Finite Group Databasebeta

← #44408051

Subgroups — #44408051

Order 29056.

582 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20300283C211yesmaximal, characteristic, complemented
2C22×C4×Dic227C211yesmaximal, characteristic, complemented
3#28408924C211yesmaximal, characteristic, complemented
4#28408922C211yesmaximal, characteristic, complemented
5#28408921C211yesmaximal, characteristic, complemented
6#28408918C211yesmaximal, characteristic, complemented
7#28408915C211yesmaximal, characteristic, complemented
8#28408913C211yesmaximal, characteristic, complemented
9#28408911C211yesmaximal, characteristic
10#28408909C211yesmaximal, characteristic
11#28408909C211yesmaximal, characteristic
12#28408907C211yesmaximal, characteristic, complemented
13#28408904C211yesmaximal, characteristic, complemented
14#28408899C211yesmaximal, characteristic, complemented
15#385×C227C211yesmaximal, characteristic, complemented
16C22×C22⋊D227C2211yescharacteristic
17C23×Dic227C2211yescharacteristic
18C23×Dic227C2211yescharacteristic
19#20300283—24no—
20C2×C4⋊Dic227C2211yescharacteristic
21C2×C4⋊Dic227C2211yescharacteristic
22C2×Dic227⋊C4C2211yescharacteristic
23C2×Dic227⋊C4C2211yescharacteristic
24C2×C22⋊Dic227C2211yescharacteristic
25C2×(C2×D227)⋊C4C2211yescharacteristic
26C2×(C2×D227)⋊C4C2211yescharacteristic
27C2×(C2×D227)⋊C4C2211yescharacteristic
28C2×(C2×D227)⋊C4C2211yescharacteristic
29C2×(C2×D227)⋊C4C2211yescharacteristic
30C2×(C2×D227)⋊C4C2211yescharacteristic
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
53C23×D227C2311yescharacteristic
54C2×C22⋊D227—22no—
55C2×C22⋊D227—22no—
56C2×C22⋊D227—22no—
57C22×Dic227—22no—
58C22×Dic227—22no—
59C22×Dic227—22no—
60C22×Dic227—22no—
61C22×Dic227—22no—
62C22×Dic227—22no—
63C22×Dic227C2311yescharacteristic
64C22×Dic227C2311yescharacteristic
65C22×Dic227C2311yescharacteristic
66C22×Dic227C2311yescharacteristic
67C22×Dic227C2311yescharacteristic
68C22×Dic227C2311yescharacteristic
69C22×Dic227C2311yescharacteristic
70C4⋊Dic227—22no—
71C4⋊Dic227—22no—
72Dic227⋊C4—22no—
73Dic227⋊C4—22no—
74C22⋊Dic227—22no—
75(C2×D227)⋊C4—22no—
76(C2×D227)⋊C4—22no—
77(C2×D227)⋊C4—22no—
78(C2×D227)⋊C4—22no—
79(C2×D227)⋊C4—22no—
80(C2×D227)⋊C4—22no—
81C4×Dic227—22no—
82C4×Dic227—22no—
83C4×Dic227—22no—
84C4×Dic227D414yescomplemented
85C24×C227C2311yescharacteristic
86C22×C4×C227—22no—
87C22×C4×C227—22no—
88C22×C4×C227—22no—
89C22×C4×C227C2311yescharacteristic
90C22×C4×C227C2311yescharacteristic
91C22×C4×C227C2311yescharacteristic
92C22×C4×C227C2311yescharacteristic
93C22×C4×C227C2311yescharacteristic
94C22×C4×C227C2311yescharacteristic
95C4⋊C4×C227—22no—
96C4⋊C4×C227—22no—
97C22⋊C4×C227—22no—
98C22⋊C4×C227—22no—
99C22×D227—21no—
100C22×D227—21no—
101C22×D227—21no—
102C22×D227—21no—
103C22×D227—21no—
104C22×D227—21no—
105C22×D227—21no—
106C22⋊D227—44no—
107C2×Dic227—28no—
108C2×Dic227—28no—
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×Dic227—21no—
118C2×Dic227—21no—
119C2×Dic227—21no—
120C2×Dic227—21no—
121C2×Dic227—21no—
122C2×Dic227—21no—
