Finite Group Databasebeta

← #44562248

Subgroups — #44562248

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#30632372C211yesmaximal, characteristic, complemented
2#30632348C211yesmaximal, characteristic
3#30632043C211yesmaximal, characteristic, complemented
4#30632026C211yesmaximal, characteristic, complemented
5#30632000C211yesmaximal, characteristic, complemented
6#30631992C211yesmaximal, characteristic, complemented
7#30631927C211yesmaximal, characteristic, complemented
8#30631916C211yesmaximal, characteristic, complemented
9#30631912C211yesmaximal, characteristic, complemented
10#30631852C211yesmaximal, characteristic, complemented
11#30631843C211yesmaximal, characteristic, complemented
12#30631779C211yesmaximal, characteristic, complemented
13C8×C22⋊D251C211yesmaximal, characteristic, complemented
14#30631682C211yesmaximal, characteristic, complemented
15C2×#103×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860800C2211yescharacteristic
18C2×#16860800C2211yescharacteristic
19C2×#16860799C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21#22548710—21no—
22#22548706C2211yescharacteristic
23#22548705—21no—
24#22548704—21no—
25#22548704C2211yescharacteristic
26#22548698—21no—
27#22548695C2211yescharacteristic
28#22548694C2211yescharacteristic
29#22548691—21no—
30#22548691C2211yescharacteristic
31#22548689—21no—
32#22548689C2211yescharacteristic
33Q8×Dic251C2211yescharacteristic
34#22548680C2211yescharacteristic
35C4×C22⋊D251—21no—
36C4×C22⋊D251C2211yescharacteristic
37#22548674—21no—
38#22548673C2211yescharacteristic
39#22548648C2211yescharacteristic
40#22548648C2211yescharacteristic
41#22548647C2211yescharacteristic
42Dic251⋊C8C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548639C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C22×Q8×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic
54C2×C4⋊C4×C251C2211yescharacteristic
55#103×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58#103×C251C2211yescharacteristic
59#16860811—22no—
60#16860811—41no—
61C2×Dic502C2311yescharacteristic
62C2×D1004—21no—
63C2×D1004C2311yescharacteristic
64C2×C22⋊D251—21no—
65C2×C22⋊D251D411yescharacteristic
66C2×C4×D251—21no—
67C2×C4×D251C2311yescharacteristic
68#16860800—22no—
69#16860800—41no—
70#16860799—41no—
71C2×C251⋊C8C2311yescharacteristic
72C2×C251⋊C8C2311yescharacteristic
73C8×D251—41no—
74C4⋊Dic251—21no—
75C4⋊Dic251—21no—
76C4⋊Dic251—21no—
77C4⋊Dic251C2311yescharacteristic
78C4⋊Dic251C2311yescharacteristic
79Dic251⋊C4—21no—
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4D411yescharacteristic
84C22⋊Dic251—21no—
85C22⋊Dic251D411yescharacteristic
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4D411yescharacteristic
93C4×Dic251—21no—
94C4×Dic251—21no—
95C4×Dic251—21no—
96C4×Dic251C2311yescharacteristic
97C2×Q8×C251—21no—
98C2×Q8×C251—22no—
99C2×Q8×C251—41no—
100C2×Q8×C251C2311yescharacteristic
101C2×Q8×C251C2311yescharacteristic
102C22×C4×C251—21no—
103C22×C4×C251—21no—
104C22×C4×C251C2311yescharacteristic
105C2×C8×C251—41no—
106C2×C8×C251C2311yescharacteristic
107C2×C8×C251C2311yescharacteristic
108C4⋊C4×C251—21no—
109C4⋊C4×C251C2311yescharacteristic
110C4⋊C4×C251C2311yescharacteristic
111C22×D251—21no—
112C22×D251C2×D411yescharacteristic
113Dic502—21no—
114Dic502—21no—
115D1004—21no—
116D1004—21no—
117D1004—42no—
118C22⋊D251—22no—
119C22⋊D251—42no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251C2×D411yescharacteristic
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C4×D251—21no—
127C4×D251—21no—
128C4×D251—42no—
129C251⋊C8—21no—
130C251⋊C8—21no—
131C23×C251C2×D411yescharacteristic
132Q8×C251—21no—
133Q8×C251—21no—
134Q8×C251—21no—
135Q8×C251—42no—
136Q8×C251#4112yes—
