Finite Group Databasebeta

← #44402657

Subgroups — #44402657

Order 29056.

660 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C22⋊D227)⋊C4C211yesmaximal, characteristic
2C2×#20300245C211yesmaximal, characteristic, complemented
3C22×Dic227⋊C4C211yesmaximal, characteristic, complemented
4#28408923C211yesmaximal, characteristic, complemented
5#28408921C211yesmaximal, characteristic, complemented
6#28408916C211yesmaximal, characteristic, complemented
7#28408914C211yesmaximal, characteristic, complemented
8#28408911C211yesmaximal, characteristic
9#28408909C211yesmaximal, characteristic
10#28408896C211yesmaximal, characteristic, complemented
11#28408891C211yesmaximal, characteristic, complemented
12#28408890C211yesmaximal, characteristic, complemented
13C4×Dic227⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic227C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C227C211yesmaximal, characteristic, complemented
16C22×C22⋊D227C2211yescharacteristic
17C23×Dic227C2211yescharacteristic
18C23×Dic227C2211yescharacteristic
19C22×C4×D227C2211yescharacteristic
20(C22⋊D227)⋊C4C418yescomplemented
21#20300245—24no—
22C2×Dic227⋊C4—24no—
23C2×Dic227⋊C4C2211yescharacteristic
24C2×Dic227⋊C4C2211yescharacteristic
25C2×Dic227⋊C4C2211yescharacteristic
26C2×Dic227⋊C4C2211yescharacteristic
27C2×Dic227⋊C4C2211yescharacteristic
28C2×Dic227⋊C4C2211yescharacteristic
29C2×Dic227⋊C4C2211yescharacteristic
30C2×C22⋊Dic227C2211yescharacteristic
31C2×(C2×D227)⋊C4C2211yescharacteristic
32C2×(C2×D227)⋊C4C2211yescharacteristic
33C2×(C2×D227)⋊C4C2211yescharacteristic
34C2×(C2×D227)⋊C4C2211yescharacteristic
35C2×(C2×D227)⋊C4C2211yescharacteristic
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
47C23×C4×C227C2211yescharacteristic
48C2×C22⋊C4×C227C2211yescharacteristic
49C2×C22⋊C4×C227C2211yescharacteristic
50C2×C42×C227C2211yescharacteristic
51C2×C42×C227C2211yescharacteristic
52#95×C227C2211yescharacteristic
53#95×C227C2211yescharacteristic
54C23×D227C2311yescharacteristic
55C2×C22⋊D227—22no—
56C2×C22⋊D227—22no—
57C2×C22⋊D227C2×C414yes—
58C22×Dic227—22no—
59C22×Dic227—22no—
60C22×Dic227—22no—
61C22×Dic227—22no—
62C22×Dic227—41no—
63C22×Dic227C2×C414yes—
64C22×Dic227C2311yescharacteristic
65C22×Dic227C2311yescharacteristic
66C22×Dic227C2311yescharacteristic
67C22×Dic227C2311yescharacteristic
68C22×Dic227C2311yescharacteristic
69C22×Dic227C2311yescharacteristic
70C22×Dic227C2311yescharacteristic
71C2×C4×D227—22no—
72C2×C4×D227—22no—
73C2×C4×D227C2×C414yescomplemented
74Dic227⋊C4—22no—
75Dic227⋊C4—22no—
76Dic227⋊C4—22no—
77Dic227⋊C4—22no—
78Dic227⋊C4—22no—
79Dic227⋊C4C2×C414yescomplemented
80Dic227⋊C4D414yescomplemented
81C22⋊Dic227—22no—
82(C2×D227)⋊C4—22no—
83(C2×D227)⋊C4—22no—
84(C2×D227)⋊C4—41no—
85(C2×D227)⋊C4—41no—
86(C2×D227)⋊C4C2×C414yescomplemented
87C4×Dic227—22no—
88C4×Dic227—41no—
89C4×Dic227C2×C414yescomplemented
90C24×C227C2311yescharacteristic
91C22×C4×C227—22no—
92C22×C4×C227—22no—
93C22×C4×C227—41no—
94C22×C4×C227C2311yescharacteristic
95C22×C4×C227C2311yescharacteristic
96C22×C4×C227C2311yescharacteristic
97C22×C4×C227C2311yescharacteristic
98C22×C4×C227C2311yescharacteristic
99C22×C4×C227C2311yescharacteristic
100C22⋊C4×C227—22no—
101C22⋊C4×C227C2×C414yes—
102C42×C227—22no—
103C42×C227—41no—
104C22×D227—21no—
105C22×D227—21no—
106C22×D227—21no—
107C22×D227—21no—
108C22×D227—21no—
109C22×D227—21no—
110C22×D227C22×C412yes—
111C22⋊D227—28no—