123C2×Dic227—21no—
124C2×Dic227—21no—
125C2×Dic227—21no—
126C2×Dic227C2×D412yes—
127C2×Dic227C2×D412yes—
128C2×Dic227#4112yes—
129C2×Dic227#4112yes—
130C23×C227—21no—
131C23×C227—21no—
132C23×C227—21no—
133C23×C227—21no—
134C23×C227#4112yes—
135C23×C227#4112yes—
136C23×C227#4112yes—
137C23×C227C2411yescharacteristic
138C2×C4×C227—28no—
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×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×C227C2×D412yes—
156C2×C4×C227#4112yes—
157C22×C227—22no—
158C22×C227—22no—
159C22×C227—22no—
160C22×C227—22no—
161C22×C227—22no—
162C22×C227—22no—
163C22×C227#12614yescomplemented
164C22×C227C22×D411yescharacteristic
165C22×C227C2×#4111yescharacteristic
166C22×C227C2×#4111yescharacteristic
167C22×C227C2×#4111yescharacteristic
168C22×C227C2×#4111yescharacteristic
169C22×C227C2×#4111yescharacteristic
170C22×C227C2×#4111yescharacteristic
171C2×D227—41no—
172C2×D227—41no—
173C2×D227—41no—
174C2×D227—41no—
175C2×D227—41no—
176C2×D227—41no—
177C2×D227—41no—
178C4×C227—22no—
179C4×C227—22no—
180C4×C227—41no—
181C4×C227—41no—
182C4×C227—41no—
183C4×C227—41no—
184Dic227—22no—
185Dic227—22no—
186Dic227—22no—
187Dic227—22no—
188Dic227—41no—
189Dic227—41no—
190Dic227—41no—
191C2×C227—24no—
192C2×C227#52511yescharacteristic
193C2×C227#52511yescharacteristic
194C2×C227#52511yescharacteristic
195C2×C227C2×#12611yescharacteristic
196C2×C227#52911yescharacteristic
197C2×C227#52911yescharacteristic
198C2×C227#52911yescharacteristic
199D227—81no—
200C227#285211yescharacteristic, complemented, Sylow
201#2852—2271nomaximal, Sylow
202C2×#126—2271no—
203C22×C42—2271no—
204#388—2271no—
205#388—2271no—
206#388—2271no—
207#387—2271no—
208#387—2271no—
209#387—2271no—
210#386—2271no—
211#386—2271no—
212#386—2271no—
213#385—2271no—
214#385—2271no—
215#385D22711yescharacteristic, complemented
216#383—2271no—
217C22×D4—2271no—
218C23×C4—2271no—
219C23×C4—2271no—
220C23×C4C2×D22711yescharacteristic
221#126—4544no—
222C2×C4⋊C4—2271no—
223C2×C4⋊C4—2271no—
224C2×C4⋊C4—2271no—
225C2×C4⋊C4—2271no—
226C2×C4⋊C4C2×D22711yescharacteristic
227C2×C4⋊C4C2×D22711yescharacteristic
228C2×C22⋊C4—2271no—
229C2×C22⋊C4—2271no—
230C2×C22⋊C4—2271no—
231C2×C22⋊C4—2271no—
232C2×C22⋊C4—2271no—
233C2×C22⋊C4—2271no—
234C2×C22⋊C4—2271no—
235C2×C22⋊C4C2×D22711yescharacteristic
236C2×C22⋊C4C2×D22711yescharacteristic
237C2×C42—4544no—
238C2×C42—2271no—
239C2×C42—2271no—
240C2×C42—2271no—
241C2×C42—2271no—
242#95—2271no—
243#95—2271no—
244#95—2271no—
245#95—2271no—
246#95—2271no—
247#95—2271no—
248#95—2271no—
249#95—2271no—
250#95—2271no—
251#95—2271no—
252#95C2×D22711yescharacteristic
253#95C2×D22711yescharacteristic
254C24—2271no—
255C24C22×D22711yescharacteristic
256C2×D4—4542no—
257C2×D4—4542no—
258C2×D4—4542no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—4542no—
263C22×C4—4542no—
264C22×C4—4542no—
265C22×C4—4542no—
266C22×C4—4542no—
267C22×C4—4542no—
268C22×C4—2271no—
269C22×C4—2271no—
270C22×C4—2271no—
271C22×C4—2271no—
272C22×C4—2271no—
273C22×C4—2271no—
274C22×C4—2271no—
