×Dic227C2311yescharacteristic
63C22×Dic227C2311yescharacteristic
64C22×Dic227C2311yescharacteristic
65C22×Dic227C2311yescharacteristic
66C4⋊Dic227—22no—
67C4⋊Dic227—41no—
68Dic227⋊C4C2×C414yescomplemented
69Dic227⋊C4C2×C414yescomplemented
70C22⋊Dic227—22no—
71C22⋊Dic227C2×C414yes—
72C4×Dic227—22no—
73C4×Dic227—22no—
74C4×Dic227—41no—
75C4×Dic227—41no—
76C4×Dic227C2×C414yescomplemented
77C4×Dic227C2×C414yescomplemented
78C24×C227C2311yescharacteristic
79C22×C4×C227—22no—
80C22×C4×C227—22no—
81C22×C4×C227—41no—
82C22×C4×C227C2311yescharacteristic
83C22×C4×C227C2311yescharacteristic
84C22×C4×C227C2311yescharacteristic
85C22×C4×C227C2311yescharacteristic
86C22×C4×C227C2311yescharacteristic
87C22×C4×C227C2311yescharacteristic
88C4⋊C4×C227—22no—
89C4⋊C4×C227—41no—
90C22⋊C4×C227—22no—
91C22⋊C4×C227C2×C414yes—
92C2×Dic227—28no—
93C2×Dic227—21no—
94C2×Dic227—21no—
95C2×Dic227—21no—
96C2×Dic227—21no—
97C2×Dic227—21no—
98C2×Dic227—21no—
99C2×Dic227—21no—
100C2×Dic227—21no—
101C2×Dic227—21no—
102C2×Dic227—21no—
103C2×Dic227—21no—
104C2×Dic227—21no—
105C2×Dic227—21no—
106C2×Dic227—21no—
107C2×Dic227—21no—
108C2×Dic227—21no—
109C2×Dic227—21no—
110C2×Dic227—21no—
111C2×Dic227—21no—
112C2×Dic227—44no—
113C2×Dic227C22×C412yes—
114C2×Dic227C22×C412yes—
115C2×Dic227C22×C412yes—
116C2×Dic227C22×C412yes—
117C2×Dic227#4112yes—
118C23×C227—21no—
119C23×C227—21no—
120C23×C227—21no—
121C23×C227—21no—
122C23×C227C22×C412yes—
123C23×C227#4112yes—
124C23×C227#4112yes—
125C23×C227C2411yescharacteristic
126C2×C4×C227—21no—
127C2×C4×C227—21no—
128C2×C4×C227—21no—
129C2×C4×C227—21no—
130C2×C4×C227—21no—
131C2×C4×C227—21no—
132C2×C4×C227—21no—
133C2×C4×C227—21no—
134C2×C4×C227—21no—
135C2×C4×C227—21no—
136C2×C4×C227—21no—
137C2×C4×C227—21no—
138C2×C4×C227—21no—
139C2×C4×C227—21no—
140C2×C4×C227—21no—
141C2×C4×C227—44no—
142C2×C4×C227C22×C412yes—
143C2×C4×C227C22×C412yes—
144C2×C4×C227#4112yes—
145C22×C227—22no—
146C22×C227—22no—
147C22×C227—22no—
148C22×C227—22no—
149C22×C227—22no—
150C22×C227—22no—
151C22×C227#11714yes—
152C22×C227C23×C411yescharacteristic
153C22×C227C2×#4111yescharacteristic
154C22×C227C2×#4111yescharacteristic
155C22×C227C2×#4111yescharacteristic
156C22×C227C2×#4111yescharacteristic
157C22×C227H(5,𝔽2)11yescharacteristic
158C22×C2272−1+411yescharacteristic
159C4×C227—22no—
160C4×C227—22no—
161C4×C227—22no—
162C4×C227—41no—
163C4×C227—41no—
164C4×C227—41no—
165Dic227—22no—
166Dic227—22no—
167Dic227—22no—
168Dic227—22no—
169Dic227—22no—
170Dic227—41no—
171Dic227—41no—
172Dic227—41no—
173C2×C227—24no—
174C2×C227C2×#11711yescharacteristic
175C2×C227C4×#4111yescharacteristic
176C2×C227#52011yescharacteristic
177C2×C227#55111yescharacteristic
178C2×C227#55111yescharacteristic
179C2×C227#55211yescharacteristic
180C2×C227#55211yescharacteristic
181C227#234311yescharacteristic, complemented, Sylow
182#2343—2271nomaximal, Sylow
183C2×#117—2271no—
184#387—2271no—
185#387—2271no—
186#387—2271no—
187#387—2271no—
188#385—2271no—
189#385D22711yescharacteristic
190#383—2271no—
191#382—2271no—
192#381—2271no—
193C4×C4⋊C4—2271no—
194C4×C22⋊C4—2271no—
195C4×C22⋊C4—2271no—
196#376—2271no—
197C2×#95—2271no—
198C23×C4—2271no—
199C23×C4—2271no—
200C23×C4C2×D22711yescharacteristic
201#117—2278no—
202C2×C4⋊C4—2271no—
203C2×C4⋊C4—2271no—
204C2×C4⋊C4—2271no—
205C2×C4⋊C4—2271no—
206C2×C4⋊C4C2×D22711yescharacteristic
207C2×C4⋊C4C2×D22711yescharacteristic
208C2×C22⋊C4—2271no—
209C2×C22⋊C4—2271no—
210C2×C22⋊C4C2×D22711yescharacteristic
211C2×C22⋊C4C2×D22711yescharacteristic
