Finite Group Databasebeta

← #44408040

Subgroups — #44408040

Order 29056.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20300278C211yesmaximal, characteristic, complemented
2C22×C4×Dic227C211yesmaximal, characteristic
3#28408929C211yesmaximal, characteristic, complemented
4#28408927C211yesmaximal, characteristic, complemented
5#28408918C211yesmaximal, characteristic, complemented
6#28408917C211yesmaximal, characteristic, complemented
7#28408914C211yesmaximal, characteristic, complemented
8#28408913C211yesmaximal, characteristic, complemented
9#28408907C211yesmaximal, characteristic
10#28408905C211yesmaximal, characteristic
11#28408904C211yesmaximal, characteristic
12#28408896C211yesmaximal, characteristic, complemented
13#28408890C211yesmaximal, characteristic
14#28408890C211yesmaximal, characteristic
15#385×C227C211yesmaximal, characteristic
16C23×Dic227C2211yescharacteristic
17C23×Dic227C2211yescharacteristic
18#20300278—24no—
19C2×C4⋊Dic227C2211yescharacteristic
20C2×C4⋊Dic227C2211yescharacteristic
21C2×Dic227⋊C4C2211yescharacteristic
22C2×Dic227⋊C4C2211yescharacteristic
23C2×Dic227⋊C4C2211yescharacteristic
24C2×Dic227⋊C4C2211yescharacteristic
25C2×Dic227⋊C4C2211yescharacteristic
26C2×Dic227⋊C4C2211yescharacteristic
27C2×Dic227⋊C4C2211yescharacteristic
28C2×Dic227⋊C4C2211yescharacteristic
29C2×C22⋊Dic227C2211yescharacteristic
30C2×C22⋊Dic227C2211yescharacteristic
31C2×C4×Dic227—24no—
32C2×C4×Dic227C2211yescharacteristic
33C2×C4×Dic227C2211yescharacteristic
34C2×C4×Dic227C2211yescharacteristic
35C2×C4×Dic227C2211yescharacteristic
36C227⋊#95C2211yescharacteristic
37C227⋊#95C2211yescharacteristic
38C227⋊#95C2211yescharacteristic
39C227⋊#95C2211yescharacteristic
40C227⋊#95C2211yescharacteristic
41C227⋊#95C2211yescharacteristic
42C227⋊#95C2211yescharacteristic
43C227⋊#95C2211yescharacteristic
44C227⋊#95C2211yescharacteristic
45C227⋊#95C2211yescharacteristic
46C23×C4×C227C2211yescharacteristic
47C2×C4⋊C4×C227C2211yescharacteristic
48C2×C4⋊C4×C227C2211yescharacteristic
49C2×C22⋊C4×C227C2211yescharacteristic
50C2×C22⋊C4×C227C2211yescharacteristic
51#95×C227C2211yescharacteristic
52#95×C227C2211yescharacteristic
53C22×Dic227—22no—
54C22×Dic227—22no—
55C22×Dic227—22no—
56C22×Dic227—22no—
57C22×Dic227—22no—
58C22×Dic227Q814yes—
59C22×Dic227C2311yescharacteristic
60C22×Dic227C2311yescharacteristic
61C22×Dic227C2311yescharacteristic
62C22×Dic227C2311yescharacteristic
63C22×Dic227C2311yescharacteristic
64C22×Dic227C2311yescharacteristic
65C22×Dic227C2311yescharacteristic
66C22×Dic227C2311yescharacteristic
67C4⋊Dic227—22no—
68C4⋊Dic227—22no—
69Dic227⋊C4—22no—
70Dic227⋊C4—22no—
71Dic227⋊C4—22no—
72Dic227⋊C4—22no—
73Dic227⋊C4—22no—
74Dic227⋊C4—22no—
75Dic227⋊C4—22no—
76Dic227⋊C4—22no—
77C22⋊Dic227—22no—
78C22⋊Dic227—22no—
79C4×Dic227—22no—
80C4×Dic227—22no—
81C4×Dic227—22no—
82C4×Dic227D414yes—
83C24×C227C2311yescharacteristic
84C22×C4×C227—22no—
85C22×C4×C227—22no—
86C22×C4×C227—22no—
87C22×C4×C227C2311yescharacteristic
