Finite Group Databasebeta

← #44402413

Subgroups — #44402413

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×#20300265C211yesmaximal, characteristic
2C2×C4×Dic454C211yesmaximal, characteristic, complemented
3C2×#20300244C211yesmaximal, characteristic
4#28408936C211yesmaximal, characteristic, complemented
5#28408929C211yesmaximal, characteristic, complemented
6#28408928C211yesmaximal, characteristic, complemented
7#28408927C211yesmaximal, characteristic, complemented
8#28408925C211yesmaximal, characteristic
9#28408920C211yesmaximal, characteristic
10#28408920C211yesmaximal, characteristic
11#28408917C211yesmaximal, characteristic, complemented
12#28408916C211yesmaximal, characteristic, complemented
13#28408890C211yesmaximal, characteristic
14#28408883C211yesmaximal, characteristic, complemented
15#388×C227C211yesmaximal, characteristic
16C22×Dic454C2211yescharacteristic
17C23×Dic227C2211yescharacteristic
18#20300265C418yescomplemented
19C4×Dic454—24no—
20#20300244—24no—
21C2×C4⋊Dic227C2211yescharacteristic
22C2×C4⋊Dic227C2211yescharacteristic
23C2×Dic227⋊C4C2211yescharacteristic
24C2×Dic227⋊C4C2211yescharacteristic
25C2×Dic227⋊C4C2211yescharacteristic
26C2×Dic227⋊C4C2211yescharacteristic
27C2×Dic227⋊C4C2211yescharacteristic
28C2×Dic227⋊C4C2211yescharacteristic
29C2×Dic227⋊C4C2211yescharacteristic
30C2×Dic227⋊C4C2211yescharacteristic
31C2×C22⋊Dic227C2211yescharacteristic
32C2×C22⋊Dic227C2211yescharacteristic
33C2×C22⋊Dic227C2211yescharacteristic
34C2×C4×Dic227C2211yescharacteristic
35C2×C4×Dic227C2211yescharacteristic
36C2×C4×Dic227C2211yescharacteristic
37C2×C4×Dic227C2211yescharacteristic
38C2×C4×Dic227C2211yescharacteristic
39C227⋊#95C2211yescharacteristic
40C227⋊#95C2211yescharacteristic
41C227⋊#95C2211yescharacteristic
42C227⋊#95C2211yescharacteristic
43C227⋊#95C2211yescharacteristic
44C227⋊#95C2211yescharacteristic
45C227⋊#95C2211yescharacteristic
46C227⋊#95C2211yescharacteristic
47C2×C4⋊C4×C227C2211yescharacteristic
48C2×C22⋊C4×C227C2211yescharacteristic
49C2×C22⋊C4×C227C2211yescharacteristic
50C2×C22⋊C4×C227C2211yescharacteristic
51C2×C42×C227C2211yescharacteristic
52#95×C227C2211yescharacteristic
53#95×C227C2211yescharacteristic
54C2×Dic454—22no—
55C2×Dic454—22no—
56C2×Dic454C2×C414yes—
57C22×Dic227—22no—
58C22×Dic227—22no—
59C22×Dic227C2×C414yes—
60C22×Dic227C2311yescharacteristic
61C22×Dic227C2311yescharacteristic
62C22×Dic227C2311yescharacteristic
63C22×Dic227C2311yescharacteristic
64C22×Dic227C2311yescharacteristic
65C22×Dic227C2311yescharacteristic
66C22×Dic227C2311yescharacteristic
67C22×Dic227C2311yescharacteristic
68C4⋊Dic227—22no—
69C4⋊Dic227C2×C414yes—
70Dic227⋊C4—22no—
71Dic227⋊C4—22no—
72Dic227⋊C4—22no—
73Dic227⋊C4—22no—
74Dic227⋊C4—22no—
75Dic227⋊C4—41no—
76Dic227⋊C4C2×C414yes—
77Dic227⋊C4C2×C414yes—
78C22⋊Dic227—22no—
79C22⋊Dic227—41no—
80C22⋊Dic227C2×C414yes—
81C4×Dic227—22no—
