Finite Group Databasebeta

← #44558061

Subgroups — #44558061

Order 32128.

472 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30631629C211yesmaximal, characteristic, complemented
2#30631611C211yesmaximal, characteristic, complemented
3#30631609C211yesmaximal, characteristic, complemented
4#30631608C211yesmaximal, characteristic, complemented
5#30631600C211yesmaximal, characteristic, complemented
6#30631600C211yesmaximal, characteristic, complemented
7#30631589C211yesmaximal, characteristic
8#30631571C211yesmaximal, characteristic, complemented
9#30631565C211yesmaximal, characteristic
10#30631565C211yesmaximal, characteristic
11#30631558C211yesmaximal, characteristic, complemented
12#30631558C211yesmaximal, characteristic, complemented
13#30631546C211yesmaximal, characteristic, complemented
14#30631544C211yesmaximal, characteristic, complemented
15#387×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C22×C4×D251C2211yescharacteristic
19C22×C4×D251C2211yescharacteristic
20C2×C4⋊Dic251C2211yescharacteristic
21C2×C4⋊Dic251C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2211yescharacteristic
27C2×(C2×D251)⋊C4C2211yescharacteristic
28C2×(C2×D251)⋊C4C2211yescharacteristic
29C2×(C2×D251)⋊C4C2211yescharacteristic
30C2×(C2×D251)⋊C4C2211yescharacteristic
31C2×(C2×D251)⋊C4C2211yescharacteristic
32C2×(C2×D251)⋊C4C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C2×C4⋊C4×C251C2211yescharacteristic
45C2×C4⋊C4×C251C2211yescharacteristic
46C2×C42×C251C2211yescharacteristic
47#95×C251C2211yescharacteristic
48#95×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51C23×D251C2311yescharacteristic
52C23×D251C2311yescharacteristic
53C2×Dic502—41no—
54C2×Dic502—41no—
55C2×Dic502—41no—
56C2×D1004—41no—
57C2×D1004—41no—
58C2×D1004Q814yescomplemented
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C22×Dic251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C2×C4×D251—22no—
66C2×C4×D251—22no—
67C2×C4×D251—41no—
68C2×C4×D251—41no—
69C2×C4×D251—41no—
70C2×C4×D251—41no—
71C4⋊Dic251—41no—
72C4⋊Dic251—41no—
73Dic251⋊C4—22no—
74Dic251⋊C4—41no—
75Dic251⋊C4—41no—
76Dic251⋊C4—41no—
77(C2×D251)⋊C4—22no—
78(C2×D251)⋊C4—22no—
79(C2×D251)⋊C4—41no—
80(C2×D251)⋊C4—41no—
81(C2×D251)⋊C4—41no—
82(C2×D251)⋊C4—41no—
83(C2×D251)⋊C4—41no—
84(C2×D251)⋊C4—41no—
85(C2×D251)⋊C4—41no—
86(C2×D251)⋊C4—41no—
87C22×C4×C251C2311yescharacteristic
88C22×C4×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C4⋊C4×C251—22no—
95C4⋊C4×C251—41no—
96C42×C251—41no—
97C22×D251—21no—
98C22×D251—21no—
99C22×D251—21no—
100C22×D251—21no—
101C22×D251—21no—
102C22×D251—21no—
103C22×D251—21no—
104C22×D251—21no—
105C22×D251—21no—
106C22×D251—21no—
107C22×D251—21no—
108C22×D251—21no—
109C22×D251C2×Q812yes—
110C22×D251C2×Q812yes—
111Dic502—44no—
112D1004—44no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251#4112yes—
131C4×D251—44no—
132C4×D251—44no—
133C23×C251C2411yescharacteristic
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×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—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251—21no—
153C2×C4×C251C2×Q812yes—
154C2×C4×C251#4112yes—
155C22×C251C22×Q811yescharacteristic
156C22×C251C2×#4111yescharacteristic
157C22×C251C2×#4111yescharacteristic
158C22×C251H(5,𝔽2)11yescharacteristic
159C22×C251H(5,𝔽2)11yescharacteristic
160C22×C251H(5,𝔽2)11yescharacteristic