88C22×C4×C227C2311yescharacteristic
89C22×C4×C227C2311yescharacteristic
90C22×C4×C227C2311yescharacteristic
91C22×C4×C227C2311yescharacteristic
92C22×C4×C227C2311yescharacteristic
93C4⋊C4×C227—22no—
94C4⋊C4×C227—22no—
95C22⋊C4×C227—22no—
96C22⋊C4×C227—22no—
97C2×Dic227—28no—
98C2×Dic227—21no—
99C2×Dic227—21no—
100C2×Dic227—21no—
101C2×Dic227—21no—
102C2×Dic227—21no—
103C2×Dic227—21no—
104C2×Dic227—28no—
105C2×Dic227—21no—
106C2×Dic227—21no—
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×Dic227C2×D412yes—
118C2×Dic227C2×D412yes—
119C2×Dic227C2×Q812yes—
120C2×Dic227C2×Q812yes—
121C2×Dic227#4112yes—
122C2×Dic227#4112yes—
123C23×C227—21no—
124C23×C227—21no—
125C23×C227—21no—
126C23×C227—21no—
127C23×C227C2×Q812yes—
128C23×C227#4112yes—
129C23×C227#4112yes—
130C23×C227C2411yescharacteristic
131C2×C4×C227—28no—
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—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×C227C2×D412yes—
149C2×C4×C227#4112yes—
150C22×C227—22no—
151C22×C227—22no—
152C22×C227—22no—
153C22×C227—22no—
154C22×C227—22no—
155C22×C227—22no—
156C22×C227#12514yescomplemented
157C22×C227C22×D411yescharacteristic
158C22×C227C22×Q811yescharacteristic
159C22×C227C2×#4111yescharacteristic
160C22×C227C2×#4111yescharacteristic
161C22×C227C2×#4111yescharacteristic
162C22×C227C2×#4111yescharacteristic
163C22×C227C2×#4111yescharacteristic
164C4×C227—22no—
165C4×C227—22no—
166C4×C227—41no—
167C4×C227—41no—
168C4×C227—41no—
169C4×C227—41no—
170Dic227—22no—
171Dic227—22no—
172Dic227—22no—
173Dic227—41no—
174Dic227—41no—
175Dic227—41no—
176Dic227—41no—
177Dic227C22⋊Q814yes—
178C2×C227—24no—
179C2×C227C2×C22⋊Q811yescharacteristic
180C2×C227#52511yescharacteristic
181C2×C227#52511yescharacteristic
182C2×C227C2×#12511yescharacteristic
183C2×C227#52911yescharacteristic
184C2×C227#52911yescharacteristic
185C2×C227#53311yescharacteristic
186C227#285111yescharacteristic, complemented, Sylow
187#2851—2271nomaximal, Sylow
188C2×#125—2271no—
189C22×C42—2271no—
190#389—2271no—
191#389—2271no—
192#387—2271no—
193#387—2271no—
194#387—2271no—
195#387—2271no—
196#385—2271no—
197#385—2271no—
198#385—2271no—
199#385D22711yescharacteristic
200#382—2271no—
201#380—2271no—
202#380—2271no—
203C23×C4—2271no—
204C23×C4—2271no—
205C23×C4C2×D22711yescharacteristic
206#125—4544no—
207C2×C4⋊C4—2271no—
208C2×C4⋊C4—2271no—
209C2×C4⋊C4—2271no—
210C2×C4⋊C4—2271no—
211C2×C4⋊C4—2271no—
212C2×C4⋊C4—2271no—
213C2×C4⋊C4—2271no—
214C2×C4⋊C4—2271no—
215C2×C4⋊C4—2271no—
216C2×C4⋊C4—2271no—
217C2×C4⋊C4C2×D22711yescharacteristic
218C2×C4⋊C4C2×D22711yescharacteristic
219C2×C22⋊C4—2271no—
220C2×C22⋊C4—2271no—
221C2×C22⋊C4C2×D22711yescharacteristic
222C2×C22⋊C4C2×D22711yescharacteristic
223C2×C42—4544no—
224C2×C42—2271no—
225C2×C42—2271no—
226C2×C42—2271no—
227C2×C42—2271no—
228#95—2271no—
229#95—2271no—
230#95—2271no—
231#95—2271no—
232#95—2271no—
233#95—2271no—
234#95—2271no—
235#95—2271no—
236#95—2271no—
237#95—2271no—
238#95C2×D22711yescharacteristic
