Finite Group Databasebeta

← #44559325

Subgroups — #44559325

Order 32128.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#16860809C211yesmaximal, characteristic
2#30631816C211yesmaximal, characteristic, complemented
3#30631815C211yesmaximal, characteristic, complemented
4#30631763C211yesmaximal, characteristic, complemented
5#30631762C211yesmaximal, characteristic, complemented
6#30631759C211yesmaximal, characteristic, complemented
7#30631758C211yesmaximal, characteristic, complemented
8#30631755C211yesmaximal, characteristic, complemented
9#30631739C211yesmaximal, characteristic, complemented
10#30631737C211yesmaximal, characteristic, complemented
11#30631737C211yesmaximal, characteristic, complemented
12#30631639C211yesmaximal, characteristic, complemented
13M16×Dic251C211yesmaximal, characteristic, complemented
14C2×C8×Dic251C211yesmaximal, characteristic, complemented
15#435×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C411yescharacteristic
17C2×#16860794C2211yescharacteristic
18C2×#16860793C2211yescharacteristic
19C2×#16860793C2211yescharacteristic
20C22×C251⋊C8C2211yescharacteristic
21C4×Dic502C2211yescharacteristic
22#22548680C411yescharacteristic
23D4×Dic251C411yescharacteristic
24C4×C22⋊D251C2211yescharacteristic
25C4×C22⋊D251C2211yescharacteristic
26(C22⋊D251)⋊C4C411yescharacteristic
27(C22⋊D251)⋊C4C411yescharacteristic
28#22548673C411yescharacteristic
29#22548671C411yescharacteristic
30#22548671C411yescharacteristic
31C2×C4×Dic251C2211yescharacteristic
32C2×C4×Dic251C2211yescharacteristic
33C42×D251C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35Dic251⋊C8C2211yescharacteristic
36Dic251⋊C8C2211yescharacteristic
37Dic251⋊C8C2211yescharacteristic
38Dic251⋊C8C2211yescharacteristic
39C22⋊(C251⋊C8)C2211yescharacteristic
40C22⋊(C251⋊C8)C2211yescharacteristic
41(C2×D251)⋊C8C2211yescharacteristic
42(C2×D251)⋊C8C2211yescharacteristic
43#22548626C2211yescharacteristic
44#22548625C2211yescharacteristic
45#22548625C2211yescharacteristic
46C8×Dic251C2211yescharacteristic
47C8×Dic251C2211yescharacteristic
48C8×Dic251C2211yescharacteristic
49C8×Dic251C2211yescharacteristic
50C8×Dic251C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×M16×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic
54C4×D4×C251C2211yescharacteristic
55C4⋊C8×C251C2211yescharacteristic
56C22⋊C8×C251C2211yescharacteristic
57C22⋊C8×C251C2211yescharacteristic
58#97×C251C2211yescharacteristic
59#16860809—24no—
60C2×Dic502C2×C411yescharacteristic
61C2×C22⋊D251C2×C411yescharacteristic
62C2×C22⋊D251C2×C411yescharacteristic
63C22×Dic251C2×C411yescharacteristic
64C22×Dic251C2×C411yescharacteristic
65C22×Dic251C2×C411yescharacteristic
66C22×Dic251C2×C411yescharacteristic
67C2×C4×D251C2×C411yescharacteristic
68C2×C4×D251C2×C411yescharacteristic
69C2×C4×D251C2311yescharacteristic
70#16860794—22no—
71#16860793—22no—
72#16860793—22no—
73C2×C251⋊C8—22no—
74C2×C251⋊C8C2311yescharacteristic
75C2×C251⋊C8C2311yescharacteristic
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C4⋊Dic251C2×C411yescharacteristic
79Dic251⋊C4C2×C411yescharacteristic
80Dic251⋊C4C2×C411yescharacteristic
81Dic251⋊C4C2×C411yescharacteristic
82Dic251⋊C4C2×C411yescharacteristic
83C22⋊Dic251C2×C411yescharacteristic
84C22⋊Dic251C2×C411yescharacteristic
85(C2×D251)⋊C4C2×C411yescharacteristic
86(C2×D251)⋊C4C2×C411yescharacteristic
87C4×Dic251C2×C411yescharacteristic
88C4×Dic251C2×C411yescharacteristic
