Finite Group Databasebeta

← #44560479

Subgroups — #44560479

Order 32128.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632222C211yesmaximal, characteristic, complemented
2C4×#16860809C211yesmaximal, characteristic, complemented
3#30632030C211yesmaximal, characteristic, complemented
4#30632013C211yesmaximal, characteristic, complemented
5#30631991C211yesmaximal, characteristic, complemented
6#30631910C211yesmaximal, characteristic, complemented
7#30631902C211yesmaximal, characteristic, complemented
8#30631850C211yesmaximal, characteristic, complemented
9#30631833C211yesmaximal, characteristic
10#30631796C211yesmaximal, characteristic, complemented
11#30631793C211yesmaximal, characteristic, complemented
12#30631786C211yesmaximal, characteristic, complemented
13#30631784C211yesmaximal, characteristic, complemented
14#30631743C211yesmaximal, characteristic, complemented
15#478×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×Dic1004C2211yescharacteristic
18C2×#16860799C2211yescharacteristic
19C2×#16860798C2211yescharacteristic
20C2×#16860795C2211yescharacteristic
21#22548710—21no—
22#22548703C2211yescharacteristic
23#22548700—21no—
24#22548695C2211yescharacteristic
25#22548690C2211yescharacteristic
26C4×Dic502C2211yescharacteristic
27#22548680C2211yescharacteristic
28#22548680C2211yescharacteristic
29D4×Dic251C2211yescharacteristic
30#22548677—21no—
31#22548677C2211yescharacteristic
32C4×C22⋊D251—21no—
33(C22⋊D251)⋊C4—21no—
34#22548673—21no—
35#22548673C2211yescharacteristic
36#22548671—21no—
37C2×C4×Dic251—21no—
38C42×D251C2211yescharacteristic
39#22548648C2211yescharacteristic
40#22548646C2211yescharacteristic
41C4⋊(C251⋊C8)C2211yescharacteristic
42#22548640C2211yescharacteristic
43#22548638C2211yescharacteristic
44#22548638C2211yescharacteristic
45#22548638C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548635C2211yescharacteristic
48(C2×D251)⋊C8C2211yescharacteristic
49(C2×D251)⋊C8C2211yescharacteristic
50#22548625C2211yescharacteristic
51#22548625C2211yescharacteristic
52#125×C251C2211yescharacteristic
53C4×D4×C251C2211yescharacteristic
54#107×C251C2211yescharacteristic
55#106×C251C2211yescharacteristic
56C4⋊C8×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59#16860809—22no—
60#16860809—22no—
61C2×Dic502C2311yescharacteristic
62C2×Dic502C2311yescharacteristic
63C2×D1004—21no—
64C2×D1004C2311yescharacteristic
65C2×C22⋊D251—21no—
66C22×Dic251—21no—
67C22×Dic251—21no—
68C2×C4×D251—21no—
69C2×C4×D251D411yescharacteristic
70C2×C4×D251D411yescharacteristic
71C2×C4×D251C2311yescharacteristic
72Dic1004—22no—
73#16860799—22no—
74#16860798—22no—
75#16860795—22no—
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C4⋊Dic251—21no—
79C4⋊Dic251C2311yescharacteristic
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83C22⋊Dic251—21no—
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90C4×Dic251—21no—
91C4×Dic251—21no—
92C4×Dic251—21no—
93C4×Dic251—21no—
94C4×Dic251D411yescharacteristic
95C4×Dic251D411yescharacteristic
96C4×Dic251C2311yescharacteristic
97C2×D4×C251C2311yescharacteristic
98C22×C4×C251—21no—
99C2×C8×C251C2311yescharacteristic
100C2×C8×C251C2311yescharacteristic
101C4⋊C4×C251—21no—
102C4⋊C4×C251—21no—
103C4⋊C4×C251C2311yescharacteristic
104C4⋊C4×C251C2311yescharacteristic
105C4⋊C4×C251C2311yescharacteristic
106C22⋊C4×C251—21no—
107C42×C251C2311yescharacteristic
108C22×D251—21no—
109C22×D251C2×D411yescharacteristic
110Dic502—21no—
111Dic502—21no—
112Dic502—21no—
113Dic502#4112yes—
114D1004—21no—
115D1004—21no—
116D1004—42no—
117C22⋊D251—24no—
118C2×Dic251—24no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—42no—
123C2×Dic251C2×D411yescharacteristic
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C4×D251—21no—
127C4×D251—21no—
128C4×D251—41no—
129C4×D251—41no—
130C4×D251—42no—
131C251⋊C8—21no—
132C251⋊C8—21no—
133C23×C251—21no—
134D4×C251—21no—
135D4×C251#4112yes—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—42no—
140C2×C4×C251C2×D411yescharacteristic
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2411yescharacteristic
143C8×C251—21no—
144C8×C251—21no—
145C22×C251—22no—
146C22×C251—41no—
147C22×C251—41no—
148C22×C251C22×D411yescharacteristic
149C2×D251—21no—
150C2×D251—21no—
151C2×D251—21no—
152C2×D251—41no—
153C2×D251—41no—
154C2×D251—41no—
155C4×C251—21no—
156C4×C251—21no—
157C4×C251—41no—
158C4×C251—41no—
159C4×C251—41no—
160C4×C251C2×#4111yescharacteristic