82C4×Dic227—22no—
83C4×Dic227—41no—
84C4×Dic227—41no—
85C4×Dic227D414yes—
86C24×C227C2311yescharacteristic
87C22×C4×C227C2311yescharacteristic
88C22×C4×C227C2311yescharacteristic
89C22×C4×C227C2311yescharacteristic
90C22×C4×C227C2311yescharacteristic
91C22×C4×C227C2311yescharacteristic
92C22×C4×C227C2311yescharacteristic
93C4⋊C4×C227—41no—
94C22⋊C4×C227—22no—
95C22⋊C4×C227—41no—
96C22⋊C4×C227C2×C414yes—
97C42×C227—22no—
98Dic454—28no—
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—28no—
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×Dic227C22×C412yes—
118C2×Dic227C22×C412yes—
119C2×Dic227C22×C412yes—
120C2×Dic227C22×C412yes—
121C2×Dic227C2×D412yes—
122C2×Dic227C2×D412yes—
123C2×Dic227#4112yes—
124C23×C227—21no—
125C23×C227—21no—
126C23×C227—21no—
127C23×C227—21no—
128C23×C227—21no—
129C23×C227—21no—
130C23×C227C22×C412yes—
131C23×C227C2411yescharacteristic
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×C227C22×C412yes—
148C2×C4×C227C22×C412yes—
149C2×C4×C227C2×D412yes—
150C22×C227—22no—
151C22×C227—22no—
152C22×C227—22no—
153C22×C227—41no—
154C22×C227—41no—
155C22×C227—41no—
156C22×C227—41no—
157C22×C227C23×C411yescharacteristic
158C22×C227C22×D411yescharacteristic
159C22×C227C2×#4111yescharacteristic
160C22×C227H(5,𝔽2)11yescharacteristic
161C22×C2272−1+411yescharacteristic
162C22×C2272−1+411yescharacteristic
163C22×C2272−1+411yescharacteristic
164C4×C227—22no—
165C4×C227—22no—
166C4×C227—22no—
167C4×C227—41no—
168C4×C227—41no—
169C4×C227—41no—
170Dic227—22no—
171Dic227—22no—
172Dic227—22no—
173Dic227—22no—
174Dic227—41no—
175Dic227—41no—
176Dic227—41no—
177Dic227C4×D414yes—
178C2×C227—42no—
179C2×C227C2×C4×D411yescharacteristic
180C2×C227#51911yescharacteristic
181C2×C227#52011yescharacteristic
182C2×C227#53611yescharacteristic
183C2×C227#53711yescharacteristic
184C2×C227#54111yescharacteristic
185C2×C227#54211yescharacteristic
186C227#232311yescharacteristic, complemented, Sylow
187#2323—2271nomaximal, Sylow
188C2×C22⋊Q8—2271no—
189C2×C4×Q8—2271no—
190C2×#117—2271no—
191#391—2271no—
192#389—2271no—
193#389—2271no—
194#389—2271no—
195#388—2271no—
196#388—2271no—
197#388—2271no—
198#388D22711yescharacteristic
199#387—2271no—
200#387—2271no—
201#380—2271no—
202C4×C4⋊C4—2271no—
203C22×Q8—2271no—
204C23×C4—2271no—
205C22⋊Q8—2278no—
206C4×Q8—4544no—
207#117—4544no—
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⋊C4—2271no—
218C2×C4⋊C4C2×D22711yescharacteristic
219C2×C22⋊C4—2271no—
220C2×C22⋊C4—2271no—
221C2×C22⋊C4—2271no—
222C2×C22⋊C4C2×D22711yescharacteristic
223C2×C22⋊C4C2×D22711yescharacteristic
224C2×C22⋊C4C2×D22711yescharacteristic
225C2×C42—2271no—
226C2×C42—2271no—
227C2×C42—2271no—
228C2×C42—2271no—
229C2×C42—2271no—
230C2×C42C2×D22711yescharacteristic
231#95—2271no—
232#95—2271no—
233#95—2271no—
234#95—2271no—
235#95—2271no—
236#95—2271no—
237#95—2271no—
238#95—2271no—
