Finite Group Databasebeta

← #44562239

Subgroups — #44562239

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#30632392C211yesmaximal, characteristic
2#30632350C211yesmaximal, characteristic
3#30632036C211yesmaximal, characteristic, complemented
4#30632024C211yesmaximal, characteristic, complemented
5#30631981C211yesmaximal, characteristic, complemented
6#30631929C211yesmaximal, characteristic, complemented
7#30631926C211yesmaximal, characteristic, complemented
8#30631896C211yesmaximal, characteristic, complemented
9#30631845C211yesmaximal, characteristic, complemented
10#30631844C211yesmaximal, characteristic, complemented
11#30631781C211yesmaximal, characteristic, complemented
12(C22⋊D251)⋊C8C211yesmaximal, characteristic
13C2×#22548638C211yesmaximal, characteristic, complemented
14#103×D251C211yesmaximal, characteristic, complemented
15#484×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic
17C2×Q8×D251C2211yescharacteristic
18C2×Dic1004C2211yescharacteristic
19C2×Dic1004C2211yescharacteristic
20C2×#16860800C2211yescharacteristic
21C22×C251⋊C8C2211yescharacteristic
22C2×C8×D251C2211yescharacteristic
23#22548714—21no—
24#22548714C2211yescharacteristic
25#22548713C2211yescharacteristic
26#22548705—21no—
27#22548695C2211yescharacteristic
28#22548695C2211yescharacteristic
29#22548694—21no—
30#22548694C2211yescharacteristic
31#22548693—21no—
32#22548693—21no—
33#22548692—21no—
34#22548680C2211yescharacteristic
35#22548680C2211yescharacteristic
36C4×C22⋊D251—21no—
37C4×C22⋊D251C2211yescharacteristic
38(C22⋊D251)⋊C4—21no—
39C4⋊C4×D251C2211yescharacteristic
40#22548648C2211yescharacteristic
41Dic251⋊C8C2211yescharacteristic
42#22548640C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548638C2211yescharacteristic
46#22548638C2211yescharacteristic
47#22548638C2211yescharacteristic
48#22548638C2211yescharacteristic
49#22548638C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C22⋊Q8×C251C2211yescharacteristic
53C2×C4⋊C4×C251C2211yescharacteristic
54#107×C251C2211yescharacteristic
55#107×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59C2×Dic502—21no—
60C2×Dic502—21no—
61C2×Dic502—22no—
62C2×Dic502—21no—
63C2×Dic502—41no—
64C2×Dic502C2311yescharacteristic
65C2×Dic502C2311yescharacteristic
66C2×Dic502C2311yescharacteristic
67Q8×D251—22no—
68Q8×D251—41no—
69C2×C22⋊D251D411yescharacteristic
70C22×Dic251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251C2311yescharacteristic
74Dic1004—22no—
75Dic1004—41no—
76#16860800—41no—
77C2×C251⋊C8—41no—
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C8×D251—41no—
81C4⋊Dic251—21no—
82C4⋊Dic251—21no—
83C4⋊Dic251C2311yescharacteristic
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88Dic251⋊C4—21no—
89Dic251⋊C4—21no—
90Dic251⋊C4—21no—
91Dic251⋊C4D411yescharacteristic
92C22⋊Dic251—21no—
93C22⋊Dic251D411yescharacteristic
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4D411yescharacteristic
97C4×Dic251—21no—
98C4×Dic251—21no—
99C4×Dic251C2311yescharacteristic
100C2×Q8×C251C2311yescharacteristic
101C22×C4×C251—21no—
102C22×C4×C251C2311yescharacteristic
103C2×C8×C251C2311yescharacteristic
104C2×C8×C251C2311yescharacteristic
105C4⋊C4×C251—21no—
106C4⋊C4×C251—21no—
107C4⋊C4×C251C2311yescharacteristic
108C4⋊C4×C251C2311yescharacteristic
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C22×D251C2×D411yescharacteristic
112Dic502—21no—
113Dic502—21no—
114Dic502—21no—
115Dic502—21no—