112C2×Dic227—21no—
113C2×Dic227—21no—
114C2×Dic227—21no—
115C2×Dic227—21no—
116C2×Dic227—21no—
117C2×Dic227—21no—
118C2×Dic227—28no—
119C2×Dic227—21no—
120C2×Dic227—21no—
121C2×Dic227—21no—
122C2×Dic227—21no—
123C2×Dic227—21no—
124C2×Dic227—21no—
125C2×Dic227—21no—
126C2×Dic227—21no—
127C2×Dic227—21no—
128C2×Dic227—44no—
129C2×Dic227C22×C412yes—
130C2×Dic227C22×C412yes—
131C2×Dic227C22×C412yes—
132C2×Dic227C2×D412yes—
133C2×Dic227C2×D412yes—
134C2×Dic227#4112yes—
135C4×D227—28no—
136C23×C227—21no—
137C23×C227—21no—
138C23×C227—21no—
139C23×C227—21no—
140C23×C227C22×C412yes—
141C23×C227#4112yes—
142C23×C227#4112yes—
143C23×C227C2411yescharacteristic
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×C227—21no—
155C2×C4×C227—21no—
156C2×C4×C227—21no—
157C2×C4×C227—21no—
158C2×C4×C227—21no—
159C2×C4×C227—44no—
160C2×C4×C227C22×C412yes—
161C2×C4×C227C22×C412yes—
162C2×C4×C227C2×D412yes—
163C22×C227—22no—
164C22×C227—22no—
165C22×C227—22no—
166C22×C227—22no—
167C22×C227—22no—
168C22×C227—22no—
169C22×C227#11714yescomplemented
170C22×C227C23×C411yescharacteristic
171C22×C227C22×D411yescharacteristic
172C22×C227C2×#4111yescharacteristic
173C22×C227C2×#4111yescharacteristic
174C22×C227C2×#4111yescharacteristic
175C22×C227H(5,𝔽2)11yescharacteristic
176C22×C2272−1+411yescharacteristic
177C2×D227—22no—
178C2×D227—22no—
179C2×D227—22no—
180C2×D227—41no—
181C2×D227—41no—
182C2×D227—41no—
183C2×D227—41no—
184C4×C227—22no—
185C4×C227—22no—
186C4×C227—22no—
187C4×C227—41no—
188C4×C227—41no—
189C4×C227—41no—
190Dic227—22no—
191Dic227—22no—
192Dic227—22no—
193Dic227—41no—
194Dic227—41no—
195Dic227—41no—
196Dic227C4×D414yes—
197C2×C227—24no—
198C2×C227C2×#11711yescharacteristic
199C2×C227C2×C4×D411yescharacteristic
200C2×C227#52011yescharacteristic
201C2×C227#54611yescharacteristic
202C2×C227#54811yescharacteristic
203C2×C227#55211yescharacteristic
204C2×C227#55311yescharacteristic
205D227—42no—
206C227#235211yescharacteristic, complemented, Sylow
207#2352—2271nomaximal, Sylow
208C2×C4×D4—2271no—
209C2×#117—2271no—
210C22×C4⋊C4—2271no—
211#388—2271no—
212#388—2271no—
213#387—2271no—
214#387—2271no—
215#386—2271no—
216#386—2271no—
217#382—2271no—
218#380—2271no—
219#380—2271no—
220C4×C4⋊C4—2271no—
221C4×C22⋊C4—2271no—
222C4×C22⋊C4D22711yescharacteristic, complemented
223C22×D4—2271no—
224C23×C4—2271no—
225C23×C4—2271no—
226C23×C4—2271no—
227C23×C4C2×D22711yescharacteristic
228C4×D4—2278no—
229#117—4544no—
230C2×C4⋊C4—4544no—
231C2×C4⋊C4—2271no—
232C2×C4⋊C4—2271no—
233C2×C4⋊C4—2271no—
234C2×C4⋊C4—2271no—
235C2×C4⋊C4—2271no—
236C2×C4⋊C4—2271no—
237C2×C4⋊C4—2271no—
238C2×C22⋊C4—2271no—
239C2×C22⋊C4—2271no—
240C2×C22⋊C4—2271no—
241C2×C22⋊C4—2271no—
242C2×C22⋊C4—2271no—
243C2×C22⋊C4—2271no—
244C2×C22⋊C4C2×D22711yescharacteristic
245C2×C22⋊C4C2×D22711yescharacteristic
246C2×C42—2271no—
247C2×C42—2271no—
248C2×C42—2271no—
249C2×C42C2×D22711yescharacteristic
250C2×C42C2×D22711yescharacteristic
251#95—2271no—
252#95—2271no—
253#95—2271no—
254#95—2271no—
255#95—2271no—
256#95—2271no—
257#95—2271no—
258#95—2271no—
259#95C2×D22711yescharacteristic
260#95C2×D22711yescharacteristic
261C24—2271no—
262C24C22×D22711yescharacteristic
263C2×D4—4542no—
264C2×D4—4542no—
265C2×D4—2274no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—9081no—
270C22×C4—4542no—
271C22×C4—4542no—
272C22×C4—4542no—
273C22×C4—4542no—
274C22×C4—4542no—
275C22×C4—4542no—
276C22×C4—2271no—
277C22×C4—2271no—
