Finite Group Databasebeta

← #44560921

Subgroups — #44560921

Order 32128.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632241C211yesmaximal, characteristic, complemented
2#30632234C211yesmaximal, characteristic
3#30632084C211yesmaximal, characteristic, complemented
4#30632058C211yesmaximal, characteristic, complemented
5#30632036C211yesmaximal, characteristic, complemented
6#30632032C211yesmaximal, characteristic, complemented
7#30631919C211yesmaximal, characteristic, complemented
8#30631917C211yesmaximal, characteristic, complemented
9#30631915C211yesmaximal, characteristic, complemented
10#30631913C211yesmaximal, characteristic, complemented
11C4×#16860799C211yesmaximal, characteristic, complemented
12C4×#16860797C211yesmaximal, characteristic, complemented
13C4×#16860798C211yesmaximal, characteristic, complemented
14C4×C8×D251C211yesmaximal, characteristic, complemented
15C4×SD16×C251C211yesmaximal, characteristic, complemented
16C2×#16860810—21no—
17C2×#16860799C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C2×#16860797C2211yescharacteristic
20C2×C8×D251C2211yescharacteristic
21C2×C8×D251C2211yescharacteristic
22#22548714C2211yescharacteristic
23#22548714C2211yescharacteristic
24#22548713C2211yescharacteristic
25#22548711C2211yescharacteristic
26#22548705—21no—
27#22548701—21no—
28#22548695C2211yescharacteristic
29#22548693—21no—
30#22548691C2211yescharacteristic
31#22548690C2211yescharacteristic
32#22548688—21no—
33#22548686C2211yescharacteristic
34C4×Dic502—21no—
35C4×Dic502C2211yescharacteristic
36C4×D1004C2211yescharacteristic
37C4×C22⋊D251—21no—
38#22548672—21no—
39C42×D251C2211yescharacteristic
40#22548647C2211yescharacteristic
41#22548646C2211yescharacteristic
42#22548646C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548638C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548635C2211yescharacteristic
49C4×C251⋊C8C2211yescharacteristic
50C8×Dic251C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×SD16×C251C2211yescharacteristic
53C4×Q8×C251C2211yescharacteristic
54C4×D4×C251C2211yescharacteristic
55#106×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C4×C8×C251C2211yescharacteristic
59#16860810—44no—
60C2×Dic502—21no—
61C2×Dic502—21no—
62C2×Dic502C2311yescharacteristic
63C2×D1004—21no—
64C2×D1004C2311yescharacteristic
65C2×C22⋊D251—21no—
66C2×C22⋊D251—21no—
67C2×C22⋊D251—21no—
68C2×C4×D251—21no—
69C2×C4×D251D411yescharacteristic
70C2×C4×D251D411yescharacteristic
71C2×C4×D251C2311yescharacteristic
72#16860799—22no—
73#16860798—22no—
74#16860797—22no—
75C2×C251⋊C8C2311yescharacteristic
76C2×C251⋊C8C2311yescharacteristic
77C8×D251—22no—
78C8×D251—22no—
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—
88C22⋊Dic251—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93C4×Dic251—21no—
94C4×Dic251D411yescharacteristic
95C4×Dic251D411yescharacteristic
96C4×Dic251C2311yescharacteristic
97C2×Q8×C251C2311yescharacteristic
98C2×D4×C251C2311yescharacteristic
99C22×C4×C251—21no—
100SD16×C251—22no—
101C2×C8×C251C2311yescharacteristic
102C2×C8×C251C2311yescharacteristic
103C4⋊C4×C251—21no—
104C4⋊C4×C251C2311yescharacteristic
105C4⋊C4×C251C2311yescharacteristic
106C22⋊C4×C251—21no—
107C42×C251—21no—
108C42×C251C2311yescharacteristic
109C22×D251—21no—
110C22×D251C2×D411yescharacteristic
111Dic502—21no—
112Dic502—21no—
113Dic502—42no—
114Dic502—42no—
115D1004—21no—
116D1004—21no—
117D1004—42no—
118C22⋊D251—42no—
119C22⋊D251—42no—
120C22⋊D251—42no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C2×Dic251C2×D411yescharacteristic
127C4×D251—21no—
128C4×D251—21no—
129C4×D251—22no—
130C4×D251—42no—
131C4×D251SD1614yescomplemented
132C251⋊C8—21no—
133C251⋊C8#4112yescomplemented
134C23×C251—21no—
135Q8×C251—21no—
136Q8×C251—21no—
137D4×C251—21no—
138D4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—42no—
143C2×C4×C251C2×D411yescharacteristic
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2411yescharacteristic
146C8×C251—21no—
147C8×C251#4112yescomplemented
148C22×C251—41no—
149C22×C251—41no—
150C22×C251—41no—
151C22×C251C22×D411yescharacteristic
152C2×D251—21no—
153C2×D251—21no—
154C2×D251—41no—
155C2×D251—41no—
156C2×D251—41no—
157C2×D251C2×SD1612yes—
158C4×C251—21no—
159C4×C251—41no—
160C4×C251—41no—
161C4×C251—41no—
162C4×C251C2×SD1612yes—
163C4×C251C2×#4111yescharacteristic
164C4×C251C2×#4111yescharacteristic
165Dic251—21no—
166Dic251—41no—
167Dic251—41no—
168Dic251—41no—
