Finite Group Databasebeta

← #44562255

Subgroups — #44562255

Order 32128.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632376C211yesmaximal, characteristic, complemented
2#30632347C211yesmaximal, characteristic
3#30632043C211yesmaximal, characteristic, complemented
4#30632002C211yesmaximal, characteristic, complemented
5#30631997C211yesmaximal, characteristic, complemented
6#30631926C211yesmaximal, characteristic, complemented
7#30631917C211yesmaximal, characteristic, complemented
8#30631912C211yesmaximal, characteristic, complemented
9#30631851C211yesmaximal, characteristic, complemented
10#30631844C211yesmaximal, characteristic, complemented
11#30631781C211yesmaximal, characteristic, complemented
12(C22⋊D251)⋊C8C211yesmaximal, characteristic, complemented
13#30631681C211yesmaximal, characteristic, complemented
14#103×D251C211yesmaximal, characteristic, complemented
15#485×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×Q8×D251C2211yescharacteristic
18C2×Dic1004C2211yescharacteristic
19C2×#16860800C2211yescharacteristic
20C2×#16860799C2211yescharacteristic
21C22×C251⋊C8C2211yescharacteristic
22C2×C8×D251C2211yescharacteristic
23#22548708—21no—
24#22548707—21no—
25#22548706C2211yescharacteristic
26#22548704C2211yescharacteristic
27#22548701—21no—
28#22548697—21no—
29#22548695C2211yescharacteristic
30#22548694C2211yescharacteristic
31#22548693—21no—
32#22548690C2211yescharacteristic
33#22548688—21no—
34#22548680C2211yescharacteristic
35#22548680C2211yescharacteristic
36#22548677—21no—
37C4×C22⋊D251C2211yescharacteristic
38#22548672—21no—
39C4⋊C4×D251C2211yescharacteristic
40#22548648C2211yescharacteristic
41Dic251⋊C8C2211yescharacteristic
42#22548640C2211yescharacteristic
43#22548639C2211yescharacteristic
44#22548638C2211yescharacteristic
45#22548638C2211yescharacteristic
46#22548638C2211yescharacteristic
47#22548637C2211yescharacteristic
48#22548635C2211yescharacteristic
49#22548635C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C22⋊Q8×C251C2211yescharacteristic
53#117×C251C2211yescharacteristic
54#107×C251C2211yescharacteristic
55#106×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59#16860810—22no—
60#16860810—41no—
61C2×Dic502—21no—
62C2×Dic502C2311yescharacteristic
63C2×Dic502C2311yescharacteristic
64Q8×D251—22no—
65Q8×D251—41no—
66C2×D1004—21no—
67C2×D1004C2311yescharacteristic
68C2×C22⋊D251—21no—
69C2×C22⋊D251D411yescharacteristic
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251C2311yescharacteristic
74Dic1004—22no—
75#16860800—41no—
76#16860799—41no—
77C2×C251⋊C8—41no—
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C8×D251—41no—
81C4⋊Dic251—21no—
82C4⋊Dic251—21no—
83C4⋊Dic251C2311yescharacteristic
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89Dic251⋊C4D411yescharacteristic
90C22⋊Dic251—21no—
91C22⋊Dic251D411yescharacteristic
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4D411yescharacteristic
97C4×Dic251—21no—
98C4×Dic251—21no—
99C4×Dic251C2311yescharacteristic
100C2×Q8×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102C2×C8×C251C2311yescharacteristic
103C2×C8×C251C2311yescharacteristic
104C4⋊C4×C251—21no—
105C4⋊C4×C251C2311yescharacteristic
106C4⋊C4×C251C2311yescharacteristic
107C4⋊C4×C251C2311yescharacteristic
108C22⋊C4×C251—21no—
109C22⋊C4×C251—21no—
110C42×C251—21no—
111C22×D251—21no—
112C22×D251C2×D411yescharacteristic
113Dic502—21no—
114Dic502—21no—
115Dic502—21no—
116Dic502—42no—
117Dic502#4112yes—
118D1004—21no—
119D1004—21no—
120D1004—42no—
121C22⋊D251—22no—
122C22⋊D251—42no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C4×D251—21no—
130C4×D251—21no—
131C4×D251—42no—
132C4×D251—42no—
133C4×D251—42no—
134C251⋊C8—21no—
135C251⋊C8—21no—
136C23×C251C2×D411yescharacteristic
137Q8×C251—21no—
138Q8×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2411yescharacteristic
147C8×C251—21no—
148C8×C251—21no—
149C22×C251—21no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C251C22×D411yescharacteristic
153C2×D251—21no—
154C2×D251—21no—
155C2×D251—41no—
156C2×D251—41no—
157C2×D251—41no—
158C2×D251#13512yes—
159C4×C251—21no—
160C4×C251—41no—
161C4×C251—41no—
162C4×C251—41no—
163C4×C251—41no—
164C4×C251C2×#4111yescharacteristic
165C4×C251H(5,𝔽2)11yescharacteristic
166Dic251—21no—
167Dic251—22no—
168Dic251—21no—