116Dic502—21no—
117Dic502—42no—
118Dic502—42no—
119Dic502—81no—
120Dic502#4112yes—
121C22⋊D251Q1614yescomplemented
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—42no—
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C4×D251—21no—
131C4×D251—21no—
132C4×D251—42no—
133C4×D251—81no—
134C251⋊C8—21no—
135C251⋊C8—21no—
136C23×C251C2×D411yescharacteristic
137Q8×C251—21no—
138Q8×C251—21no—
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—81no—
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—41no—
159C4×C251—41no—
160C4×C251—41no—
161C4×C251—41no—
162C4×C251C2×#4111yescharacteristic
163C4×C251H(5,𝔽2)11yescharacteristic
164Dic251—21no—
165Dic251—22no—
166Dic251—21no—
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#484D25111yescharacteristic, complemented
183#477—2511no—
184#467—2511no—
185#467—2511no—
186#462—2511no—
187#451—2511no—
188#451—2511no—
189C4×Q16—2511no—
190C8×D4—2511no—
191C2×#103—2511no—
192C2×#103—2511no—
193C22×Q8—2511no—
194C22×Q8—2511no—
195C2×Q16—2511no—
196C2×Q16—2511no—
197C2×Q16—2511no—
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⋊C4C2×D25111yescharacteristic
219#107—2511no—
220#107C2×D25111yescharacteristic
221#107C2×D25111yescharacteristic
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—10041no—
237C2×Q8—10041no—
238C2×Q8—5022no—
239C2×Q8—5021no—
240C2×Q8—5021no—
241C2×Q8—5022no—
242C2×Q8—5021no—
243C2×Q8—2511no—
244C2×Q8—2511no—
245C2×Q8—2511no—
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—10041no—
255Q16—10041no—
256Q16—5022no—
257C2×C8—10041no—
258C2×C8—10041no—
259C2×C8—2511no—
260C2×C8—2511no—
261C2×C8C22×D25111yescharacteristic
262C2×C8C22×D25111yescharacteristic
263C4⋊C4—21no—
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—5021no—
273C4⋊C4—5021no—
274C4⋊C4—2511no—
275C4⋊C4—2511no—
276C4⋊C4C22×D25111yescharacteristic
277C4⋊C4C22×D25111yescharacteristic
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—20081no—
293Q8—10042no—
294Q8—10042no—
295Q8—5021no—
296Q8—5021no—
297Q8—5021no—
298Q8—5021no—
299Q8—5021no—
300Q8—2512no—
301D4—2514no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—81no—
309C2×C4—20081no—
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—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—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—10041no—
340C4—10041no—
341C4—10041no—
342C4—5021no—
343C4—5022no—
344C4—5021no—
345C4—2512no—
346C4#41×D25111yescharacteristic
347C4#2254876111yescharacteristic
348C2—22no—
349C2—5022no—
350C2#3063232911yescharacteristic
351C2C2×Q16×D25111yescharacteristic
352C2#3063249711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#484×C251C2
socleC22×C251C22×D4
prime radicalC2×C22⋊D251D4
hypercenter#484D251
hypocenterC251#3299
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #30632036C2semidirect
4. #30632024C2semidirect
5. #30631981C2semidirect
6. #30631929C2semidirect
7. #30631926C2semidirect
8. #30631896C2semidirect
9. #30631845C2semidirect
10. #30631844C2semidirect
11. #30631781C2semidirect
13. C2×#22548638C2semidirect
14. #103×D251C2semidirect
15. #484×C251C2semidirect
121. C22⋊D251Q16semidirect
176. C251#3299semidirect
182. #484D251semidirect

Maximal subgroups

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

subgroupindex
#306323922
#306323502
#306320362
#306320242
#306319812
#306319292
#306319262
#306318962
#306318452
#306318442
#306317812
(C22⋊D251)⋊C82
C2×#225486382
#103×D2512
#484×C2512
#3299251