Finite Group Databasebeta

← #44562254

Subgroups — #44562254

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#30632377C211yesmaximal, characteristic, complemented
2#30632346C211yesmaximal, characteristic
3#30632042C211yesmaximal, characteristic, complemented
4#30632029C211yesmaximal, characteristic, complemented
5#30632001C211yesmaximal, characteristic, complemented
6#30631995C211yesmaximal, characteristic, complemented
7#30631928C211yesmaximal, characteristic, complemented
8#30631916C211yesmaximal, characteristic, complemented
9#30631910C211yesmaximal, characteristic, complemented
10#30631851C211yesmaximal, characteristic, complemented
11#30631845C211yesmaximal, characteristic, complemented
12C4×Dic1004C211yesmaximal, characteristic, complemented
13#30631684C211yesmaximal, characteristic, complemented
14C2×#22548640C211yesmaximal, characteristic, complemented
15C8×D4×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C22×Dic502C2211yescharacteristic
18C2×Dic1004C2211yescharacteristic
19C2×Dic1004C2211yescharacteristic
20C2×#16860799C2211yescharacteristic
21#22548708—21no—
22#22548707C2211yescharacteristic
23#22548705—21no—
24#22548703C2211yescharacteristic
25#22548701—21no—
26#22548696—21no—
27#22548694C2211yescharacteristic
28#22548694C2211yescharacteristic
29#22548694C2211yescharacteristic
30#22548693—21no—
31#22548689C2211yescharacteristic
32#22548687—21no—
33C4×Dic502C2211yescharacteristic
34C4×Dic502C2211yescharacteristic
35(C22⋊D251)⋊C4—21no—
36#22548674C2211yescharacteristic
37#22548671—21no—
38C2×C4⋊Dic251C2211yescharacteristic
39#22548649C2211yescharacteristic
40#22548649C2211yescharacteristic
41#22548647C2211yescharacteristic
42#22548640C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548640C2211yescharacteristic
45#22548640C2211yescharacteristic
46#22548640C2211yescharacteristic
47#22548640C2211yescharacteristic
48#22548640C2211yescharacteristic
49#22548637C2211yescharacteristic
50#22548637C2211yescharacteristic
51#22548637C2211yescharacteristic
52C22×C8×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic
54C4×D4×C251C2211yescharacteristic
55C4⋊C8×C251C2211yescharacteristic
56C22⋊C8×C251C2211yescharacteristic
57C22⋊C8×C251C2211yescharacteristic
58C4×C8×C251C2211yescharacteristic
59#16860810—22no—
60#16860810—41no—
61C2×Dic502—22no—
62C2×Dic502—21no—
63C2×Dic502—41no—
64C2×Dic502C2311yescharacteristic
65C2×Dic502C2311yescharacteristic
66C2×Dic502C2311yescharacteristic
67C2×D1004C2311yescharacteristic
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C22×Dic251—21no—
71C22×Dic251—21no—
72C2×C4×D251—21no—
73Dic1004—22no—
74Dic1004—41no—
75#16860799—41no—
76C4⋊Dic251—21no—
77C4⋊Dic251C2311yescharacteristic
78C4⋊Dic251C2311yescharacteristic
79C4⋊Dic251C2311yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88C22⋊Dic251—21no—
89C22⋊Dic251—21no—
90C22⋊Dic251—21no—
91C22⋊Dic251—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95C4×Dic251—21no—
96C4×Dic251—21no—
97C4×Dic251—21no—
98C2×D4×C251D411yescharacteristic
99C22×C4×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C2×C8×C251—41no—
102C2×C8×C251—41no—
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251D411yescharacteristic
108C22⋊C4×C251D411yescharacteristic
109C22⋊C4×C251D411yescharacteristic
110C42×C251C2311yescharacteristic
111C22×D251—21no—
112Dic502—21no—
113Dic502—21no—
114Dic502—21no—
115Dic502—21no—
116Dic502—21no—
117Dic502—42no—
118Dic502#4112yes—
119D1004—21no—
120D1004—21no—
121C22⋊D251—42no—
122C22⋊D251—42no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—42no—
131C2×Dic251—42no—
132C4×D251—42no—
133C23×C251C2×D411yescharacteristic
134C23×C251C2×D411yescharacteristic
135D4×C251—22no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251C2×D411yescharacteristic
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2×D411yescharacteristic
143C2×C4×C251C2×D411yescharacteristic
144C2×C4×C251C2411yescharacteristic
