Finite Group Databasebeta

← #44562226

Subgroups — #44562226

Order 32128.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D251C211yesmaximal, characteristic, complemented
2#30632338C211yesmaximal, characteristic
3#30632036C211yesmaximal, characteristic, complemented
4#30632017C211yesmaximal, characteristic, complemented
5#30632016C211yesmaximal, characteristic, complemented
6#30631970C211yesmaximal, characteristic, complemented
7#30631919C211yesmaximal, characteristic, complemented
8#30631916C211yesmaximal, characteristic, complemented
9#30631891C211yesmaximal, characteristic, complemented
10#30631827C211yesmaximal, characteristic, complemented
11#30631826C211yesmaximal, characteristic, complemented
12#30631772C211yesmaximal, characteristic, complemented
13C8×C22⋊D251C211yesmaximal, characteristic, complemented
14#103×D251C211yesmaximal, characteristic, complemented
15C2×#103×C251C211yesmaximal, characteristic, complemented
16C2×#16860810—21no—
17C2×Q8×D251C2211yescharacteristic
18C2×#16860799C2211yescharacteristic
19C2×#16860797C2211yescharacteristic
20C2×#16860797C2211yescharacteristic
21C2×C8×D251C2211yescharacteristic
22#22548714C2211yescharacteristic
23#22548713—21no—
24#22548713C2211yescharacteristic
25#22548700—21no—
26#22548698—21no—
27#22548694—21no—
28#22548694C2211yescharacteristic
29#22548693—21no—
30#22548691—21no—
31#22548691C2211yescharacteristic
32#22548690C2211yescharacteristic
33#22548689C2211yescharacteristic
34Q8×Dic251C2211yescharacteristic
35#22548677C2211yescharacteristic
36C4×C22⋊D251—21no—
37C4×C22⋊D251C2211yescharacteristic
38C4⋊C4×D251C2211yescharacteristic
39#22548647C2211yescharacteristic
40#22548646C2211yescharacteristic
41#22548646C2211yescharacteristic
42Dic251⋊C8C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548639C2211yescharacteristic
46#22548638C2211yescharacteristic
47#22548638C2211yescharacteristic
48#22548637C2211yescharacteristic
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#16860810—44no—
60C2×Dic502—21no—
61C2×Dic502—21no—
62C2×Dic502C2311yescharacteristic
63Q8×D251—22no—
64Q8×D251—41no—
65C2×D1004—21no—
66C2×D1004C2311yescharacteristic
67C2×C22⋊D251—21no—
68C2×C22⋊D251D411yescharacteristic
69C2×C4×D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251C2311yescharacteristic
73#16860799—41no—
74#16860797—22no—
75#16860797—41no—
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C8×D251—41no—
79C4⋊Dic251—21no—
80C4⋊Dic251C2311yescharacteristic
81C4⋊Dic251C2311yescharacteristic
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4D411yescharacteristic
89C22⋊Dic251—21no—
90C22⋊Dic251D411yescharacteristic
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4D411yescharacteristic
96C4×Dic251—21no—
97C4×Dic251C2311yescharacteristic
98C2×Q8×C251—21no—
99C2×Q8×C251—22no—
100C2×Q8×C251—41no—
101C2×Q8×C251C2311yescharacteristic
102C2×Q8×C251C2311yescharacteristic
103C22×C4×C251—21no—
104C22×C4×C251—21no—
105C22×C4×C251C2311yescharacteristic
106C2×C8×C251—41no—
107C2×C8×C251C2311yescharacteristic
108C2×C8×C251C2311yescharacteristic
109C4⋊C4×C251—21no—
110C4⋊C4×C251C2311yescharacteristic
111C4⋊C4×C251C2311yescharacteristic
112C22×D251—21no—
113C22×D251C2×D411yescharacteristic
114Dic502—21no—
115Dic502—21no—
116Dic502—42no—
117Dic502—42no—
118D1004—21no—
119D1004—21no—
120D1004—42no—
121C22⋊D251—42no—
122C22⋊D251SD1614yescomplemented
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—
139Q8×C251—21no—
140Q8×C251—42no—
141Q8×C251#4112yescomplemented
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—42no—
149C2×C4×C251—42no—
150C2×C4×C251C2×D411yescharacteristic
151C2×C4×C251C2411yescharacteristic
152C8×C251—21no—
153C8×C251—21no—
154C22×C251—21no—
155C22×C251—21no—
156C22×C251C2×SD1612yes—
157C22×C251C22×D411yescharacteristic
158C2×D251—21no—
159C2×D251—21no—
160C2×D251—41no—
161C2×D251—41no—
162C2×D251—41no—
163C2×D251C2×SD1612yes—
164C4×C251—21no—
165C4×C251—22no—
166C4×C251—41no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251C2×#4111yescharacteristic
170C4×C251H(5,𝔽2)11yescharacteristic
171Dic251—21no—
172Dic251—21no—
173Dic251—41no—
174Dic251—41no—
175Dic251—41no—
176Dic251C2×SD1612yes—
177C2×C251—22no—
178C2×C251#54611yescharacteristic
179C2×C251C22×SD1611yescharacteristic
180C2×C251#57811yescharacteristic
