Finite Group Databasebeta

← #44562234

Subgroups — #44562234

Order 32128.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632395C211yesmaximal, characteristic
2Q8×C22⋊D251C211yesmaximal, characteristic
3#30632036C211yesmaximal, characteristic, complemented
4#30632023C211yesmaximal, characteristic, complemented
5#30632021C211yesmaximal, characteristic, complemented
6#30631980C211yesmaximal, characteristic, complemented
7#30631926C211yesmaximal, characteristic, complemented
8#30631925C211yesmaximal, characteristic, complemented
9#30631895C211yesmaximal, characteristic, complemented
10#30631841C211yesmaximal, characteristic, complemented
11Q16×Dic251C211yesmaximal, characteristic, complemented
12(C22⋊D251)⋊C8C211yesmaximal, characteristic
13C2×#22548639C211yesmaximal, characteristic, complemented
14#103×D251C211yesmaximal, characteristic, complemented
15#451×C251C211yesmaximal, characteristic, complemented
16C2×Q8×D251C2211yescharacteristic
17C2×#16860800C2211yescharacteristic
18C22×C251⋊C8C2211yescharacteristic
19C2×C8×D251C2211yescharacteristic
20#22548714C2211yescharacteristic
21#22548713—21no—
22#22548713C2211yescharacteristic
23#22548705—21no—
24#22548695—21no—
25#22548694—21no—
26#22548694C2211yescharacteristic
27#22548692—21no—
28#22548691—21no—
29#22548691C2211yescharacteristic
30#22548691C2211yescharacteristic
31Q8×Dic251C2211yescharacteristic
32Q8×Dic251C2211yescharacteristic
33C4×C22⋊D251—21no—
34C4×C22⋊D251C2211yescharacteristic
35(C22⋊D251)⋊C4—21no—
36C2×C4⋊Dic251C2211yescharacteristic
37C4⋊C4×D251C2211yescharacteristic
38#22548649C2211yescharacteristic
39#22548649C2211yescharacteristic
40#22548648C2211yescharacteristic
41Dic251⋊C8C2211yescharacteristic
42#22548640C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548639C2211yescharacteristic
46#22548639C2211yescharacteristic
47#22548639C2211yescharacteristic
48#22548639C2211yescharacteristic
49#22548638C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C22×Q8×C251C2211yescharacteristic
53C2×Q16×C251C2211yescharacteristic
54C2×Q16×C251C2211yescharacteristic
55C22⋊Q8×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59C2×Dic502—21no—
60C2×Dic502—21no—
61C2×Dic502C2311yescharacteristic
62Q8×D251—22no—
63Q8×D251—41no—
64C2×C22⋊D251D411yescharacteristic
65C22×Dic251—21no—
66C2×C4×D251—21no—
67C2×C4×D251—21no—
68C2×C4×D251C2311yescharacteristic
69#16860800—41no—
70C2×C251⋊C8—41no—
71C2×C251⋊C8C2311yescharacteristic
72C2×C251⋊C8C2311yescharacteristic
73C8×D251—41no—
74C4⋊Dic251—21no—
75C4⋊Dic251—21no—
76C4⋊Dic251—21no—
77C4⋊Dic251C2311yescharacteristic
78C4⋊Dic251C2311yescharacteristic
79C4⋊Dic251C2311yescharacteristic
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4—21no—
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4D411yescharacteristic
88C22⋊Dic251—21no—
89C22⋊Dic251D411yescharacteristic
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4D411yescharacteristic
93C4×Dic251—21no—
94C4×Dic251—21no—
95C4×Dic251C2311yescharacteristic
96C2×Q8×C251—21no—
97C2×Q8×C251—22no—
98C2×Q8×C251—41no—
99C2×Q8×C251C2311yescharacteristic
100C2×Q8×C251C2311yescharacteristic
101C2×Q8×C251C2311yescharacteristic
102C22×C4×C251—21no—
103C22×C4×C251C2311yescharacteristic
104Q16×C251—22no—
105Q16×C251—41no—
106C2×C8×C251C2311yescharacteristic
107C2×C8×C251C2311yescharacteristic
108C4⋊C4×C251—21no—
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C22×D251C2×D411yescharacteristic
112Dic502—21no—
113Dic502—21no—
114Dic502—42no—
115Dic502—81no—
116C22⋊D251Q1614yescomplemented
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—81no—
122C2×Dic251C2×D411yescharacteristic
123C2×Dic251C2×D411yescharacteristic
124C2×Dic251C2×D411yescharacteristic
125C4×D251—21no—
126C4×D251—21no—
127C4×D251—42no—
128C4×D251—81no—
129C251⋊C8—21no—
130C251⋊C8—21no—
131C23×C251C2×D411yescharacteristic
132Q8×C251—21no—
133Q8×C251—21no—
134Q8×C251—21no—
135Q8×C251—21no—
136Q8×C251—21no—
137Q8×C251—42no—
138Q8×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—21no—
145C2×C4×C251—42no—
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2411yescharacteristic
148C8×C251—21no—
149C8×C251—21no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C251C2×Q1612yes—
153C22×C251C22×D411yescharacteristic
154C2×D251—21no—
155C2×D251—21no—
156C2×D251C2×Q1612yes—
157C4×C251—21no—
158C4×C251—22no—
