Finite Group Databasebeta

← #44562246

Subgroups — #44562246

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#30632346C211yesmaximal, characteristic
2#30632044C211yesmaximal, characteristic, complemented
3#30632028C211yesmaximal, characteristic, complemented
4#30632000C211yesmaximal, characteristic, complemented
5#30631991C211yesmaximal, characteristic, complemented
6#30631926C211yesmaximal, characteristic, complemented
7#30631918C211yesmaximal, characteristic, complemented
8#30631909C211yesmaximal, characteristic, complemented
9#30631847C211yesmaximal, characteristic, complemented
10#30631842C211yesmaximal, characteristic, complemented
11C4×#16860800C211yesmaximal, characteristic, complemented
12D4×C251⋊C8C211yesmaximal, characteristic, complemented
13#30631681C211yesmaximal, characteristic, complemented
14C2×#22548638C211yesmaximal, characteristic, complemented
15#552×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C22×Dic502C2211yescharacteristic
18C2×#16860800C2211yescharacteristic
19C2×#16860800C2211yescharacteristic
20C2×#16860797C2211yescharacteristic
21C22×C251⋊C8C2211yescharacteristic
22C22×C251⋊C8C2211yescharacteristic
23#22548703C2211yescharacteristic
24#22548701—21no—
25#22548694C2211yescharacteristic
26#22548693—21no—
27#22548689C2211yescharacteristic
28#22548687—21no—
29C4×Dic502C2211yescharacteristic
30(C22⋊D251)⋊C4—21no—
31#22548648C2211yescharacteristic
32#22548648C2211yescharacteristic
33#22548646C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35#22548639C2211yescharacteristic
36#22548638C2211yescharacteristic
37#22548638C2211yescharacteristic
38#22548638C2211yescharacteristic
39#22548638C2211yescharacteristic
40#22548638C2211yescharacteristic
41#22548638C2211yescharacteristic
42#22548635C2211yescharacteristic
43#22548635C2211yescharacteristic
44#22548635C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46C22⋊(C251⋊C8)C2211yescharacteristic
47C4×C251⋊C8C2211yescharacteristic
48#126×C251—21no—
49#125×C251—21no—
50#125×C251C2211yescharacteristic
51#123×C251—21no—
52C22⋊Q8×C251C2211yescharacteristic
53C22⋊Q8×C251C2211yescharacteristic
54C4×Q8×C251C2211yescharacteristic
55C4×D4×C251C2211yescharacteristic
56#117×C251—21no—
57#117×C251C2211yescharacteristic
58C2×C4⋊C4×C251C2211yescharacteristic
59#16860810—22no—
60#16860810—41no—
61C2×Dic502—22no—
62C2×Dic502—21no—
63C2×Dic502—41no—
64C2×Dic502C2311yescharacteristic
65C2×Dic502C2311yescharacteristic
66C2×D1004C2311yescharacteristic
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69C22×Dic251—21no—
70C2×C4×D251—21no—
71#16860800—22no—
72#16860800—41no—
73#16860797—41no—
74C2×C251⋊C8—41no—
75C2×C251⋊C8—41no—
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83C22⋊Dic251—21no—
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87C4×Dic251—21no—
88C2×Q8×C251C2311yescharacteristic
89C2×D4×C251D411yescharacteristic
90C22×C4×C251—21no—
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C4⋊C4×C251—21no—
94C4⋊C4×C251—21no—
95C4⋊C4×C251—21no—
96C4⋊C4×C251—21no—
97C4⋊C4×C251—21no—
98C4⋊C4×C251—21no—
99C4⋊C4×C251D411yescharacteristic
100C4⋊C4×C251C2311yescharacteristic
101C4⋊C4×C251C2311yescharacteristic
102C4⋊C4×C251C2311yescharacteristic
103C22⋊C4×C251—21no—
104C22⋊C4×C251—21no—
105C22⋊C4×C251—21no—
106C22⋊C4×C251D411yescharacteristic
107C22⋊C4×C251D411yescharacteristic
108C42×C251—21no—
109C42×C251—21no—
110C42×C251C2311yescharacteristic
111C22×D251—21no—
112Dic502—21no—
113Dic502—21no—
114Dic502—21no—
115Dic502—42no—
116Dic502#4112yes—
117D1004—21no—
118D1004—21no—
119C22⋊D251—42no—
120C22⋊D251—42no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—42no—
125C4×D251—42no—
126C251⋊C8—21no—
127C251⋊C8—21no—
128C251⋊C8—21no—
129C251⋊C8—21no—
130C23×C251C2×D411yescharacteristic
131C23×C251C2×D411yescharacteristic
132Q8×C251—21no—
133Q8×C251—21no—
134D4×C251—22no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—42no—
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2411yescharacteristic
149C22×C251—21no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C251—21no—
153C22×C251—21no—
154C22×C251#13512yes—
155C22×C251C22×D411yescharacteristic
156C2×D251—41no—
157C2×D251—41no—
158C2×D251—41no—
159C4×C251—21no—