89C4×Dic251C2×C411yescharacteristic
90C4×Dic251C2×C411yescharacteristic
91C4×Dic251C2×C411yescharacteristic
92C4×Dic251C2×C411yescharacteristic
93C4×Dic251C2311yescharacteristic
94C4×Dic251C2311yescharacteristic
95C4×Dic251C2311yescharacteristic
96C2×D4×C251C2×C411yescharacteristic
97C22×C4×C251C2311yescharacteristic
98C22×C4×C251C2311yescharacteristic
99M16×C251—22no—
100C2×C8×C251—22no—
101C2×C8×C251C2311yescharacteristic
102C2×C8×C251C2311yescharacteristic
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C4⋊C4×C251C2×C411yescharacteristic
106C22⋊C4×C251C2×C411yescharacteristic
107C22⋊C4×C251C2×C411yescharacteristic
108C42×C251C2311yescharacteristic
109C22×D251C22×C411yescharacteristic
110Dic502—22no—
111C22⋊D251—22no—
112C22⋊D251—22no—
113C2×Dic251—22no—
114C2×Dic251—22no—
115C2×Dic251—22no—
116C2×Dic251M1614yescomplemented
117C2×Dic251C22×C411yescharacteristic
118C2×Dic251C22×C411yescharacteristic
119C2×Dic251C22×C411yescharacteristic
120C2×Dic251C22×C411yescharacteristic
121C2×Dic251C22×C411yescharacteristic
122C2×Dic251C22×C411yescharacteristic
123C2×Dic251C22×C411yescharacteristic
124C4×D251—21no—
125C4×D251—21no—
126C4×D251—22no—
127C4×D251—22no—
128C251⋊C8—21no—
129C251⋊C8—21no—
130C251⋊C8—21no—
131C251⋊C8#4112yescomplemented
132C23×C251C22×C411yescharacteristic
133C23×C251C22×C411yescharacteristic
134D4×C251—22no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251C22×C411yescharacteristic
140C2×C4×C251C22×C411yescharacteristic
141C2×C4×C251C22×C411yescharacteristic
142C2×C4×C251C22×C411yescharacteristic
143C2×C4×C251C2411yescharacteristic
144C8×C251—21no—
145C8×C251—21no—
146C8×C251—21no—
147C8×C251#4112yescomplemented
148C22×C251—21no—
149C22×C251—21no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C251—21no—
153C22×C251C2×M1612yes—
154C22×C251C23×C411yescharacteristic
155C2×D251—21no—
156C2×D251—21no—
157C2×D251—21no—
158C4×C251—21no—
159C4×C251—21no—
160C4×C251—21no—
161C4×C251—21no—
162C4×C251C2×#4111yescharacteristic
163C4×C251C2×#4111yescharacteristic
164Dic251—21no—
165Dic251—21no—
166Dic251—21no—
167Dic251—21no—
168Dic251C2×M1612yes—
169Dic251C2×M1612yes—
170Dic251#13112yes—
171C2×C251—22no—
172C2×C251—41no—
173C2×C251C4×#4111yescharacteristic
174C2×C251C22×M1611yescharacteristic
175C2×C251C2×#13111yescharacteristic
176D251—41no—
177C251#296511yescharacteristic, complemented, Sylow
178#2965—2511nomaximal, Sylow
179C4×#41—2511no—
180#446—2511no—
181#446—2511no—
182#436—2511no—
183#436—2511no—
184#435—2511no—
185#435—2511no—
186#435—2511no—
187#435D25111yescharacteristic, complemented
188#431—2511no—
189#431—2511no—
190#431—2511no—
191C4×M16—2511no—
192C4×M16—2511no—
193C2×C4×C8—2511no—
194C2×#41—2511no—
195C2×M16—2511no—
196C2×M16—2511no—
197C2×M16—2511no—
198C2×M16C2×D25111yescharacteristic
199C22×C8—2511no—
200C22×C8C2×D25111yescharacteristic
201C4×Q8—2511no—
202C4×Q8—2511no—
203C4×D4—2511no—
204C4×D4—2511no—
205C4×D4—2511no—
206C4×D4—2511no—
207C4×D4—2511no—
208C4×D4C2×D25111yescharacteristic
209#117—2511no—
210#117—2511no—
211#117—2511no—
212C2×C42—2511no—
213C2×C42—2511no—
214C2×C42—2511no—
215C4⋊C8—2511no—
216C4⋊C8—2511no—
217C4⋊C8—2511no—
218C4⋊C8—2511no—
219C4⋊C8—2511no—
220C4⋊C8C2×D25111yescharacteristic
221C22⋊C8—2511no—
222C22⋊C8—2511no—
223C22⋊C8—2511no—
224C22⋊C8—2511no—
225C22⋊C8C2×D25111yescharacteristic
226C22⋊C8C2×D25111yescharacteristic
227#97—2511no—
228#97—2511no—
229#97—2511no—