159C4×C251—41no—
160C4×C251—41no—
161C4×C251—41no—
162C4×C251C2×#4111yescharacteristic
163C4×C251H(5,𝔽2)11yescharacteristic
164Dic251—21no—
165Dic251—21no—
166Dic251—41no—
167Dic251—41no—
168Dic251—41no—
169Dic251—41no—
170Dic251C2×Q1612yes—
171C2×C251—22no—
172C2×C251#54611yescharacteristic
173C2×C251C22×Q1611yescharacteristic
174C2×C251#57711yescharacteristic
175D251—22no—
176C251#329911yescharacteristic, complemented, Sylow
177#3299—2511nomaximal, Sylow
178#554—2511no—
179D4×Q8—2511no—
180#487—2511no—
181#484—2511no—
182#484—2511no—
183#477—2511no—
184#467—2511no—
185#467—2511no—
186#462—2511no—
187#451—2511no—
188#451D25111yescharacteristic, complemented
189C4×Q16—2511no—
190C8×D4—2511no—
191C2×#103—2511no—
192C2×#103—2511no—
193C22×Q8—2511no—
194C22×Q8C2×D25111yescharacteristic
195C2×Q16—2511no—
196C2×Q16C2×D25111yescharacteristic
197C2×Q16C2×D25111yescharacteristic
198C22×C8—2511no—
199C22×C8—2511no—
200C4⋊Q8—5021no—
201C4⋊Q8—2511no—
202C4⋊Q8—2511no—
203#125—5021no—
204C22⋊Q8—5021no—
205C22⋊Q8—5021no—
206C22⋊Q8—5021no—
207C22⋊Q8—5021no—
208C22⋊Q8—2511no—
209C22⋊Q8—2511no—
210C22⋊Q8—2511no—
211C22⋊Q8C2×D25111yescharacteristic
212C4×Q8—2511no—
213C4×Q8—2511no—
214C4×D4—5021no—
215C4×D4—5021no—
216C4×D4—2511no—
217C2×C4⋊C4—2511no—
218C2×C4⋊C4—2511no—
219#107—2511no—
220#107—2511no—
221#107—2511no—
222C4⋊C8—2511no—
223#103—2511no—
224#103—2511no—
225#103—2511no—
226#103—2511no—
227#103—2511no—
228#103—2511no—
229#103—2511no—
230#103—2511no—
231#103C2×D25111yescharacteristic
232#103C2×D25111yescharacteristic
233C22⋊C8—2511no—
234C22⋊C8C2×D25111yescharacteristic
235C4×C8—2511no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—10041no—
240C2×Q8—5021no—
241C2×Q8—5021no—
242C2×Q8—5022no—
243C2×Q8—2511no—
244C2×Q8C22×D25111yescharacteristic
245C2×Q8C22×D25111yescharacteristic
246C2×Q8C22×D25111yescharacteristic
247C2×D4—2511no—
248C22×C4—21no—
249C22×C4—5021no—
250C22×C4—5021no—
251C22×C4—5021no—
252C22×C4—2511no—
253C22×C4C22×D25111yescharacteristic
254Q16—22no—
255Q16—41no—
256Q16—10041no—
257C2×C8—10041no—
258C2×C8—10041no—
259C2×C8—2511no—
260C2×C8—2511no—
261C2×C8C22×D25111yescharacteristic
262C2×C8C22×D25111yescharacteristic
263C4⋊C4—21no—
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—5021no—
273C4⋊C4—5021no—
274C4⋊C4—2511no—
275C4⋊C4—2511no—
276C4⋊C4—2511no—
277C4⋊C4—2511no—
278C4⋊C4C22×D25111yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—2511no—
284C22⋊C4—2511no—
285C42—5021no—
286C42—5021no—
287C42—2511no—
288C23—2511no—
289C23D4×D25111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—20081no—
297Q8—10042no—
298Q8—5021no—
299Q8—5021no—
300Q8#1686080912yes—
301D4—2514no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—20081no—
310C2×C4—20081no—
311C2×C4—10042no—
312C2×C4—5021no—
313C2×C4—5021no—
314C2×C4—5021no—
315C2×C4—5021no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—2511no—
319C2×C4—2511no—
320C2×C4—2511no—
321C2×C4D4×D25111yescharacteristic
322C2×C4C23×D25111yescharacteristic
323C8—21no—
324C8—21no—
325C8—5021no—
326C8—5021no—
327C22—21no—
328C22—21no—
329C22—5021no—
330C22—5021no—
331C22—2512no—
332C22Q16×D25112yes—
333C22C2×D4×D25111yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—10041no—
340C4—10041no—
341C4—10041no—
342C4—10041no—
343C4—5021no—
344C4—5021no—
345C4—2512no—
346C4C2×#1686080911yescharacteristic
347C4#2254876011yescharacteristic
348C2—22no—
349C2—5022no—
350C2#3063232411yescharacteristic
351C2C2×Q16×D25111yescharacteristic
352C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#451×C251C2
socleC22×C251C22×D4
prime radicalC2×C22⋊D251D4
hypercenter#451D251
hypocenterC251#3299
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #30632036C2semidirect
4. #30632023C2semidirect
5. #30632021C2semidirect
6. #30631980C2semidirect
7. #30631926C2semidirect
8. #30631925C2semidirect
9. #30631895C2semidirect
10. #30631841C2semidirect
11. Q16×Dic251C2semidirect
13. C2×#22548639C2semidirect
14. #103×D251C2semidirect
15. #451×C251C2semidirect
116. C22⋊D251Q16semidirect
176. C251#3299semidirect
188. #451D251semidirect

Maximal subgroups

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

subgroupindex
#306323952
Q8×C22⋊D2512
#306320362
#306320232
#306320212
#306319802
#306319262
#306319252
#306318952
#306318412
Q16×Dic2512
(C22⋊D251)⋊C82
C2×#225486392
#103×D2512
#451×C2512
#3299251