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×C251—42no—
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2411yescharacteristic
147C8×C251—21no—
148C8×C251—21no—
149C22×C251—21no—
150C22×C251—21no—
151C22×C251#13512yes—
152C22×C251C22×D411yescharacteristic
153C2×D251—21no—
154C2×D251—21no—
155C2×D251—21no—
156C2×D251—41no—
157C2×D251—41no—
158C2×D251—41no—
159C4×C251—21no—
160C4×C251—22no—
161C4×C251—41no—
162C4×C251—41no—
163C4×C251—41no—
164C4×C251C2×#4111yescharacteristic
165C4×C251H(5,𝔽2)11yescharacteristic
166Dic251—21no—
167Dic251—21no—
168Dic251—21no—
169Dic251—41no—
170Dic251—41no—
171Dic251—41no—
172C2×C251—22no—
173C2×C251#54611yescharacteristic
174C2×C251C2×#13511yescharacteristic
175C2×C251#57811yescharacteristic
176D251—41no—
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×D4—2511no—
193#416—2511no—
194C2×#103D25111yescharacteristic, complemented
195C2×#41—2511no—
196C22×Q8C2×D25111yescharacteristic
197C2×Q16—2511no—
198C2×Q16—2511no—
199C2×SD16—2511no—
200C22×C8—2511no—
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×D4—2511no—
218#117—5021no—
219#117—2511no—
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#103C2×D25111yescharacteristic
229#103C2×D25111yescharacteristic
230#103C2×D25111yescharacteristic
231#103C2×D25111yescharacteristic
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—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—2511no—
244C2×Q8C22×D25111yescharacteristic
245C2×Q8C22×D25111yescharacteristic
246C2×D4—5021no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4—2511no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—5021no—
253C22×C4—2511no—
254C22×C4C22×D25111yescharacteristic
255Q16—10041no—
256Q16—5022no—
257SD16—10041no—
258C2×C8—41no—
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⋊C4—2511no—
275C4⋊C4C22×D25111yescharacteristic
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—5021no—
278C22⋊C4—5021no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4—2511no—
285C22⋊C4—2511no—
286C42—5021no—
287C42—5021no—
288C42—5021no—
289C42—2511no—
290C23—5021no—
291C23—2511no—
292C23D4×D25111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—5021no—
298Q8—5021no—
299Q8#1686081012yes—
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—42no—
312C2×C4—42no—
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—10041no—
331C22—10041no—
332C22—10041no—
333C22—5021no—
334C22—5021no—
335C22—5021no—
336C22#2254873612yes—
337C22C2×D4×D25111yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—10041no—
344C4—10041no—
345C4—10041no—
346C4—5021no—
347C4—5021no—
348C4—5021no—
349C4C2×#1686081011yescharacteristic
350C4#2254876011yescharacteristic
351C2—22no—
352C2—20081no—
353C2—10041no—
354C2#3063232711yescharacteristic
355C2C2×#2254873611yescharacteristic
356C2#3063250011yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#103×C251C2
socleC22×C251C22×D4
prime radical#30632348C2
hypercenterC2×#103D251
hypocenterC251#3300
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. #30632372C2semidirect
3. #30632043C2semidirect
4. #30632026C2semidirect
5. #30632000C2semidirect
6. #30631992C2semidirect
7. #30631927C2semidirect
8. #30631916C2semidirect
9. #30631912C2semidirect
10. #30631852C2semidirect
11. #30631843C2semidirect
12. #30631779C2semidirect
13. C8×C22⋊D251C2semidirect
14. #30631682C2semidirect
15. C2×#103×C251C2semidirect
178. C251#3300semidirect
194. C2×#103D251semidirect

Maximal subgroups

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

subgroupindex
#306323722
#306323482
#306320432
#306320262
#306320002
#306319922
#306319272
#306319162
#306319122
#306318522
#306318432
#306317792
C8×C22⋊D2512
#306316822
C2×#103×C2512
#3300251