161C4×C251C2×#4111yescharacteristic
162Dic251—21no—
163Dic251—22no—
164Dic251—21no—
165Dic251—21no—
166Dic251—41no—
167Dic251—41no—
168C2×C251—42no—
169C2×C251#52511yescharacteristic
170C2×C251#57511yescharacteristic
171C2×C251#57511yescharacteristic
172D251—41no—
173D251—81no—
174C251#311111yescharacteristic, complemented, Sylow
175#3111—2511nomaximal, Sylow
176#529—2511no—
177C4×#41—2511no—
178#485—2511no—
179#482—2511no—
180#479—2511no—
181#478D25111yescharacteristic, complemented
182#464—2511no—
183#463—2511no—
184#452—2511no—
185#449—2511no—
186#442—2511no—
187#441—2511no—
188#440—2511no—
189#440—2511no—
190#432—2511no—
191C2×#41—2511no—
192C2×Q16—2511no—
193C2×SD16—2511no—
194C2×SD16—2511no—
195C2×D8—2511no—
196#126—5021no—
197#125C2×D25111yescharacteristic
198#124—2511no—
199#123—5021no—
200C22⋊Q8—2511no—
201#121—2511no—
202C4×Q8—2511no—
203C4×Q8—2511no—
204C4×Q8—2511no—
205C4×D4—5021no—
206C4×D4—5021no—
207C4×D4—5021no—
208C4×D4—2511no—
209C4×D4—2511no—
210C4×D4C2×D25111yescharacteristic
211#117—5021no—
212#117—5021no—
213#117—2511no—
214C2×C42—5021no—
215C2×C42—2511no—
216#107—2511no—
217#107C2×D25111yescharacteristic
218#106—2511no—
219#106C2×D25111yescharacteristic
220C4⋊C8—2511no—
221C4⋊C8C2×D25111yescharacteristic
222#103—2511no—
223#103—2511no—
224#103—2511no—
225#103—2511no—
226#102—2511no—
227#102—2511no—
228#102C2×D25111yescharacteristic
229#102C2×D25111yescharacteristic
230C22⋊C8—2511no—
231C22⋊C8—2511no—
232#97—2511no—
233#97—2511no—
234#41—5022no—
235#41—5022no—
236C2×Q8—2511no—
237C2×Q8—2511no—
238C2×D4—5021no—
239C2×D4—5021no—
240C2×D4—2511no—
241C2×D4C22×D25111yescharacteristic
242C22×C4—21no—
243C22×C4—5021no—
244C22×C4—5021no—
245C22×C4—5021no—
246C22×C4—2511no—
247C22×C4—2511no—
248C22×C4—2511no—
249Q16—5022no—
250SD16—5022no—
251SD16—5022no—
252D8—5022no—
253C2×C8—2511no—
254C2×C8—2511no—
255C2×C8C22×D25111yescharacteristic
256C2×C8C22×D25111yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—5021no—
260C4⋊C4—5021no—
261C4⋊C4—5021no—
262C4⋊C4—5021no—
263C4⋊C4—2511no—
264C4⋊C4C22×D25111yescharacteristic
265C4⋊C4C22×D25111yescharacteristic
266C4⋊C4C22×D25111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—5021no—
269C22⋊C4—5021no—
270C22⋊C4—5021no—
271C22⋊C4—5021no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4—5021no—
275C42—5021no—
276C42—5021no—
277C42—5021no—
278C42—5021no—
279C42—2511no—
280C42—2511no—
281C42—2511no—
282C42C22×D25111yescharacteristic
283C23—21no—
284C23—5021no—
285C23—2511no—
286Q8—5021no—
287Q8—5021no—
288Q8—5021no—
289Q8—2512no—
290D4—21no—
291D4—10042no—
292D4—5024no—
293D4—5021no—
294D4—5021no—
295D4#1686081112yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—10041no—
301C2×C4—10041no—
302C2×C4—10042no—
303C2×C4—10042no—
304C2×C4—5021no—
305C2×C4—5021no—
306C2×C4—5024no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—2511no—
311C2×C4—2511no—
312C2×C4—2511no—
313C2×C4D4×D25111yescharacteristic
314C2×C4D4×D25111yescharacteristic
315C2×C4C23×D25111yescharacteristic
316C8—21no—
317C8—21no—
318C8—5021no—
319C8—5021no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—10041no—
324C22—10041no—
325C22—10041no—
326C22—5021no—
327C22—5021no—
328C22—5021no—
329C22C2×D4×D25111yescharacteristic
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—10041no—
336C4—10041no—
337C4—5021no—
338C4—5022no—
339C4—5021no—
340C4—5021no—
341C4#41×D25111yescharacteristic
342C4C2×#1686081111yescharacteristic
343C2—42no—
344C2—20081no—
345C2—10041no—
346C2#3063220011yescharacteristic
347C2#3063248311yescharacteristic
348C2#3063248411yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#478×C251C2
socleC22×C251C22×D4
prime radical#30631833C2
hypercenter#478D251
hypocenterC251#3111
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #30632222C2semidirect
2. C4×#16860809C2semidirect
3. #30632030C2semidirect
4. #30632013C2semidirect
5. #30631991C2semidirect
6. #30631910C2semidirect
7. #30631902C2semidirect
8. #30631850C2semidirect
10. #30631796C2semidirect
11. #30631793C2semidirect
12. #30631786C2semidirect
13. #30631784C2semidirect
14. #30631743C2semidirect
15. #478×C251C2semidirect
174. C251#3111semidirect
181. #478D251semidirect

Maximal subgroups

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

subgroupindex
#306322222
C4×#168608092
#306320302
#306320132
#306319912
#306319102
#306319022
#306318502
#306318332
#306317962
#306317932
#306317862
#306317842
#306317432
#478×C2512
#3111251