Finite Group Databasebeta

← #44560626

Subgroups — #44560626

Order 32128.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632306C211yesmaximal, characteristic, complemented
2#30632169C211yesmaximal, characteristic, complemented
3#30632021C211yesmaximal, characteristic, complemented
4#30631999C211yesmaximal, characteristic, complemented
5#30631982C211yesmaximal, characteristic, complemented
6#30631968C211yesmaximal, characteristic, complemented
7#30631899C211yesmaximal, characteristic, complemented
8#30631848C211yesmaximal, characteristic, complemented
9#30631831C211yesmaximal, characteristic
10#30631790C211yesmaximal, characteristic, complemented
11#30631788C211yesmaximal, characteristic, complemented
12Q16×Dic251C211yesmaximal, characteristic, complemented
13#30631767C211yesmaximal, characteristic, complemented
14#30631747C211yesmaximal, characteristic, complemented
15#464×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860797C2211yescharacteristic
18C2×#16860795C2211yescharacteristic
19#22548715C2211yescharacteristic
20#22548708—21no—
21#22548701—21no—
22#22548699—21no—
23#22548692—21no—
24#22548691C2211yescharacteristic
25#22548686C2211yescharacteristic
26Q8×Dic251C2211yescharacteristic
27Q8×Dic251C2211yescharacteristic
28C4×D1004—21no—
29C4×D1004C2211yescharacteristic
30D4×Dic251C2211yescharacteristic
31#22548677—21no—
32#22548677C2211yescharacteristic
33#22548672—21no—
34#22548672C2211yescharacteristic
35C4⋊C4×D251—21no—
36C4⋊C4×D251C2211yescharacteristic
37#22548649C2211yescharacteristic
38#22548649C2211yescharacteristic
39#22548648C2211yescharacteristic
40#22548648C2211yescharacteristic
41C4⋊(C251⋊C8)C2211yescharacteristic
42#22548639C2211yescharacteristic
43#22548639C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548636C2211yescharacteristic
48(C2×D251)⋊C8C2211yescharacteristic
49(C2×D251)⋊C8C2211yescharacteristic
50#22548625C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×Q16×C251C2211yescharacteristic
53C2×SD16×C251C2211yescharacteristic
54#124×C251C2211yescharacteristic
55C4×Q8×C251C2211yescharacteristic
56C4⋊C8×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59#16860811—22no—
60#16860811—22no—
61C2×Dic502—21no—
62C2×D1004—21no—
63C2×D1004C2311yescharacteristic
64C2×C22⋊D251—21no—
65C22×Dic251—21no—
66C2×C4×D251—21no—
67C2×C4×D251—21no—
68C2×C4×D251—21no—
69C2×C4×D251—21no—
70C2×C4×D251C2311yescharacteristic
71#16860797—22no—
72#16860795—22no—
73C2×C251⋊C8C2311yescharacteristic
74C2×C251⋊C8C2311yescharacteristic
75C4⋊Dic251—21no—
76C4⋊Dic251—21no—
77C4⋊Dic251—21no—
78C4⋊Dic251C2311yescharacteristic
79C4⋊Dic251C2311yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4D411yescharacteristic
84Dic251⋊C4D411yescharacteristic
85C22⋊Dic251—21no—
86C22⋊Dic251—21no—
87(C2×D251)⋊C4—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4D411yescharacteristic
91(C2×D251)⋊C4D411yescharacteristic
92C4×Dic251—21no—
93C4×Dic251—21no—
94C4×Dic251C2311yescharacteristic
95C2×Q8×C251C2311yescharacteristic
96C2×Q8×C251C2311yescharacteristic
97C2×D4×C251C2311yescharacteristic
98Q16×C251—22no—
99SD16×C251—22no—
100C2×C8×C251C2311yescharacteristic
101C2×C8×C251C2311yescharacteristic
102C4⋊C4×C251—21no—
103C4⋊C4×C251C2311yescharacteristic
104C22⋊C4×C251—21no—
105C22⋊C4×C251—21no—
106C42×C251—21no—
107C42×C251C2311yescharacteristic
108C22×D251—21no—
109C22×D251C2×D411yescharacteristic
110Dic502—81no—
111D1004—24no—
112D1004—21no—
113D1004#4112yes—
114C22⋊D251—81no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—81no—
119C2×Dic251C2×D411yescharacteristic
120C2×Dic251C2×D411yescharacteristic
121C2×Dic251C2×D411yescharacteristic
122C4×D251—24no—
123C4×D251—21no—
124C4×D251—21no—
125C4×D251—81no—
126C4×D251—81no—
127C4×D251—81no—
128C251⋊C8—21no—
129C251⋊C8—21no—
130C23×C251—21no—
131Q8×C251—21no—
132Q8×C251—21no—
133Q8×C251—21no—
134Q8×C251#4112yes—
135D4×C251—21no—
136D4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251C2×D411yescharacteristic
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2411yescharacteristic
143C8×C251—21no—
144C8×C251—21no—
145C22×C251—41no—
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—
155C4×C251—22no—
156C4×C251—21no—
157C4×C251—21no—
158C4×C251—41no—
159C4×C251—41no—
160C4×C251C2×#4111yescharacteristic
161C4×C251C2×#4111yescharacteristic
162Dic251—21no—
163Dic251—21no—
164Dic251—21no—
165Dic251—41no—