275C22×C4C22×D22711yescharacteristic
276C22×C4C22×D22711yescharacteristic
277C22×C4C22×D22711yescharacteristic
278C22×C4C22×D22711yescharacteristic
279C22×C4C22×D22711yescharacteristic
280C22×C4C22×D22711yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—4542no—
284C4⋊C4—4542no—
285C4⋊C4—4542no—
286C4⋊C4—4542no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—4542no—
290C22⋊C4—4542no—
291C22⋊C4—4542no—
292C22⋊C4—4542no—
293C22⋊C4—4542no—
294C22⋊C4—4542no—
295C22⋊C4—4542no—
296C42—4542no—
297C42—4542no—
298C42—4542no—
299C42—2274no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—4541no—
305C23—4541no—
306C23—4541no—
307C23—4541no—
308C23—4541no—
309C23—4541no—
310C23—4541no—
311C23#1681388312yes—
312C23#1681388412yes—
313C23#1681388512yes—
314C23C23×D22711yescharacteristic
315D4—9084no—
316C2×C4—28no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—4541no—
334C2×C4—4541no—
335C2×C4—4541no—
336C2×C4—4541no—
337C2×C4—4541no—
338C2×C4—4541no—
339C2×C4—4541no—
340C2×C4—4541no—
341C2×C4—4541no—
342C2×C4—4548no—
343C2×C4—4548no—
344C2×C4—4541no—
345C2×C4—4541no—
346C2×C4—4541no—
347C2×C4—4541no—
348C2×C4—4541no—
349C2×C4—4541no—
350C2×C4—4541no—
351C2×C4—4541no—
352C2×C4—2272no—
353C2×C4—2272no—
354C2×C4—2272no—
355C2×C4—2272no—
356C2×C4D4×D22712yes—
357C2×C4#1681388312yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—9081no—
365C22—9081no—
366C22—9081no—
367C22—9081no—
368C22—9081no—
369C22—9081no—
370C22—9081no—
371C22#2030028314yescomplemented
372C22C2×D4×D22711yescharacteristic
373C22#41×D22711yescharacteristic
374C22#41×D22711yescharacteristic
375C22C2×#1681388311yescharacteristic
376C22C2×#1681388311yescharacteristic
377C22C2×#1681388411yescharacteristic
378C22C2×#1681388511yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—9081no—
386C4—9081no—
387C4—9081no—
388C4—4542no—
389C4—4542no—
390C4—4542no—
391C4—4542no—
392C2—24no—
393C2—18161no—
394C2#2840953811yescharacteristic
395C2#2840954111yescharacteristic
396C2#2840954511yescharacteristic
397C2C2×#2030028311yescharacteristic
398C2#2840955911yescharacteristic
399C2#2840956211yescharacteristic
400C2#2840956811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting#385×C227C2
prime radicalC22×C22⋊D227C22
hypercenter#385D227
hypocenterC227#2852
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#20300283C2semidirect
2. C22×C4×Dic227C2semidirect
3. #28408924C2semidirect
4. #28408922C2semidirect
5. #28408921C2semidirect
6. #28408918C2semidirect
7. #28408915C2semidirect
8. #28408913C2semidirect
12. #28408907C2semidirect
13. #28408904C2semidirect
14. #28408899C2semidirect
15. #385×C227C2semidirect
84. C4×Dic227D4semidirect
163. C22×C227#126semidirect
200. C227#2852semidirect
215. #385D227semidirect
371. C22#20300283semidirect

Maximal subgroups

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

subgroupindex
C2×#203002832
C22×C4×Dic2272
#284089242
#284089222
#284089212
#284089182
#284089152
#284089132
#284089112
#284089092
#284089072
#284089042
#284088992
#385×C2272
#2852227