Finite Group Databasebeta

← #44562278

Subgroups — #44562278

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#30632405C211yesmaximal, characteristic, complemented
2#30632336C211yesmaximal, characteristic
3#30632036C211yesmaximal, characteristic, complemented
4#30632028C211yesmaximal, characteristic, complemented
5#30632027C211yesmaximal, characteristic, complemented
6#30631965C211yesmaximal, characteristic, complemented
7#30631928C211yesmaximal, characteristic, complemented
8#30631927C211yesmaximal, characteristic, complemented
9#30631896C211yesmaximal, characteristic, complemented
10#30631850C211yesmaximal, characteristic, complemented
11#30631849C211yesmaximal, characteristic, complemented
12#30631778C211yesmaximal, characteristic, complemented
13C8×C22⋊D251C211yesmaximal, characteristic, complemented
14#30631680C211yesmaximal, characteristic, complemented
15#416×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×Dic1004C2211yescharacteristic
18C2×#16860800C2211yescharacteristic
19C2×#16860798C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21#22548714C2211yescharacteristic
22#22548713C2211yescharacteristic
23#22548709—21no—
24#22548708—21no—
25#22548701—21no—
26#22548698—21no—
27#22548696—21no—
28#22548695C2211yescharacteristic
29#22548694C2211yescharacteristic
30#22548694C2211yescharacteristic
31#22548691C2211yescharacteristic
32#22548687—21no—
33C4×Dic502—21no—
34#22548680C2211yescharacteristic
35D4×Dic251C2211yescharacteristic
36C4×C22⋊D251C2211yescharacteristic
37#22548673C2211yescharacteristic
38C2×Dic251⋊C4—21no—
39#22548649C2211yescharacteristic
40#22548648C2211yescharacteristic
41#22548646C2211yescharacteristic
42Dic251⋊C8C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548640C2211yescharacteristic
45#22548639C2211yescharacteristic
46#22548638C2211yescharacteristic
47#22548638C2211yescharacteristic
48#22548636C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×#41×C251C2211yescharacteristic
53C22×C8×C251C2211yescharacteristic
54#117×C251C2211yescharacteristic
55#103×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59#16860809—22no—
60#16860809—41no—
61C2×Dic502—21no—
62C2×Dic502C2311yescharacteristic
63C2×Dic502C2311yescharacteristic
64C2×C22⋊D251—21no—
65C2×C22⋊D251D411yescharacteristic
66C22×Dic251—21no—
67C22×Dic251—21no—
68C2×C4×D251C2311yescharacteristic
69Dic1004—41no—
70#16860800—41no—
71#16860798—22no—
72C2×C251⋊C8C2311yescharacteristic
73C2×C251⋊C8C2311yescharacteristic
74C8×D251—41no—
75C4⋊Dic251—21no—
76C4⋊Dic251C2311yescharacteristic
77C4⋊Dic251C2311yescharacteristic
78Dic251⋊C4—21no—
79Dic251⋊C4—21no—
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4D411yescharacteristic
86C22⋊Dic251—21no—
87C22⋊Dic251—21no—
88C22⋊Dic251—21no—
89C22⋊Dic251D411yescharacteristic
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4D411yescharacteristic
94C4×Dic251—21no—
95C4×Dic251—21no—
96C4×Dic251C2311yescharacteristic
97#41×C251—22no—
98#41×C251—41no—
99C2×Q8×C251C2311yescharacteristic
100C2×D4×C251—21no—
101C2×D4×C251C2311yescharacteristic
102C22×C4×C251—21no—
103C22×C4×C251C2311yescharacteristic
104C2×C8×C251—41no—
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251C2311yescharacteristic
108C4⋊C4×C251C2311yescharacteristic
109C22⋊C4×C251—21no—
110C42×C251—21no—
111C22×D251C2×D411yescharacteristic
112Dic502—21no—
113Dic502—21no—
114Dic502—21no—
115Dic502—21no—
116Dic502—42no—
117C22⋊D251—22no—
118C22⋊D251—42no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—42no—
124C2×Dic251—42no—
125C2×Dic251C2×D411yescharacteristic
126C2×Dic251C2×D411yescharacteristic
127C2×Dic251C2×D411yescharacteristic
128C4×D251—21no—
129C4×D251—21no—
130C251⋊C8—21no—
131C251⋊C8—21no—
132C23×C251—21no—
133C23×C251C2×D411yescharacteristic
134Q8×C251—21no—
135Q8×C251—21no—
136D4×C251—21no—
137D4×C251—42no—
138D4×C251#4112yes—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—42no—
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2411yescharacteristic
147C8×C251—21no—
148C8×C251—21no—
149C22×C251—21no—
150C22×C251—22no—
151C22×C251—21no—