169Dic251—21no—
170Dic251—41no—
171Dic251—41no—
172C2×C251—41no—
173C2×C251#54611yescharacteristic
174C2×C251C2×#13511yescharacteristic
175C2×C251#57811yescharacteristic
176D251—22no—
177D251—81no—
178C251#330011yescharacteristic, complemented, Sylow
179#3300—2511nomaximal, Sylow
180#552—2511no—
181#548—2511no—
182#488—2511no—
183#485D25111yescharacteristic, complemented
184#480—2511no—
185#479—2511no—
186#467—2511no—
187#465—2511no—
188#464—2511no—
189#452—2511no—
190#451—2511no—
191C4×Q16—2511no—
192C8×D4—2511no—
193#416—2511no—
194C2×#103—2511no—
195C2×#41—2511no—
196C22×Q8—2511no—
197C2×Q16—2511no—
198C2×Q16—2511no—
199C2×SD16—2511no—
200C22×C8—2511no—
201C22×C8—2511no—
202#126—5021no—
203#125—5021no—
204#125—2511no—
205#124—5021no—
206#124—2511no—
207#123—5021no—
208C22⋊Q8—5021no—
209C22⋊Q8—2511no—
210C22⋊Q8—2511no—
211C22⋊Q8C2×D25111yescharacteristic
212#121—5021no—
213#121—2511no—
214C4×Q8—2511no—
215C4×Q8—2511no—
216C4×D4—5021no—
217C4×D4—2511no—
218#117—5021no—
219#117C2×D25111yescharacteristic
220C2×C4⋊C4—2511no—
221#107—2511no—
222#107C2×D25111yescharacteristic
223#106C2×D25111yescharacteristic
224C4⋊C8—2511no—
225#103—2511no—
226#103—2511no—
227#103—2511no—
228#103—2511no—
229#103—2511no—
230#103C2×D25111yescharacteristic
231#103C2×D25111yescharacteristic
232#102—2511no—
233#102—2511no—
234#102—2511no—
235C22⋊C8—2511no—
236C22⋊C8C2×D25111yescharacteristic
237C4×C8—2511no—
238#41—10041no—
239#41—5022no—
240C2×Q8—10041no—
241C2×Q8—5021no—
242C2×Q8—5022no—
243C2×Q8—2511no—
244C2×Q8—2511no—
245C2×Q8C22×D25111yescharacteristic
246C2×D4—5021no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4—2511no—
250C22×C4—5021no—
251C22×C4—5021no—
252C22×C4—5021no—
253C22×C4—2511no—
254C22×C4C22×D25111yescharacteristic
255Q16—10041no—
256Q16—5022no—
257SD16—10041no—
258C2×C8—10041no—
259C2×C8—10041no—
260C2×C8—2511no—
261C2×C8—2511no—
262C2×C8C22×D25111yescharacteristic
263C2×C8C22×D25111yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—5021no—
266C4⋊C4—5021no—
267C4⋊C4—5021no—
268C4⋊C4—5021no—
269C4⋊C4—5021no—
270C4⋊C4—5021no—
271C4⋊C4—5021no—
272C4⋊C4—2511no—
273C4⋊C4—2511no—
274C4⋊C4C22×D25111yescharacteristic
275C4⋊C4C22×D25111yescharacteristic
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4—2511no—
285C22⋊C4—2511no—
286C42—21no—
287C42—5021no—
288C42—5021no—
289C42—2511no—
290C23—5021no—
291C23—2511no—
292C23D4×D25111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—10042no—
296Q8—5021no—
297Q8—5021no—
298Q8—5021no—
299Q8—2512no—
300D4—10042no—
301D4—10042no—
302D4—5022no—
303D4—5021no—
304D4—5021no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—10042no—
312C2×C4—10042no—
313C2×C4—10042no—
314C2×C4—5021no—
315C2×C4—5021no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4—2511no—
320C2×C4—2511no—
321C2×C4—2511no—
322C2×C4D4×D25111yescharacteristic
323C2×C4C23×D25111yescharacteristic
324C8—21no—
325C8—21no—
326C8—5021no—
327C8—5021no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—10041no—
332C22—10041no—
333C22—10041no—
334C22—5021no—
335C22—5021no—
336C22—2512no—
337C22C2×D4×D25111yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—10041no—
344C4—10041no—
345C4—5021no—
346C4—5021no—
347C4—5022no—
348C4—5021no—
349C4#41×D25111yescharacteristic
350C4#2254876111yescharacteristic
351C2—41no—
352C2—20081no—
353C2—5022no—
354C2#3063232911yescharacteristic
355C2#135×D25111yescharacteristic
356C2#3063250311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#485×C251C2
socleC22×C251C22×D4
prime radical#30632347C2
hypercenter#485D251
hypocenterC251#3300
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #30632376C2semidirect
3. #30632043C2semidirect
4. #30632002C2semidirect
5. #30631997C2semidirect
6. #30631926C2semidirect
7. #30631917C2semidirect
8. #30631912C2semidirect
9. #30631851C2semidirect
10. #30631844C2semidirect
11. #30631781C2semidirect
12. (C22⋊D251)⋊C8C2semidirect
13. #30631681C2semidirect
14. #103×D251C2semidirect
15. #485×C251C2semidirect
178. C251#3300semidirect
183. #485D251semidirect

Maximal subgroups

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

subgroupindex
#306323762
#306323472
#306320432
#306320022
#306319972
#306319262
#306319172
#306319122
#306318512
#306318442
#306317812
(C22⋊D251)⋊C82
#306316812
#103×D2512
#485×C2512
#3300251