169Dic251C2×SD1612yes—
170Dic251#13512yes—
171C2×C251—81no—
172C2×C251#53211yescharacteristic
173C2×C251C22×SD1611yescharacteristic
174C2×C251C2×#13511yescharacteristic
175D251—22no—
176D251—81no—
177C251#316711yescharacteristic, complemented, Sylow
178#3167—2511nomaximal, Sylow
179#533—2511no—
180#532—2511no—
181#498—2511no—
182#492—2511no—
183#487—2511no—
184#486—2511no—
185#465—2511no—
186#465—2511no—
187#465—2511no—
188#465—2511no—
189C4×SD16—2511no—
190C4×SD16—2511no—
191C4×SD16—2511no—
192C4×SD16D25111yescharacteristic, complemented
193C2×C4×C8—2511no—
194C2×#41—5021no—
195C2×SD16—2511no—
196C2×SD16—2511no—
197C2×SD16—2511no—
198C2×SD16C2×D25111yescharacteristic
199C22×C8—2511no—
200C22×C8—2511no—
201C4⋊Q8—2511no—
202C4⋊Q8—2511no—
203C4⋊Q8—2511no—
204#127—2511no—
205#125—5021no—
206#124—5021no—
207C22⋊Q8—5021no—
208C22⋊Q8—2511no—
209C22⋊Q8—2511no—
210#121—5021no—
211#121—2511no—
212#121—2511no—
213C4×Q8—5021no—
214C4×Q8—2511no—
215C4×Q8C2×D25111yescharacteristic
216C4×D4—5021no—
217C4×D4—2511no—
218C4×D4C2×D25111yescharacteristic
219#117—5021no—
220C2×C42—2511no—
221#106—2511no—
222#106—2511no—
223#106—2511no—
224#106C2×D25111yescharacteristic
225#103—2511no—
226#103—2511no—
227#103—2511no—
228#103C2×D25111yescharacteristic
229#102—2511no—
230#102—2511no—
231#102—2511no—
232#102C2×D25111yescharacteristic
233C4×C8—2511no—
234C4×C8—2511no—
235C4×C8—2511no—
236C4×C8C2×D25111yescharacteristic
237#41—10044no—
238C2×Q8—5021no—
239C2×Q8—5021no—
240C2×Q8—2511no—
241C2×Q8C22×D25111yescharacteristic
242C2×D4—5021no—
243C2×D4—5021no—
244C2×D4—5021no—
245C2×D4—5021no—
246C2×D4—2511no—
247C2×D4C22×D25111yescharacteristic
248C22×C4—21no—
249C22×C4—5021no—
250C22×C4—2511no—
251C22×C4—2511no—
252C22×C4—2511no—
253SD16—22no—
254SD16—5022no—
255SD16—5022no—
256SD16—5022no—
257C2×C8—5022no—
258C2×C8—5022no—
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—2511no—
272C4⋊C4—2511no—
273C4⋊C4C22×D25111yescharacteristic
274C4⋊C4C22×D25111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—5021no—
277C22⋊C4—5021no—
278C22⋊C4—5021no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C42—21no—
282C42—5021no—
283C42—2511no—
284C42—2511no—
285C42—2511no—
286C42C22×D25111yescharacteristic
287C23—21no—
288C23—5021no—
289C23—2511no—
290Q8—21no—
291Q8—21no—
292Q8—10042no—
293Q8—10042no—
294Q8—5021no—
295Q8—5021no—
296D4—21no—
297D4—21no—
298D4—10042no—
299D4—10042no—
300D4—10042no—
301D4—10042no—
302D4—5021no—
303D4—5021no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—10042no—
309C2×C4—5022no—
310C2×C4—5021no—
311C2×C4—5021no—
312C2×C4—5021no—
313C2×C4—5021no—
314C2×C4—5021no—
315C2×C4—2511no—
316C2×C4—2511no—
317C2×C4—2511no—
318C2×C4—2514no—
319C2×C4D4×D25111yescharacteristic
320C2×C4D4×D25111yescharacteristic
321C2×C4C23×D25111yescharacteristic
322C8—21no—
323C8—5021no—
324C8—2512no—
325C8#1686081012yescomplemented
326C22—41no—
327C22—41no—
328C22—41no—
329C22—10041no—
330C22—10041no—
331C22—10041no—
332C22—5021no—
333C22—5021no—
334C22—2512no—
335C22C2×D4×D25111yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—10041no—
341C4—10041no—
342C4—10041no—
343C4—5021no—
344C4—2512no—
345C4—2512no—
346C4SD16×D25112yes—
347C4#41×D25111yescharacteristic
348C4C2×#1686081011yescharacteristic
349C2—81no—
350C2—20081no—
351C2—5022no—
352C2#3063223411yescharacteristic
353C2C2×SD16×D25111yescharacteristic
354C2#135×D25111yescharacteristic
subgroupG/H
centerC2×C4D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC4×SD16×C251C2
socleC22×C251C22×D4
prime radical#30632234C2
hypercenterC4×SD16D251
hypocenterC251#3167
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #30632241C2semidirect
3. #30632084C2semidirect
4. #30632058C2semidirect
5. #30632036C2semidirect
6. #30632032C2semidirect
7. #30631919C2semidirect
8. #30631917C2semidirect
9. #30631915C2semidirect
10. #30631913C2semidirect
11. C4×#16860799C2semidirect
12. C4×#16860797C2semidirect
13. C4×#16860798C2semidirect
14. C4×C8×D251C2semidirect
15. C4×SD16×C251C2semidirect
131. C4×D251SD16semidirect
133. C251⋊C8#41semidirect
147. C8×C251#41semidirect
177. C251#3167semidirect
192. C4×SD16D251semidirect
325. C8#16860810semidirect

Maximal subgroups

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

subgroupindex
#306322412
#306322342
#306320842
#306320582
#306320362
#306320322
#306319192
#306319172
#306319152
#306319132
C4×#168607992
C4×#168607972
C4×#168607982
C4×C8×D2512
C4×SD16×C2512
#3167251