166Dic251—41no—
167Dic251—41no—
168C2×C251—81no—
169C2×C251#52511yescharacteristic
170C2×C251#57611yescharacteristic
171C2×C251#57811yescharacteristic
172D251—41no—
173D251—42no—
174C251#312411yescharacteristic, complemented, Sylow
175#3124—2511nomaximal, Sylow
176#542—2511no—
177#520—2511no—
178#484—2511no—
179#480—2511no—
180#477—2511no—
181#474—2511no—
182#464D25111yescharacteristic, complemented
183#463—2511no—
184#452—2511no—
185#449—2511no—
186#440—2511no—
187#440—2511no—
188C4×Q16—2511no—
189C4×D8—2511no—
190#433—2511no—
191C2×#41—2511no—
192C2×Q16C2×D25111yescharacteristic
193C2×SD16—2511no—
194C2×SD16C2×D25111yescharacteristic
195C2×D8—2511no—
196C4⋊Q8—2511no—
197#126—5021no—
198#124—5021no—
199#124C2×D25111yescharacteristic
200#123—5021no—
201C22⋊Q8—5021no—
202C22⋊Q8—2511no—
203#121—2511no—
204C4×Q8—2511no—
205C4×Q8—2511no—
206C4×Q8C2×D25111yescharacteristic
207C4×D4—5021no—
208C4×D4—5021no—
209C4×D4—2511no—
210C4×D4—2511no—
211C4×D4—2511no—
212#117—5021no—
213#117—2511no—
214C2×C4⋊C4—5021no—
215C2×C4⋊C4—2511no—
216#107—2511no—
217#107—2511no—
218#107—2511no—
219#107—2511no—
220C4⋊C8—2511no—
221C4⋊C8C2×D25111yescharacteristic
222#103—2511no—
223#103—2511no—
224#103—2511no—
225#103C2×D25111yescharacteristic
226#102—2511no—
227#102—2511no—
228#102—2511no—
229#102C2×D25111yescharacteristic
230C22⋊C8—2511no—
231C22⋊C8—2511no—
232#97—2511no—
233C4×C8—2511no—
234#41—5022no—
235#41—5022no—
236C2×Q8—5021no—
237C2×Q8C22×D25111yescharacteristic
238C2×Q8C22×D25111yescharacteristic
239C2×D4—5021no—
240C2×D4—5021no—
241C2×D4—2511no—
242C2×D4C22×D25111yescharacteristic
243C22×C4—5021no—
244C22×C4—5021no—
245C22×C4—5021no—
246C22×C4—5021no—
247C22×C4—5021no—
248C22×C4—2511no—
249Q16—22no—
250SD16—22no—
251SD16—5022no—
252D8—5022no—
253C2×C8—2511no—
254C2×C8—2511no—
255C2×C8C22×D25111yescharacteristic
256C2×C8C22×D25111yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—5021no—
259C4⋊C4—5021no—
260C4⋊C4—5021no—
261C4⋊C4—5021no—
262C4⋊C4—5021no—
263C4⋊C4—2511no—
264C4⋊C4—2511no—
265C4⋊C4—2511no—
266C4⋊C4—2511no—
267C4⋊C4—2511no—
268C4⋊C4C22×D25111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—5021no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4—5021no—
275C22⋊C4—5021no—
276C22⋊C4—2511no—
277C22⋊C4—2511no—
278C42—21no—
279C42—5021no—
280C42—5021no—
281C42—2511no—
282C42C22×D25111yescharacteristic
283C23—21no—
284C23—5021no—
285C23—2511no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—20081no—
290Q8#1686080912yes—
291D4—21no—
292D4—21no—
293D4—20081no—
294D4—5024no—
295D4—5021no—
296D4—2512no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—20081no—
301C2×C4—20081no—
302C2×C4—20081no—
303C2×C4—20081no—
304C2×C4—5024no—
305C2×C4—5021no—
306C2×C4—5021no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—2511no—
311C2×C4—2511no—
312C2×C4—2511no—
313C2×C4D4×D25111yescharacteristic
314C2×C4D4×D25111yescharacteristic
315C2×C4C23×D25111yescharacteristic
316C8—21no—
317C8—21no—
318C8—5021no—
319C8—5021no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—10041no—
324C22—10041no—
325C22—5021no—
326C22—5022no—
327C22—5021no—
328C22—5021no—
329C22C2×D4×D25111yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—10041no—
336C4—10041no—
337C4—10041no—
338C4—5021no—
339C4—5021no—
340C4—5021no—
341C4#41×D25111yescharacteristic
342C4C2×#1686080911yescharacteristic
343C2—81no—
344C2—10042no—
345C2—10041no—
346C2#3063219711yescharacteristic
347C2#3063248911yescharacteristic
348C2#3063250311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#464×C251C2
socleC22×C251C22×D4
prime radical#30631831C2
hypercenter#464D251
hypocenterC251#3124
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #30632306C2semidirect
2. #30632169C2semidirect
3. #30632021C2semidirect
4. #30631999C2semidirect
5. #30631982C2semidirect
6. #30631968C2semidirect
7. #30631899C2semidirect
8. #30631848C2semidirect
10. #30631790C2semidirect
11. #30631788C2semidirect
12. Q16×Dic251C2semidirect
13. #30631767C2semidirect
14. #30631747C2semidirect
15. #464×C251C2semidirect
174. C251#3124semidirect
182. #464D251semidirect

Maximal subgroups

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

subgroupindex
#306323062
#306321692
#306320212
#306319992
#306319822
#306319682
#306318992
#306318482
#306318312
#306317902
#306317882
Q16×Dic2512
#306317672
#306317472
#464×C2512
#3124251