Finite Group Databasebeta

← #44560615

Subgroups — #44560615

Order 32128.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632298C211yesmaximal, characteristic, complemented
2#30632159C211yesmaximal, characteristic
3#30632027C211yesmaximal, characteristic, complemented
4#30632026C211yesmaximal, characteristic, complemented
5#30631988C211yesmaximal, characteristic, complemented
6#30631987C211yesmaximal, characteristic, complemented
7#30631889C211yesmaximal, characteristic, complemented
8#30631836C211yesmaximal, characteristic
9#30631836C211yesmaximal, characteristic
10#30631790C211yesmaximal, characteristic, complemented
11#30631788C211yesmaximal, characteristic, complemented
12SD16×Dic251C211yesmaximal, characteristic, complemented
13#30631772C211yesmaximal, characteristic, complemented
14#30631747C211yesmaximal, characteristic, complemented
15#461×C251C211yesmaximal, characteristic, complemented
16C2×D4×D251C2211yescharacteristic
17C2×#16860797C2211yescharacteristic
18C2×#16860797C2211yescharacteristic
19#22548710—21no—
20#22548710—21no—
21#22548707C2211yescharacteristic, complemented
22#22548705—21no—
23#22548705—21no—
24#22548691C2211yescharacteristic
25#22548691C2211yescharacteristic
26Q8×Dic251C2211yescharacteristic, complemented
27Q8×Dic251C2211yescharacteristic, complemented
28C4×D1004C2211yescharacteristic
29D4×Dic251C2211yescharacteristic
30#22548677C2211yescharacteristic
31C4×C22⋊D251—21no—
32(C22⋊D251)⋊C4—21no—
33#22548673C2211yescharacteristic
34#22548672C2211yescharacteristic
35C2×(C2×D251)⋊C4—21no—
36C22⋊C4×D251—21no—
37#22548649C2211yescharacteristic, complemented
38#22548648C2211yescharacteristic, complemented
39#22548647C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41C4⋊(C251⋊C8)C2211yescharacteristic, complemented
42#22548639C2211yescharacteristic, complemented
43#22548639C2211yescharacteristic, complemented
44#22548637C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548636C2211yescharacteristic
48(C2×D251)⋊C8C2211yescharacteristic
49(C2×D251)⋊C8C2211yescharacteristic
50#22548625C2211yescharacteristic, complemented
51C8×Dic251C2211yescharacteristic, complemented
52C2×SD16×C251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic
54C4⋊Q8×C251C2211yescharacteristic, complemented
55C4×D4×C251C2211yescharacteristic
56C4⋊C8×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58#103×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60C2×D1004C2311yescharacteristic
61C2×C22⋊D251—21no—
62D4×D251—22no—
63D4×D251—22no—
64C22×Dic251—21no—
65C2×C4×D251—21no—
66C2×C4×D251—21no—
67C2×C4×D251C2311yescharacteristic
68#16860797—22no—
69#16860797—22no—
70C2×C251⋊C8C2311yescharacteristic
71C2×C251⋊C8C2311yescharacteristic
72C4⋊Dic251—21no—
73C4⋊Dic251—21no—
74C4⋊Dic251C2311yescharacteristic
75C4⋊Dic251C2311yescharacteristic
76C4⋊Dic251C2311yescharacteristic
77Dic251⋊C4—21no—
78Dic251⋊C4—21no—
79Dic251⋊C4—21no—
80Dic251⋊C4—21no—
81Dic251⋊C4D411yescharacteristic
82Dic251⋊C4D411yescharacteristic
83C22⋊Dic251—21no—
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4D411yescharacteristic
90(C2×D251)⋊C4D411yescharacteristic
91C4×Dic251—21no—
92C4×Dic251—21no—
93C4×Dic251—21no—
94C4×Dic251C2311yescharacteristic