152C22×C251—21no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251C22×D411yescharacteristic
156C2×D251—21no—
157C2×D251—21no—
158C2×D251—21no—
159C4×C251—21no—
160C4×C251—41no—
161C4×C251—41no—
162C4×C251—41no—
163C4×C251C2×#4111yescharacteristic
164C4×C251H(5,𝔽2)11yescharacteristic
165Dic251—21no—
166Dic251—21no—
167Dic251—41no—
168Dic251—41no—
169Dic251—41no—
170Dic251—41no—
171Dic251#13512yes—
172C2×C251—41no—
173C2×C251—42no—
174C2×C251#54611yescharacteristic
175C2×C251C2×#13511yescharacteristic
176C2×C251#57811yescharacteristic
177D251—41no—
178C251#330211yescharacteristic, complemented, Sylow
179#3302—2511nomaximal, Sylow
180#556—2511no—
181#547—2511no—
182#487—2511no—
183#485—2511no—
184#485—2511no—
185#474—2511no—
186#467—2511no—
187#467—2511no—
188#462—2511no—
189#452—2511no—
190#452—2511no—
191C4×SD16—2511no—
192C8×D4—2511no—
193#416—2511no—
194#416D25111yescharacteristic, complemented
195C2×#41—2511no—
196C2×#41C2×D25111yescharacteristic
197C2×Q16—2511no—
198C2×Q16—2511no—
199C2×SD16—2511no—
200C22×C8—2511no—
201C22×C8C2×D25111yescharacteristic
202C4⋊Q8—2511no—
203C4⋊Q8—2511no—
204#126—5021no—
205#126—5021no—
206#124—5021no—
207#123—5021no—
208#123—5021no—
209C22⋊Q8—2511no—
210C22⋊Q8—2511no—
211C22⋊Q8—2511no—
212C22⋊Q8—2511no—
213#121—5021no—
214C4×Q8—5021no—
215C4×Q8—2511no—
216C4×D4—2511no—
217C4×D4—2511no—
218#117—2511no—
219#117C2×D25111yescharacteristic
220C2×C4⋊C4—5021no—
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#103C2×D25111yescharacteristic
231#103C2×D25111yescharacteristic
232#102—2511no—
233#102C2×D25111yescharacteristic
234#102C2×D25111yescharacteristic
235C22⋊C8—2511no—
236C22⋊C8—2511no—
237C4×C8—2511no—
238#41—22no—
239#41—41no—
240#41—10041no—
241#41—5022no—
242C2×Q8—5021no—
243C2×Q8—2511no—
244C2×Q8—2511no—
245C2×Q8C22×D25111yescharacteristic
246C2×D4—21no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4C22×D25111yescharacteristic
250C22×C4—21no—
251C22×C4—5021no—
252C22×C4—5021no—
253C22×C4—2511no—
254C22×C4C22×D25111yescharacteristic
255Q16—10041no—
256Q16—10041no—
257SD16—5022no—
258C2×C8—41no—
259C2×C8—10041no—
260C2×C8—2511no—
261C2×C8—2511no—
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⋊C4C22×D25111yescharacteristic
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—21no—
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—21no—
287C42—5021no—
288C42—5021no—
289C42—2511no—
290C23—21no—
291C23—2511no—
292C23D4×D25111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—10042no—
296Q8—5021no—
297Q8—5021no—
298Q8—5021no—
299Q8—5021no—
300D4—21no—
301D4—42no—
302D4—10042no—
303D4—5022no—
304D4#1686081012yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—10042no—
312C2×C4—10042no—
313C2×C4—5021no—
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—22no—
332C22—41no—
333C22—41no—
334C22—5021no—
335C22—5021no—
336C22—5021no—
337C22C2×D4×D25111yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—10041no—
343C4—10041no—
344C4—10041no—
345C4—10041no—
346C4—5021no—
347C4—5021no—
348C4—2512no—
349C4C2×#1686081011yescharacteristic
350C4#2254876011yescharacteristic
351C2—41no—
352C2—42no—
353C2—10041no—
354C2#3063232711yescharacteristic
355C2#135×D25111yescharacteristic
356C2#3063250111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#416×C251C2
socleC22×C251C22×D4
prime radical#30632336C2
hypercenter#416D251
hypocenterC251#3302
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #30632405C2semidirect
3. #30632036C2semidirect
4. #30632028C2semidirect
5. #30632027C2semidirect
6. #30631965C2semidirect
7. #30631928C2semidirect
8. #30631927C2semidirect
9. #30631896C2semidirect
10. #30631850C2semidirect
11. #30631849C2semidirect
12. #30631778C2semidirect
13. C8×C22⋊D251C2semidirect
14. #30631680C2semidirect
15. #416×C251C2semidirect
178. C251#3302semidirect
194. #416D251semidirect

Maximal subgroups

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

subgroupindex
#306324052
#306323362
#306320362
#306320282
#306320272
#306319652
#306319282
#306319272
#306318962
#306318502
#306318492
#306317782
C8×C22⋊D2512
#306316802
#416×C2512
#3302251