Finite Group Databasebeta

← #44402616

Subgroups — #44402616

Order 29056.

504 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20300244C211yesmaximal, characteristic
2#28408929C211yesmaximal, characteristic, complemented
3#28408927C211yesmaximal, characteristic, complemented
4#28408926C211yesmaximal, characteristic
5#28408925C211yesmaximal, characteristic
6#28408920C211yesmaximal, characteristic
7#28408897C211yesmaximal, characteristic, complemented
8#28408896C211yesmaximal, characteristic, complemented
9#28408890C211yesmaximal, characteristic
10C4×Dic227⋊C4C211yesmaximal, characteristic, complemented
11#28408883C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic227C211yesmaximal, characteristic
13#28408878C211yesmaximal, characteristic, complemented
14#28408877C211yesmaximal, characteristic, complemented
15#388×C227C211yesmaximal, characteristic
16C23×Dic227C2211yescharacteristic
17#20300244C418yescomplemented
18C2×C4⋊Dic227C2211yescharacteristic
19C2×Dic227⋊C4C2211yescharacteristic
20C2×Dic227⋊C4C2211yescharacteristic
21C2×Dic227⋊C4C2211yescharacteristic
22C2×Dic227⋊C4C2211yescharacteristic
23C2×Dic227⋊C4C2211yescharacteristic
24C2×Dic227⋊C4C2211yescharacteristic
25C2×C22⋊Dic227C2211yescharacteristic
26C2×C22⋊Dic227C2211yescharacteristic
27C2×C22⋊Dic227C2211yescharacteristic
28C2×C4×Dic227C2211yescharacteristic
29C2×C4×Dic227C2211yescharacteristic
30C2×C4×Dic227C2211yescharacteristic
31C2×C4×Dic227C2211yescharacteristic
32C2×C4×Dic227C2211yescharacteristic
33C2×C4×Dic227C2211yescharacteristic
34C2×C4×Dic227C2211yescharacteristic
35C227⋊#95C2211yescharacteristic
36C227⋊#95C2211yescharacteristic
37C227⋊#95C2211yescharacteristic
38C227⋊#95C2211yescharacteristic
39C227⋊#95C2211yescharacteristic
40C227⋊#95C2211yescharacteristic
41C227⋊#95C2211yescharacteristic
42C227⋊#95C2211yescharacteristic
43C227⋊#95C2211yescharacteristic
44C227⋊#95C2211yescharacteristic
45C2×C4⋊C4×C227C2211yescharacteristic
46C2×C22⋊C4×C227C2211yescharacteristic
47C2×C22⋊C4×C227C2211yescharacteristic
48C2×C22⋊C4×C227C2211yescharacteristic
49C2×C42×C227C2211yescharacteristic
50#95×C227C2211yescharacteristic
51#95×C227C2211yescharacteristic
52C22×Dic227—22no—
53C22×Dic227—22no—
54C22×Dic227C2×C414yes—
55C22×Dic227C2311yescharacteristic
56C22×Dic227C2311yescharacteristic
57C22×Dic227C2311yescharacteristic
58C22×Dic227C2311yescharacteristic
59C22×Dic227C2311yescharacteristic
60C22×Dic227C2311yescharacteristic
61C22×Dic227C2311yescharacteristic
62C22×Dic227C2311yescharacteristic
63C4⋊Dic227—41no—
64Dic227⋊C4—22no—
65Dic227⋊C4—22no—
66Dic227⋊C4—41no—
67Dic227⋊C4—41no—
68Dic227⋊C4C2×C414yes—
69Dic227⋊C4C2×C414yes—
70C22⋊Dic227—22no—
71C22⋊Dic227—41no—
72C22⋊Dic227C2×C414yes—
73C4×Dic227—22no—
74C4×Dic227—22no—
75C4×Dic227—22no—
76C4×Dic227—22no—
77C4×Dic227—22no—
78C4×Dic227C2×C414yes—
79C4×Dic227C2×C414yes—
80C24×C227C2311yescharacteristic
81C22×C4×C227C2311yescharacteristic
82C22×C4×C227C2311yescharacteristic
83C22×C4×C227C2311yescharacteristic
84C22×C4×C227C2311yescharacteristic
85C22×C4×C227C2311yescharacteristic
86C22×C4×C227C2311yescharacteristic
87C4⋊C4×C227—41no—
88C22⋊C4×C227—22no—
89C22⋊C4×C227—41no—
90C22⋊C4×C227C2×C414yes—
91C42×C227—22no—
92C2×Dic227—21no—
93C2×Dic227—21no—
94C2×Dic227—21no—
95C2×Dic227—21no—
96C2×Dic227—21no—
97C2×Dic227—21no—
98C2×Dic227—21no—
99C2×Dic227—21no—
100C2×Dic227—28no—
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×Dic227C22×C412yes—
111C2×Dic227C22×C412yes—
112C2×Dic227C22×C412yes—
113C2×Dic227C22×C412yes—
114C2×Dic227#4112yes—
115C2×Dic227#4112yes—
116C2×Dic227#4112yes—
117C23×C227—21no—
118C23×C227—21no—
119C23×C227—21no—
120C23×C227—21no—
121C23×C227—21no—
122C23×C227—21no—
123C23×C227C22×C412yes—
124C23×C227C2411yescharacteristic
125C2×C4×C227—21no—
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×C227C22×C412yes—
141C2×C4×C227C22×C412yes—
142C2×C4×C227#4112yes—
143C22×C227—22no—
144C22×C227—22no—
145C22×C227—22no—
146C22×C227—41no—
147C22×C227—41no—
148C22×C227—41no—
149C22×C227—41no—
150C22×C227C23×C411yescharacteristic
151C22×C227C2×#4111yescharacteristic
152C22×C227C2×#4111yescharacteristic
153C22×C227C2×#4111yescharacteristic
154C22×C227C2×#4111yescharacteristic
155C22×C227H(5,𝔽2)11yescharacteristic