230#97C2×D25111yescharacteristic
231C4×C8—2511no—
232C4×C8—2511no—
233C4×C8—2511no—
234C4×C8—2511no—
235C4×C8—2511no—
236C4×C8—2511no—
237#41—5024no—
238C2×Q8—2511no—
239C2×D4—2511no—
240C2×D4—2511no—
241C2×D4C4×D25111yescharacteristic
242C22×C4—2511no—
243C22×C4—2511no—
244C22×C4—2511no—
245C22×C4—2511no—
246C22×C4—2511no—
247C22×C4—2511no—
248C22×C4—2511no—
249C22×C4C22×D25111yescharacteristic
250C22×C4C22×D25111yescharacteristic
251M16—22no—
252M16—5022no—
253M16—5022no—
254M16—5022no—
255C2×C8—22no—
256C2×C8—5022no—
257C2×C8—2511no—
258C2×C8—2511no—
259C2×C8—2511no—
260C2×C8—2511no—
261C2×C8C22×D25111yescharacteristic
262C2×C8C22×D25111yescharacteristic
263C2×C8C22×D25111yescharacteristic
264C2×C8C22×D25111yescharacteristic
265C4⋊C4—2511no—
266C4⋊C4—2511no—
267C4⋊C4—2511no—
268C4⋊C4—2511no—
269C4⋊C4—2511no—
270C4⋊C4C4×D25111yescharacteristic
271C22⋊C4—2511no—
272C22⋊C4—2511no—
273C22⋊C4—2511no—
274C22⋊C4—2511no—
275C22⋊C4C4×D25111yescharacteristic
276C22⋊C4C4×D25111yescharacteristic
277C42—2511no—
278C42—2511no—
279C42—2511no—
280C42—2511no—
281C42—2511no—
282C42—2511no—
283C42—2511no—
284C42—2511no—
285C42—2511no—
286C42C22×D25111yescharacteristic
287C23—2511no—
288C23C2×C4×D25111yescharacteristic
289C23C2×C4×D25111yescharacteristic
290Q8—5022no—
291D4—22no—
292D4—5022no—
293D4—5022no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—5022no—
299C2×C4—5022no—
300C2×C4—5022no—
301C2×C4—5022no—
302C2×C4—5022no—
303C2×C4—5021no—
304C2×C4—5021no—
305C2×C4—2511no—
306C2×C4—2511no—
307C2×C4—2511no—
308C2×C4—2511no—
309C2×C4—2511no—
310C2×C4—2511no—
311C2×C4—2511no—
312C2×C4—2514no—
313C2×C4C2×C4×D25111yescharacteristic
314C2×C4C2×C4×D25111yescharacteristic
315C2×C4C2×C4×D25111yescharacteristic
316C2×C4C2×C4×D25111yescharacteristic
317C2×C4C23×D25111yescharacteristic
318C8—21no—
319C8—21no—
320C8—21no—
321C8—5021no—
322C8—5021no—
323C8—5021no—
324C8—2512no—
325C8#1686080912yescomplemented
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—5021no—
332C22—5021no—
333C22—5021no—
334C22M16×D25112yes—
335C22C22×C4×D25111yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—5021no—
341C4—5021no—
342C4—5021no—
343C4—5021no—
344C4—2512no—
345C4—2512no—
346C4—2512no—
347C4#41×D25111yescharacteristic
348C4C2×#1686080911yescharacteristic
349C2—22no—
350C2—41no—
351C2—10041no—
352C2C4×#1686080911yescharacteristic
353C2C2×M16×D25111yescharacteristic
354C2#131×D25111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D251
commutator, socleC22×C251C23×C4
Fitting#435×C251C2
prime radicalC2×#16860809C4
cosocleC2×C4×C251C24
hypercenter#435D251
hypocenterC251#2965
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #30631816C2semidirect
3. #30631815C2semidirect
4. #30631763C2semidirect
5. #30631762C2semidirect
6. #30631759C2semidirect
7. #30631758C2semidirect
8. #30631755C2semidirect
9. #30631739C2semidirect
10. #30631737C2semidirect
11. #30631737C2semidirect
12. #30631639C2semidirect
13. M16×Dic251C2semidirect
14. C2×C8×Dic251C2semidirect
15. #435×C251C2semidirect
116. C2×Dic251M16semidirect
131. C251⋊C8#41semidirect
147. C8×C251#41semidirect
177. C251#2965semidirect
187. #435D251semidirect
325. C8#16860809semidirect

Maximal subgroups

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

subgroupindex
C4×#168608092
#306318162
#306318152
#306317632
#306317622
#306317592
#306317582
#306317552
#306317392
#306317372
#306316392
M16×Dic2512
C2×C8×Dic2512
#435×C2512
#2965251