212C2×C42—2271no—
213C2×C42—2271no—
214C2×C42—2271no—
215C2×C42—2271no—
216C2×C42—2271no—
217C2×C42—2271no—
218#95—2271no—
219#95—2271no—
220#95—2271no—
221#95—2271no—
222#95—2271no—
223#95—2271no—
224#95—2271no—
225#95—2271no—
226#95—2271no—
227#95—2271no—
228#95—2271no—
229#95—2271no—
230#95—2271no—
231#95—2271no—
232#95C2×D22711yescharacteristic
233#95C2×D22711yescharacteristic
234C24C22×D22711yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—9081no—
239C22×C4—4542no—
240C22×C4—4542no—
241C22×C4—4542no—
242C22×C4—4542no—
243C22×C4—2271no—
244C22×C4—2271no—
245C22×C4—2271no—
246C22×C4—2274no—
247C22×C4—2271no—
248C22×C4—2271no—
249C22×C4—2271no—
250C22×C4—2271no—
251C22×C4—2271no—
252C22×C4C22×D22711yescharacteristic
253C22×C4C22×D22711yescharacteristic
254C22×C4C22×D22711yescharacteristic
255C22×C4C22×D22711yescharacteristic
256C22×C4C22×D22711yescharacteristic
257C22×C4C22×D22711yescharacteristic
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—9081no—
261C4⋊C4—4542no—
262C4⋊C4—2274no—
263C4⋊C4—2274no—
264C22⋊C4—22no—
265C22⋊C4—4542no—
266C22⋊C4—2274no—
267C22⋊C4C4×D22714yes—
268C42—9081no—
269C42—9081no—
270C42—4542no—
271C42—4542no—
272C42—2274no—
273C42—2274no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D22712yes—
279C23#1681388312yes—
280C23#1681388512yes—
281C23C23×D22711yescharacteristic
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—44no—
298C2×C4—9084no—
299C2×C4—4541no—
300C2×C4—4541no—
301C2×C4—4541no—
302C2×C4—4541no—
303C2×C4—4541no—
304C2×C4—4541no—
305C2×C4—4541no—
306C2×C4—4541no—
307C2×C4—4541no—
308C2×C4—4541no—
309C2×C4—4541no—
310C2×C4—4541no—
311C2×C4—4541no—
312C2×C4—4541no—
313C2×C4—4541no—
314C2×C4—4541no—
315C2×C4—4541no—
316C2×C4—4541no—
317C2×C4—4541no—
318C2×C4—4548no—
319C2×C4—2272no—
320C2×C4—2272no—
321C2×C4—2272no—
322C2×C4—2272no—
323C2×C4—2272no—
324C2×C4C2×C4×D22712yes—
325C2×C4C2×C4×D22712yes—
326C2×C4#1681388312yes—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22#2030024614yes—
334C22C22×C4×D22711yescharacteristic
335C22#41×D22711yescharacteristic
336C22C2×#1681388311yescharacteristic
337C22C2×#1681388311yescharacteristic
338C22C2×#1681388511yescharacteristic
339C22#2030033311yescharacteristic
340C22#2030033511yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—9081no—
348C4—9081no—
349C4—9081no—
350C4—4542no—
351C4—4542no—
352C4—4542no—
353C4—4542no—
354C4—4542no—
355C2—24no—
356C2C2×#2030024611yescharacteristic
357C2C4×#1681388311yescharacteristic
358C2#2840951111yescharacteristic
359C2#2840970911yescharacteristic
360C2#2840971311yescharacteristic
361C2#2840971911yescharacteristic
362C2#2840972411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutatorC22×C227C23×C4
Fitting#385×C227C2
socle, cosocleC23×C227C24
prime radicalC24×C227C23
hypercenter#385D227
hypocenterC227#2343
LCSG ▷ C22×C227 ▷ C227
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #28408919C2semidirect
3. #28408919C2semidirect
4. #28408913C2semidirect
5. #28408913C2semidirect
7. #28408899C2semidirect
8. #28408897C2semidirect
10. C4⋊C4×Dic227C2semidirect
13. #28408877C2semidirect
18. #20300244C4semidirect
68. Dic227⋊C4C2×C4semidirect
69. Dic227⋊C4C2×C4semidirect
76. C4×Dic227C2×C4semidirect
77. C4×Dic227C2×C4semidirect
181. C227#2343semidirect

Maximal subgroups

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

subgroupindex
C2×#203002442
#284089192
#284089132
#284089072
#284088992
#284088972
#284088932
C4⋊C4×Dic2272
C22⋊C4×Dic2272
C4×C22⋊Dic2272
#284088772
C2×C227⋊#952
#385×C2272
#2343227