145C8×C251—21no—
146C8×C251—21no—
147C8×C251—21no—
148C8×C251—21no—
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×C251C2×#4111yescharacteristic
164C4×C251H(5,𝔽2)11yescharacteristic
165Dic251—22no—
166Dic251—41no—
167Dic251—41no—
168Dic251—41no—
169Dic251—41no—
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#552—2511no—
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×D4D25111yescharacteristic, complemented
193#416—2511no—
194C2×#103—2511no—
195C2×#41—2511no—
196C22×Q8—2511no—
197C2×Q16—2511no—
198C2×Q16—2511no—
199C2×SD16—2511no—
200C22×C8C2×D25111yescharacteristic
201C22×C8C2×D25111yescharacteristic
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⋊Q8—2511no—
212#121—5021no—
213#121—2511no—
214C4×Q8—2511no—
215C4×Q8—2511no—
216C4×D4—5021no—
217C4×D4C2×D25111yescharacteristic
218#117—5021no—
219#117—2511no—
220C2×C4⋊C4—2511no—
221#107—2511no—
222#107—2511no—
223#106—2511no—
224C4⋊C8C2×D25111yescharacteristic
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⋊C8C2×D25111yescharacteristic
236C22⋊C8C2×D25111yescharacteristic
237C4×C8C2×D25111yescharacteristic
238#41—10041no—
239#41—5022no—
240C2×Q8—10041no—
241C2×Q8—5022no—
242C2×Q8—5021no—
243C2×Q8—2511no—
244C2×Q8—2511no—
245C2×Q8—2511no—
246C2×D4—5021no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4D100411yescharacteristic
250C22×C4—5021no—
251C22×C4—5021no—
252C22×C4—5021no—
253C22×C4C22×D25111yescharacteristic
254C22×C4C22×D25111yescharacteristic
255Q16—10041no—
256Q16—5022no—
257SD16—10041no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8C22×D25111yescharacteristic
261C2×C8C22×D25111yescharacteristic
262C2×C8C22×D25111yescharacteristic
263C2×C8C22×D25111yescharacteristic
264C4⋊C4—5021no—
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⋊C4—2511no—
275C4⋊C4—2511no—
276C4⋊C4D100411yescharacteristic
277C22⋊C4—5021no—
278C22⋊C4—5021no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4D100411yescharacteristic
285C22⋊C4D100411yescharacteristic
286C42—5021no—
287C42—5021no—
288C42—5021no—
289C42C22×D25111yescharacteristic
290C23—5021no—
291C23C2×D100411yescharacteristic
292C23C2×D100411yescharacteristic
293Q8—10042no—
294Q8—5021no—
295Q8—5021no—
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—10042no—
310C2×C4—10042no—
311C2×C4—10042no—
312C2×C4—5021no—
313C2×C4—5021no—
314C2×C4—5021no—
315C2×C4—5021no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4C2×D100411yescharacteristic
320C2×C4C2×D100411yescharacteristic
321C2×C4C2×D100411yescharacteristic
322C2×C4C2×D100411yescharacteristic
323C2×C4C23×D25111yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—10041no—
334C22—10041no—
335C22—10041no—
336C22#2254873712yes—
337C22C22×D100411yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—10041no—
343C4—10041no—
344C4—10041no—
345C4—10041no—
346C4—10041no—
347C4—10041no—
348C4—5022no—
349C4#41×D25111yescharacteristic
350C4#2254876011yescharacteristic
351C2—22no—
352C2—41no—
353C2—20081no—
354C2#3063233211yescharacteristic
355C2C2×#2254873711yescharacteristic
356C2#3063250411yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC8×D4×C251C2
socleC22×C251C22×D4
prime radical#30632346C2
hypercenterC8×D4D251
hypocenterC251#3300
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #30632377C2semidirect
3. #30632042C2semidirect
4. #30632029C2semidirect
5. #30632001C2semidirect
6. #30631995C2semidirect
7. #30631928C2semidirect
8. #30631916C2semidirect
9. #30631910C2semidirect
10. #30631851C2semidirect
11. #30631845C2semidirect
12. C4×Dic1004C2semidirect
13. #30631684C2semidirect
14. C2×#22548640C2semidirect
15. C8×D4×C251C2semidirect
178. C251#3300semidirect
192. C8×D4D251semidirect

Maximal subgroups

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

subgroupindex
#306323772
#306323462
#306320422
#306320292
#306320012
#306319952
#306319282
#306319162
#306319102
#306318512
#306318452
C4×Dic10042
#306316842
C2×#225486402
C8×D4×C2512
#3300251