239#95C2×D22711yescharacteristic
240#95C2×D22711yescharacteristic
241C24C22×D22711yescharacteristic
242C2×Q8—4542no—
243C2×Q8—4542no—
244C2×Q8—2274no—
245C22×C4—4542no—
246C22×C4—4542no—
247C22×C4—2271no—
248C22×C4—2271no—
249C22×C4—2271no—
250C22×C4—2274no—
251C22×C4—2271no—
252C22×C4—2271no—
253C22×C4—2271no—
254C22×C4—2271no—
255C22×C4—2271no—
256C22×C4C22×D22711yescharacteristic
257C22×C4C22×D22711yescharacteristic
258C22×C4C22×D22711yescharacteristic
259C22×C4C22×D22711yescharacteristic
260C22×C4C22×D22711yescharacteristic
261C22×C4C22×D22711yescharacteristic
262C4⋊C4—41no—
263C4⋊C4—9081no—
264C4⋊C4—4542no—
265C4⋊C4—4542no—
266C4⋊C4—4542no—
267C4⋊C4—4542no—
268C4⋊C4—4542no—
269C4⋊C4—4542no—
270C4⋊C4—2274no—
271C4⋊C4—2274no—
272C4⋊C4—2274no—
273C22⋊C4—22no—
274C22⋊C4—41no—
275C22⋊C4—9081no—
276C22⋊C4—4542no—
277C22⋊C4—2274no—
278C22⋊C4C4×D22714yes—
279C42—22no—
280C42—9081no—
281C42—9081no—
282C42—4542no—
283C42—4542no—
284C42—2274no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23C2×C4×D22712yes—
292C23C23×D22711yescharacteristic
293Q8—4548no—
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—4541no—
322C2×C4—4541no—
323C2×C4—4541no—
324C2×C4—4541no—
325C2×C4—4541no—
326C2×C4—4548no—
327C2×C4—2272no—
328C2×C4—2272no—
329C2×C4—2272no—
330C2×C4—2272no—
331C2×C4—2272no—
332C2×C4—2272no—
333C2×C4—2272no—
334C2×C4C2×C4×D22712yes—
335C2×C4C2×C4×D22712yes—
336C2×C4D4×D22712yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22C22×C4×D22711yescharacteristic
345C22C2×D4×D22711yescharacteristic
346C22#41×D22711yescharacteristic
347C22#2030033311yescharacteristic
348C22#2030033511yescharacteristic
349C22#2030033511yescharacteristic
350C22#2030033511yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—9081no—
358C4—9081no—
359C4—9081no—
360C4—4542no—
361C4—4542no—
362C4—4542no—
363C4—4542no—
364C4—2274no—
365C2—42no—
366C2C4×D4×D22711yescharacteristic
367C2#2840950911yescharacteristic
368C2#2840951111yescharacteristic
369C2#2840960411yescharacteristic
370C2#2840961111yescharacteristic
371C2#2840963911yescharacteristic
372C2#2840964611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutatorC22×C227C23×C4
Fitting#388×C227C2
socle, cosocleC23×C227C24
prime radicalC24×C227C23
hypercenter#388D227
hypocenterC227#2323
LCSG ▷ C22×C227 ▷ C227
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic454C2semidirect
4. #28408936C2semidirect
5. #28408929C2semidirect
6. #28408928C2semidirect
7. #28408927C2semidirect
11. #28408917C2semidirect
12. #28408916C2semidirect
14. #28408883C2semidirect
18. #20300265C4semidirect
186. C227#2323semidirect

Maximal subgroups

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

subgroupindex
C2×#203002652
C2×C4×Dic4542
C2×#203002442
#284089362
#284089292
#284089282
#284089272
#284089252
#284089202
#284089172
#284089162
#284088902
#284088832
#388×C2272
#2323227