160C4×C251—21no—
161C4×C251—21no—
162C4×C251—21no—
163C4×C251—41no—
164C4×C251—41no—
165C4×C251—41no—
166C4×C251—41no—
167C4×C251C2×#4111yescharacteristic
168C4×C251H(5,𝔽2)11yescharacteristic
169Dic251—22no—
170Dic251—41no—
171Dic251—41no—
172C2×C251—22no—
173C2×C251—41no—
174C2×C251#54611yescharacteristic
175C2×C251C2×#13511yescharacteristic
176C2×C251#57811yescharacteristic
177D251—81no—
178C251#330011yescharacteristic, complemented, Sylow
179#3300—2511nomaximal, Sylow
180#552D25111yescharacteristic, complemented
181#548—2511no—
182#488—2511no—
183#485—2511no—
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—21no—
203#125—21no—
204#125C2×D25111yescharacteristic
205#124—5021no—
206#124—2511no—
207#123—21no—
208C22⋊Q8—5021no—
209C22⋊Q8—2511no—
210C22⋊Q8C2×D25111yescharacteristic
211C22⋊Q8C2×D25111yescharacteristic
212#121—5021no—
213#121—2511no—
214C4×Q8—2511no—
215C4×Q8C2×D25111yescharacteristic
216C4×D4—5021no—
217C4×D4C2×D25111yescharacteristic
218#117—21no—
219#117C2×D25111yescharacteristic
220C2×C4⋊C4C2×D25111yescharacteristic
221#107—2511no—
222#107—2511no—
223#106—2511no—
224C4⋊C8—2511no—
225#103—2511no—
226#103—2511no—
227#103—2511no—
228#103—2511no—
229#103—2511no—
230#103—2511no—
231#103—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235C22⋊C8—2511no—
236C22⋊C8—2511no—
237C4×C8—2511no—
238#41—10041no—
239#41—5022no—
240C2×Q8—10041no—
241C2×Q8—5022no—
242C2×Q8—5021no—
243C2×Q8—2511no—
244C2×Q8—2511no—
245C2×Q8C22×D25111yescharacteristic
246C2×D4—5021no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4C22⋊D25111yescharacteristic
250C22×C4—21no—
251C22×C4—5021no—
252C22×C4—5021no—
253C22×C4C22×D25111yescharacteristic
254C22×C4C22×D25111yescharacteristic
255Q16—10041no—
256Q16—5022no—
257SD16—10041no—
258C2×C8—10041no—
259C2×C8—10041no—
260C2×C8—2511no—
261C2×C8—2511no—
262C2×C8—2511no—
263C2×C8—2511no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—5021no—
271C4⋊C4—5021no—
272C4⋊C4—2511no—
273C4⋊C4C22⋊D25111yescharacteristic
274C4⋊C4C22×D25111yescharacteristic
275C4⋊C4C22×D25111yescharacteristic
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4C22⋊D25111yescharacteristic
285C22⋊C4C22⋊D25111yescharacteristic
286C42—21no—
287C42—21no—
288C42—5021no—
289C42C22×D25111yescharacteristic
290C23—5021no—
291C23C2×C22⋊D25111yescharacteristic
292C23C2×C22⋊D25111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—10042no—
296Q8—5021no—
297Q8—5021no—
298Q8—5021no—
299Q8—2512no—
300D4—22no—
301D4—10042no—
302D4—10042no—
303D4—5021no—
304D4—5021no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—10042no—
315C2×C4—10042no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4C2×C22⋊D25111yescharacteristic
320C2×C4C2×C22⋊D25111yescharacteristic
321C2×C4C2×C22⋊D25111yescharacteristic
322C2×C4C2×C22⋊D25111yescharacteristic
323C2×C4C23×D25111yescharacteristic
324C8—5021no—
325C8—5021no—
326C8—5021no—
327C8—5021no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—10041no—
334C22—10041no—
335C22—10041no—
336C22#2254873512yes—
337C22C22×C22⋊D25111yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—10041no—
347C4—10041no—
348C4—5022no—
349C4#41×D25111yescharacteristic
350C4#2254876111yescharacteristic
351C2—22no—
352C2—41no—
353C2—20081no—
354C2#3063232511yescharacteristic
355C2C2×#2254873511yescharacteristic
356C2#3063250211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#552×C251C2
socleC22×C251C22×D4
prime radical#30632346C2
hypercenter#552D251
hypocenterC251#3300
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #30632044C2semidirect
3. #30632028C2semidirect
4. #30632000C2semidirect
5. #30631991C2semidirect
6. #30631926C2semidirect
7. #30631918C2semidirect
8. #30631909C2semidirect
9. #30631847C2semidirect
10. #30631842C2semidirect
11. C4×#16860800C2semidirect
12. D4×C251⋊C8C2semidirect
13. #30631681C2semidirect
14. C2×#22548638C2semidirect
15. #552×C251C2semidirect
178. C251#3300semidirect
180. #552D251semidirect

Maximal subgroups

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

subgroupindex
#306323462
#306320442
#306320282
#306320002
#306319912
#306319262
#306319182
#306319092
#306318472
#306318422
C4×#168608002
D4×C251⋊C82
#306316812
C2×#225486382
#552×C2512
#3300251