95C2×Q8×C251C2311yescharacteristic
96C2×Q8×C251C2311yescharacteristic
97C2×D4×C251C2311yescharacteristic
98C22×C4×C251—21no—
99SD16×C251—22no—
100SD16×C251—22no—
101C2×C8×C251C2311yescharacteristic
102C2×C8×C251C2311yescharacteristic
103C4⋊C4×C251—21no—
104C4⋊C4×C251—21no—
105C4⋊C4×C251C2311yescharacteristic
106C22⋊C4×C251—21no—
107C42×C251C2311yescharacteristic
108C22×D251—21no—
109C22×D251—24no—
110C22×D251—81no—
111C22×D251—81no—
112C22×D251C2×D411yescharacteristic
113D1004—21no—
114D1004#4112yes—
115C22⋊D251—24no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—81no—
120C2×Dic251C2×D411yescharacteristic
121C2×Dic251C2×D411yescharacteristic
122C2×Dic251C2×D411yescharacteristic
123C4×D251—21no—
124C4×D251—21no—
125C4×D251—81no—
126C4×D251—81no—
127C251⋊C8—21no—
128C251⋊C8—21no—
129C23×C251—21no—
130Q8×C251—21no—
131Q8×C251—21no—
132Q8×C251—21no—
133Q8×C251—21no—
134D4×C251—21no—
135D4×C251#4112yes—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—81no—
140C2×C4×C251C2×D411yescharacteristic
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2411yescharacteristic
143C8×C251—21no—
144C8×C251—21no—
145C22×C251—22no—
146C22×C251—41no—
147C22×C251—41no—
148C22×C251C22×D411yescharacteristic
149C2×D251—21no—
150C2×D251—22no—
151C2×D251—21no—
152C2×D251—21no—
153C2×D251—41no—
154C2×D251—41no—
155C2×D251—84no—
156C4×C251—21no—
157C4×C251—21no—
158C4×C251—41no—
159C4×C251—41no—
160C4×C251—41no—
161C4×C251C2×#4111yescharacteristic
162C4×C251C2×#4111yescharacteristic
163Dic251—21no—
164Dic251—21no—
165Dic251—21no—
166Dic251—41no—
167Dic251—41no—
168Dic251—41no—
169C2×C251—42no—
170C2×C251#52511yescharacteristic
171C2×C251#57711yescharacteristic
172C2×C251#57711yescharacteristic
173D251—41no—
174D251—42no—
175C251#312311yescharacteristic, complemented, Sylow
176#3123—2511nomaximal, Sylow
177#541—2511no—
178#518—2511no—
179#485—2511no—
180#485—2511no—
181#478—2511no—
182#478—2511no—
183#461—2511no—
184#461D25111yescharacteristic, complemented
185#450—2511no—
186#450—2511no—
187#440—2511no—
188#440—2511no—
189C4×SD16—2511no—
190C4×SD16—2511no—
191#433—2511no—
192C22×D4—2511no—
193C2×SD16—2511no—
194C2×SD16—2511no—
195C2×SD16C2×D25111yescharacteristic
196C2×SD16C2×D25111yescharacteristic
197C4⋊Q8C2×D25111yescharacteristic, complemented
198#126—5021no—
199#126—5021no—
200#125—5021no—
201#125—5021no—
202#125—2511no—
203C22⋊Q8—2511no—
204C22⋊Q8—2511no—
205C4×Q8—2511no—
206C4×Q8—2511no—
207C4×D4—5021no—
208C4×D4—5021no—
209C4×D4—2511no—
210C4×D4—2511no—
211C4×D4—2511no—
212C4×D4C2×D25111yescharacteristic
213#117—2511no—
214#117—2511no—
215C2×C22⋊C4—5021no—
216C2×C22⋊C4—5021no—
217#107—2511no—
218#107—2511no—
219#106—2511no—
220#106—2511no—
221C4⋊C8—2511no—
222C4⋊C8C2×D25111yescharacteristic, complemented
223#103—2511no—
224#103—2511no—
225#103C2×D25111yescharacteristic, complemented
226#103C2×D25111yescharacteristic, complemented
227#102—2511no—
228#102—2511no—
229#102—2511no—
230#102—2511no—
231C22⋊C8—2511no—
232C22⋊C8—2511no—
233#97—2511no—
234C4×C8—2511no—
235C24—5021no—
236C2×Q8C22×D25111yescharacteristic
237C2×Q8C22×D25111yescharacteristic
238C2×D4—5022no—
239C2×D4—5022no—
240C2×D4—5021no—
241C2×D4—2511no—