278C22×C4—2271no—
279C22×C4—2274no—
280C22×C4—2271no—
281C22×C4—2271no—
282C22×C4—2271no—
283C22×C4—2274no—
284C22×C4—2271no—
285C22×C4C22×D22711yescharacteristic
286C22×C4C22×D22711yescharacteristic
287C22×C4C22×D22711yescharacteristic
288C22×C4C22×D22711yescharacteristic
289C22×C4C22×D22711yescharacteristic
290C22×C4C22×D22711yescharacteristic
291C4⋊C4—4542no—
292C4⋊C4—4542no—
293C4⋊C4—4542no—
294C4⋊C4—4542no—
295C4⋊C4—4542no—
296C4⋊C4—2274no—
297C4⋊C4—2274no—
298C22⋊C4—22no—
299C22⋊C4—9081no—
300C22⋊C4—9081no—
301C22⋊C4—4542no—
302C22⋊C4—4542no—
303C22⋊C4—4542no—
304C22⋊C4—2274no—
305C22⋊C4C4×D22714yes—
306C42—22no—
307C42—41no—
308C42—9081no—
309C42—4542no—
310C42—2274no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—4541no—
316C23—4541no—
317C23—4541no—
318C23—4541no—
319C23—4541no—
320C23—4541no—
321C23—2272no—
322C23C2×C4×D22712yes—
323C23#1681388412yes—
324C23#1681388412yes—
325C23C23×D22711yescharacteristic
326D4—4548no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—9084no—
344C2×C4—4548no—
345C2×C4—4541no—
346C2×C4—4541no—
347C2×C4—4541no—
348C2×C4—4541no—
349C2×C4—4541no—
350C2×C4—4541no—
351C2×C4—4541no—
352C2×C4—4541no—
353C2×C4—4541no—
354C2×C4—4541no—
355C2×C4—4541no—
356C2×C4—4541no—
357C2×C4—4541no—
358C2×C4—4541no—
359C2×C4—4541no—
360C2×C4—4548no—
361C2×C4—2272no—
362C2×C4—2272no—
363C2×C4—2272no—
364C2×C4—2272no—
365C2×C4—2272no—
366C2×C4—2272no—
367C2×C4C2×C4×D22712yes—
368C2×C4C2×C4×D22712yes—
369C2×C4D4×D22712yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—9081no—
377C22—9081no—
378C22—9081no—
379C22—9081no—
380C22—4542no—
381C22—4542no—
382C22—4542no—
383C22#2030024514yescomplemented
384C22C22×C4×D22711yescharacteristic
385C22C2×D4×D22711yescharacteristic
386C22#41×D22711yescharacteristic
387C22C2×#1681388411yescharacteristic
388C22C2×#1681388411yescharacteristic
389C22#2030033311yescharacteristic
390C22#2030033511yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—9081no—
398C4—9081no—
399C4—9081no—
400C4—4542no—
401C4—4542no—
402C4—4542no—
403C4—2274no—
404C2—24no—
405C2—9082no—
406C2C2×#2030024511yescharacteristic
407C2C4×D4×D22711yescharacteristic
408C2#2840951111yescharacteristic
409C2#2840967211yescharacteristic
410C2#2840969111yescharacteristic
411C2#2840971511yescharacteristic
412C2#2840973111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutatorC22×C227C23×C4
FittingC4×C22⋊C4×C227C2
socle, cosocleC23×C227C24
prime radicalC22×C22⋊D227C22
hypercenterC4×C22⋊C4D227
hypocenterC227#2352
LCSG ▷ C22×C227 ▷ C227
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#20300245C2semidirect
3. C22×Dic227⋊C4C2semidirect
4. #28408923C2semidirect
5. #28408921C2semidirect
6. #28408916C2semidirect
7. #28408914C2semidirect
10. #28408896C2semidirect
11. #28408891C2semidirect
12. #28408890C2semidirect
13. C4×Dic227⋊C4C2semidirect
14. C22⋊C4×Dic227C2semidirect
15. C4×C22⋊C4×C227C2semidirect
20. (C22⋊D227)⋊C4C4semidirect
73. C2×C4×D227C2×C4semidirect
79. Dic227⋊C4C2×C4semidirect
80. Dic227⋊C4D4semidirect
86. (C2×D227)⋊C4C2×C4semidirect
89. C4×Dic227C2×C4semidirect
169. C22×C227#117semidirect
206. C227#2352semidirect
222. C4×C22⋊C4D227semidirect
383. C22#20300245semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D227)⋊C42
C2×#203002452
C22×Dic227⋊C42
#284089232
#284089212
#284089162
#284089142
#284089112
#284089092
#284088962
#284088912
#284088902
C4×Dic227⋊C42
C22⋊C4×Dic2272
C4×C22⋊C4×C2272
#2352227