156C22×C2272−1+411yescharacteristic
157C4×C227—22no—
158C4×C227—22no—
159C4×C227—22no—
160C4×C227—41no—
161C4×C227—41no—
162C4×C227—41no—
163Dic227—22no—
164Dic227—22no—
165Dic227—22no—
166Dic227—22no—
167Dic227—41no—
168Dic227—41no—
169Dic227—41no—
170Dic227#11714yes—
171C2×C227—42no—
172C2×C227C2×#11711yescharacteristic
173C2×C227C4×#4111yescharacteristic
174C2×C227#52011yescharacteristic
175C2×C227#55211yescharacteristic
176C2×C227#55211yescharacteristic
177C2×C227#55511yescharacteristic
178C2×C227#55511yescharacteristic
179C227#234611yescharacteristic, complemented, Sylow
180#2346—2271nomaximal, Sylow
181C2×#117—2271no—
182#389—2271no—
183#389—2271no—
184#388—2271no—
185#388—2271no—
186#388—2271no—
187#388D22711yescharacteristic
188#382—2271no—
189#382—2271no—
190#380—2271no—
191C4×C4⋊C4—2271no—
192C4×C4⋊C4—2271no—
193C4×C22⋊C4—2271no—
194#376—2271no—
195#376—2271no—
196C23×C4—2271no—
197#117—2278no—
198C2×C4⋊C4—2271no—
199C2×C4⋊C4—2271no—
200C2×C4⋊C4—2271no—
201C2×C4⋊C4—2271no—
202C2×C4⋊C4—2271no—
203C2×C4⋊C4—2271no—
204C2×C4⋊C4—2271no—
205C2×C4⋊C4C2×D22711yescharacteristic
206C2×C22⋊C4—2271no—
207C2×C22⋊C4—2271no—
208C2×C22⋊C4—2271no—
209C2×C22⋊C4C2×D22711yescharacteristic
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—
218C2×C42—2271no—
219C2×C42C2×D22711yescharacteristic
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#95C2×D22711yescharacteristic
231#95C2×D22711yescharacteristic
232C24C22×D22711yescharacteristic
233C22×C4—4542no—
234C22×C4—4542no—
235C22×C4—2271no—
236C22×C4—2271no—
237C22×C4—2271no—
238C22×C4—2274no—
239C22×C4—2271no—
240C22×C4—2271no—
241C22×C4—2271no—
242C22×C4—2271no—
243C22×C4—2271no—
244C22×C4C22×D22711yescharacteristic
245C22×C4C22×D22711yescharacteristic
246C22×C4C22×D22711yescharacteristic
247C22×C4C22×D22711yescharacteristic
248C22×C4C22×D22711yescharacteristic
249C22×C4C22×D22711yescharacteristic
250C4⋊C4—41no—
251C4⋊C4—9081no—
252C4⋊C4—9081no—
253C4⋊C4—9081no—
254C4⋊C4—4542no—
255C4⋊C4—4542no—
256C4⋊C4—2274no—
257C4⋊C4—2274no—
258C22⋊C4—22no—
259C22⋊C4—41no—
260C22⋊C4—9081no—
261C22⋊C4—4542no—
262C22⋊C4—2274no—
263C22⋊C4C4×D22714yes—
264C42—22no—
265C42—4542no—
266C42—4542no—
267C42—4542no—
268C42—4542no—
269C42—4542no—
270C42—2274no—
271C42—2274no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D22712yes—
279C23C23×D22711yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
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—4541no—
296C2×C4—4541no—
297C2×C4—4541no—
298C2×C4—4541no—
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—4548no—
313C2×C4—2272no—
314C2×C4—2272no—
315C2×C4—2272no—
316C2×C4—2272no—
317C2×C4—2272no—
318C2×C4—2272no—
319C2×C4—2272no—
320C2×C4C2×C4×D22712yes—
321C2×C4C2×C4×D22712yes—
322C2×C4#1681388312yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×C4×D22711yescharacteristic
331C22#41×D22711yescharacteristic
332C22#41×D22711yescharacteristic
333C22#41×D22711yescharacteristic
334C22C2×#1681388311yescharacteristic
335C22#2030033311yescharacteristic
336C22#2030033511yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—9081no—
344C4—9081no—
345C4—9081no—
346C4—4542no—
347C4—4542no—
348C4—4542no—
349C4—4542no—
350C4—2274no—
351C2—42no—
352C2#117×D22711yescharacteristic
353C2C4×#1681388311yescharacteristic
354C2#2840951111yescharacteristic
355C2#2840971811yescharacteristic
356C2#2840971911yescharacteristic
357C2#2840974411yescharacteristic
358C2#2840974611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutatorC22×C227C23×C4
Fitting#388×C227C2
socle, cosocleC23×C227C24
prime radicalC24×C227C23
hypercenter#388D227
hypocenterC227#2346
LCSG ▷ C22×C227 ▷ C227
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #28408929C2semidirect
3. #28408927C2semidirect
7. #28408897C2semidirect
8. #28408896C2semidirect
10. C4×Dic227⋊C4C2semidirect
11. #28408883C2semidirect
13. #28408878C2semidirect
14. #28408877C2semidirect
17. #20300244C4semidirect
179. C227#2346semidirect

Maximal subgroups

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

subgroupindex
C2×#203002442
#284089292
#284089272
#284089262
#284089252
#284089202
#284088972
#284088962
#284088902
C4×Dic227⋊C42
#284088832
C22⋊C4×Dic2272
#284088782
#284088772
#388×C2272
#2346227