239#95C2×D22711yescharacteristic
240C24C22×D22711yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—4542no—
245C22×C4—4542no—
246C22×C4—4542no—
247C22×C4—4542no—
248C22×C4—4542no—
249C22×C4—2271no—
250C22×C4—2271no—
251C22×C4—2271no—
252C22×C4—2271no—
253C22×C4—2274no—
254C22×C4—2271no—
255C22×C4—2271no—
256C22×C4—2271no—
257C22×C4—2271no—
258C22×C4C22×D22711yescharacteristic
259C22×C4C22×D22711yescharacteristic
260C22×C4C22×D22711yescharacteristic
261C22×C4C22×D22711yescharacteristic
262C22×C4C22×D22711yescharacteristic
263C22×C4C22×D22711yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—4542no—
267C4⋊C4—4542no—
268C4⋊C4—4542no—
269C4⋊C4—4542no—
270C4⋊C4—4542no—
271C4⋊C4—4542no—
272C4⋊C4—4542no—
273C4⋊C4—4542no—
274C4⋊C4—4542no—
275C4⋊C4—4542no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—4542no—
279C22⋊C4—4542no—
280C42—4542no—
281C42—4542no—
282C42—4542no—
283C42—2274no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23Q8×D22712yes—
289C23#1681388312yes—
290C23#1681388412yes—
291C23C23×D22711yescharacteristic
292C2×C4—28no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
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—4541no—
319C2×C4—4541no—
320C2×C4—4541no—
321C2×C4—4548no—
322C2×C4—4541no—
323C2×C4—4541no—
324C2×C4—4541no—
325C2×C4—4541no—
326C2×C4—4541no—
327C2×C4—4541no—
328C2×C4—4548no—
329C2×C4—2272no—
330C2×C4—2272no—
331C2×C4—2272no—
332C2×C4—2272no—
333C2×C4—2272no—
334C2×C4—2272no—
335C2×C4D4×D22712yes—
336C2×C4#1681388312yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#2030027814yescomplemented
344C22C2×D4×D22711yescharacteristic
345C22C2×Q8×D22711yescharacteristic
346C22#41×D22711yescharacteristic
347C22#41×D22711yescharacteristic
348C22C2×#1681388311yescharacteristic
349C22C2×#1681388311yescharacteristic
350C22C2×#1681388411yescharacteristic
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—9081no—
358C4—9081no—
359C4—9081no—
360C4—9081no—
361C4—4542no—
362C4—4542no—
363C4—4542no—
364C4—2274no—
365C2—24no—
366C2C22⋊Q8×D22711yescharacteristic
367C2#2840953811yescharacteristic
368C2#2840954111yescharacteristic
369C2C2×#2030027811yescharacteristic
370C2#2840955911yescharacteristic
371C2#2840956211yescharacteristic
372C2#2840958411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting#385×C227C2
prime radicalC24×C227C23
hypercenter#385D227
hypocenterC227#2851
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#20300278C2semidirect
3. #28408929C2semidirect
4. #28408927C2semidirect
5. #28408918C2semidirect
6. #28408917C2semidirect
7. #28408914C2semidirect
8. #28408913C2semidirect
12. #28408896C2semidirect
156. C22×C227#125semidirect
186. C227#2851semidirect
343. C22#20300278semidirect

Maximal subgroups

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

subgroupindex
C2×#203002782
C22×C4×Dic2272
#284089292
#284089272
#284089182
#284089172
#284089142
#284089132
#284089072
#284089052
#284089042
#284088962
#284088902
#385×C2272
#2851227