181D251—22no—
182D251—81no—
183C251#329811yescharacteristic, complemented, Sylow
184#3298—2511nomaximal, Sylow
185D4×Q8—2511no—
186#547—2511no—
187#487—2511no—
188#483—2511no—
189#483—2511no—
190#475—2511no—
191#465—2511no—
192#465—2511no—
193#461—2511no—
194#448—2511no—
195#448—2511no—
196C4×SD16—2511no—
197C8×D4—2511no—
198C2×#103—2511no—
199C2×#103D25111yescharacteristic, complemented
200C2×#41—5021no—
201C22×Q8—2511no—
202C22×Q8C2×D25111yescharacteristic
203C2×SD16—2511no—
204C2×SD16—2511no—
205C2×SD16—2511no—
206C22×C8—2511no—
207C22×C8C2×D25111yescharacteristic
208C4⋊Q8—5021no—
209C4⋊Q8—2511no—
210C4⋊Q8—2511no—
211#123—5021no—
212#123—5021no—
213C22⋊Q8—5021no—
214C22⋊Q8—5021no—
215C22⋊Q8—5021no—
216C22⋊Q8—2511no—
217C22⋊Q8—2511no—
218#121—2511no—
219#121—2511no—
220C4×Q8—2511no—
221C4×D4—5021no—
222C4×D4—2511no—
223C4×D4—2511no—
224C2×C4⋊C4—2511no—
225C2×C4⋊C4C2×D25111yescharacteristic
226#106—2511no—
227#106—2511no—
228#106—2511no—
229C4⋊C8—2511no—
230#103—2511no—
231#103—2511no—
232#103—2511no—
233#103—2511no—
234#103—2511no—
235#103C2×D25111yescharacteristic
236#103C2×D25111yescharacteristic
237#103C2×D25111yescharacteristic
238#103C2×D25111yescharacteristic
239#102—2511no—
240C22⋊C8—2511no—
241C22⋊C8—2511no—
242C4×C8—2511no—
243#41—10044no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—10041no—
248C2×Q8—5021no—
249C2×Q8—5021no—
250C2×Q8—5022no—
251C2×Q8—2511no—
252C2×Q8C22×D25111yescharacteristic
253C2×Q8C22×D25111yescharacteristic
254C2×D4—5021no—
255C2×D4—5021no—
256C2×D4—2511no—
257C2×D4—2511no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—5021no—
261C22×C4—5021no—
262C22×C4—5021no—
263C22×C4—2511no—
264C22×C4C22×D25111yescharacteristic
265SD16—10041no—
266SD16—10041no—
267SD16—5022no—
268C2×C8—41no—
269C2×C8—10041no—
270C2×C8—2511no—
271C2×C8—2511no—
272C2×C8C22×D25111yescharacteristic
273C2×C8C22×D25111yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—5021no—
276C4⋊C4—5021no—
277C4⋊C4—5021no—
278C4⋊C4—5021no—
279C4⋊C4—5021no—
280C4⋊C4—5021no—
281C4⋊C4—5021no—
282C4⋊C4—2511no—
283C4⋊C4—2511no—
284C4⋊C4—2511no—
285C4⋊C4C22×D25111yescharacteristic
286C4⋊C4C22×D25111yescharacteristic
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—5021no—
292C22⋊C4—2511no—
293C22⋊C4—2511no—
294C42—5021no—
295C42—2511no—
296C23—5021no—
297C23—2511no—
298C23D4×D25111yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—10042no—
304Q8—10042no—
305Q8—5021no—
306Q8—5021no—
307Q8#1686081012yescomplemented
308D4—10042no—
309D4—10042no—
310D4—5021no—
311D4—5021no—
312D4—2514no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—10042no—
322C2×C4—10042no—
323C2×C4—10042no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4—2511no—
330C2×C4—2511no—
331C2×C4—2511no—
332C2×C4D4×D25111yescharacteristic
333C2×C4C23×D25111yescharacteristic
334C8—21no—
335C8—21no—
336C8—5021no—
337C8—5021no—
338C22—21no—
339C22—21no—
340C22—10041no—
341C22—10041no—
342C22—10041no—
343C22—5021no—
344C22—5021no—
345C22—2512no—
346C22SD16×D25112yes—
347C22C2×D4×D25111yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—10041no—
354C4—10041no—
355C4—10041no—
356C4—5021no—
357C4—5021no—
358C4—2512no—
359C4C2×#1686081011yescharacteristic
360C4#2254876011yescharacteristic
361C2—22no—
362C2—20081no—
363C2—5022no—
364C2#3063232711yescharacteristic
365C2C2×SD16×D25111yescharacteristic
366C2#3063250011yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#103×C251C2
socleC22×C251C22×D4
prime radical#30632338C2
hypercenterC2×#103D251
hypocenterC251#3298
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. Q8×C22⋊D251C2semidirect
3. #30632036C2semidirect
4. #30632017C2semidirect
5. #30632016C2semidirect
6. #30631970C2semidirect
7. #30631919C2semidirect
8. #30631916C2semidirect
9. #30631891C2semidirect
10. #30631827C2semidirect
11. #30631826C2semidirect
12. #30631772C2semidirect
13. C8×C22⋊D251C2semidirect
14. #103×D251C2semidirect
15. C2×#103×C251C2semidirect
122. C22⋊D251SD16semidirect
141. Q8×C251#41semidirect
183. C251#3298semidirect
199. C2×#103D251semidirect
307. Q8#16860810semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D2512
#306323382
#306320362
#306320172
#306320162
#306319702
#306319192
#306319162
#306318912
#306318272
#306318262
#306317722
C8×C22⋊D2512
#103×D2512
C2×#103×C2512
#3298251