242C2×D4C22×D25111yescharacteristic
243C22×C4—21no—
244C22×C4—5021no—
245C22×C4—5021no—
246C22×C4—5021no—
247C22×C4—2511no—
248SD16—22no—
249SD16—22no—
250SD16—5022no—
251SD16—5022no—
252C2×C8—2511no—
253C2×C8—2511no—
254C2×C8C22×D25111yescharacteristic
255C2×C8C22×D25111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—5021no—
259C4⋊C4—5021no—
260C4⋊C4—5021no—
261C4⋊C4—5021no—
262C4⋊C4—5021no—
263C4⋊C4—5021no—
264C4⋊C4—2511no—
265C4⋊C4—2511no—
266C4⋊C4—2511no—
267C4⋊C4—2511no—
268C4⋊C4—2511no—
269C4⋊C4C22×D25111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—5021no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4—5021no—
275C22⋊C4—5021no—
276C22⋊C4—5021no—
277C22⋊C4—2511no—
278C22⋊C4—2511no—
279C42—5021no—
280C42—5021no—
281C42—5021no—
282C42—2511no—
283C42C22×D25111yescharacteristic
284C23—21no—
285C23—20081no—
286C23—20081no—
287C23—5024no—
288C23—5021no—
289C23—2511no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294D4—21no—
295D4—5024no—
296D4—5021no—
297D4—2512no—
298D4#1686080912yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—20081no—
304C2×C4—20081no—
305C2×C4—20081no—
306C2×C4—5021no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—5021no—
311C2×C4—2511no—
312C2×C4—2511no—
313C2×C4—2511no—
314C2×C4D4×D25111yescharacteristic
315C2×C4D4×D25111yescharacteristic
316C2×C4C23×D25111yescharacteristic
317C8—21no—
318C8—21no—
319C8—5021no—
320C8—5021no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—20084no—
325C22—10041no—
326C22—10041no—
327C22—5021no—
328C22—5022no—
329C22—5021no—
330C22—5021no—
331C22C2×D4×D25111yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—10041no—
338C4—10041no—
339C4—10041no—
340C4—5021no—
341C4—5021no—
342C4—5021no—
343C4#41×D25111yescharacteristic
344C4C2×#1686080911yescharacteristic
345C2—42no—
346C2—10041no—
347C2—10042no—
348C2#3063219711yescharacteristic
349C2#3063249311yescharacteristic
350C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#461×C251C2
socleC22×C251C22×D4
prime radicalC2×D4×D251C22
hypercenter#461D251
hypocenterC251#3123
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #30632298C2semidirect
3. #30632027C2semidirect
4. #30632026C2semidirect
5. #30631988C2semidirect
6. #30631987C2semidirect
7. #30631889C2semidirect
10. #30631790C2semidirect
11. #30631788C2semidirect
12. SD16×Dic251C2semidirect
13. #30631772C2semidirect
14. #30631747C2semidirect
15. #461×C251C2semidirect
21. #22548707C22semidirect
26. Q8×Dic251C22semidirect
27. Q8×Dic251C22semidirect
37. #22548649C22semidirect
38. #22548648C22semidirect
39. #22548647C22semidirect
40. #22548646C22semidirect
41. C4⋊(C251⋊C8)C22semidirect
42. #22548639C22semidirect
43. #22548639C22semidirect
50. #22548625C22semidirect
51. C8×Dic251C22semidirect
54. C4⋊Q8×C251C22semidirect
56. C4⋊C8×C251C22semidirect
57. #103×C251C22semidirect
58. #103×C251C22semidirect
175. C251#3123semidirect
184. #461D251semidirect
197. C4⋊Q8C2×D251semidirect
222. C4⋊C8C2×D251semidirect
225. #103C2×D251semidirect
226. #103C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322982
#306321592
#306320272
#306320262
#306319882
#306319872
#306318892
#306318362
#306317902
#306317882
SD16×Dic2512
#306317722
#306317472
#461×C2512
#3123251