161C22×C251H(5,𝔽2)11yescharacteristic
162C2×D251—22no—
163C2×D251—22no—
164C2×D251—22no—
165C2×D251—22no—
166C2×D251—22no—
167C2×D251—22no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—41no—
172C2×D251—41no—
173C2×D251—41no—
174C2×D251—41no—
175C2×D251—41no—
176C4×C251—22no—
177C4×C251—22no—
178C4×C251—41no—
179C4×C251—41no—
180C4×C251—41no—
181C4×C251—41no—
182C4×C251—41no—
183Dic251—22no—
184Dic251—41no—
185Dic251—41no—
186Dic251—41no—
187Dic251—41no—
188Dic251—41no—
189C2×C251#53811yescharacteristic
190C2×C251#53811yescharacteristic
191C2×C251#54311yescharacteristic
192C2×C251#55411yescharacteristic
193C2×C251#55411yescharacteristic
194C2×C251#55511yescharacteristic
195C2×C251#56011yescharacteristic
196D251—42no—
197D251—42no—
198C251#278411yescharacteristic, complemented, Sylow
199#2784—2511nomaximal, Sylow
200#400—2511no—
201#396—2511no—
202#396—2511no—
203#395—2511no—
204#393—2511no—
205#393—2511no—
206#390—2511no—
207#387—2511no—
208#387D25111yescharacteristic, complemented
209#386—2511no—
210#386—2511no—
211#385—2511no—
212#385—2511no—
213#380—2511no—
214#380—2511no—
215C22×Q8—2511no—
216C22×D4—2511no—
217C23×C4—2511no—
218C23×C4—2511no—
219C2×C4⋊C4—2511no—
220C2×C4⋊C4—2511no—
221C2×C4⋊C4—2511no—
222C2×C4⋊C4—2511no—
223C2×C4⋊C4—2511no—
224C2×C4⋊C4—2511no—
225C2×C4⋊C4C2×D25111yescharacteristic
226C2×C4⋊C4C2×D25111yescharacteristic
227C2×C22⋊C4—2511no—
228C2×C22⋊C4—2511no—
229C2×C22⋊C4—2511no—
230C2×C22⋊C4—2511no—
231C2×C22⋊C4—2511no—
232C2×C22⋊C4—2511no—
233C2×C22⋊C4—2511no—
234C2×C22⋊C4—2511no—
235C2×C22⋊C4—2511no—
236C2×C22⋊C4—2511no—
237C2×C42C2×D25111yescharacteristic
238#95—2511no—
239#95—2511no—
240#95—2511no—
241#95—2511no—
242#95—2511no—
243#95—2511no—
244#95—2511no—
245#95—2511no—
246#95C2×D25111yescharacteristic
247#95C2×D25111yescharacteristic
248#95C2×D25111yescharacteristic
249#95C2×D25111yescharacteristic
250C24—2511no—
251C24—2511no—
252C2×Q8—10041no—
253C2×Q8—10041no—
254C2×Q8—10041no—
255C2×D4—10041no—
256C2×D4—10041no—
257C2×D4—2514no—
258C22×C4—10041no—
259C22×C4—10041no—
260C22×C4—10041no—
261C22×C4—10041no—
262C22×C4—5022no—
263C22×C4—5022no—
264C22×C4—2511no—
265C22×C4—2511no—
266C22×C4—2511no—
267C22×C4—2511no—
268C22×C4—2511no—
269C22×C4—2511no—
270C22×C4C22×D25111yescharacteristic
271C22×C4C22×D25111yescharacteristic
272C22×C4C22×D25111yescharacteristic
273C22×C4C22×D25111yescharacteristic
274C22×C4C22×D25111yescharacteristic
275C22×C4C22×D25111yescharacteristic
276C22×C4C22×D25111yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—10041no—
280C4⋊C4—10041no—
281C4⋊C4—10041no—
282C4⋊C4—10041no—
283C4⋊C4—10041no—
284C4⋊C4—5022no—
285C22⋊C4—10041no—
286C22⋊C4—10041no—
287C22⋊C4—10041no—
288C22⋊C4—10041no—
289C22⋊C4—10041no—
290C22⋊C4—10041no—
291C22⋊C4—10041no—
292C22⋊C4—10041no—
293C22⋊C4—5022no—
294C22⋊C4—5022no—
295C42—41no—
296C23—5021no—
297C23—5021no—
298C23—5021no—
299C23—5021no—
300C23—5021no—
301C23—5021no—
302C23—5021no—
303C23—5021no—
304C23—5021no—
305C23—5021no—
306C23—5021no—
307C23—5021no—
308C23—2512no—
309C23—2512no—
310C23C23×D25111yescharacteristic
311Q8—10044no—
312D4—10044no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—10044no—
333C2×C4—10044no—
334C2×C4—5021no—
335C2×C4—5021no—
336C2×C4—5021no—
337C2×C4—5021no—
338C2×C4—5021no—
339C2×C4—5021no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4—5021no—
345C2×C4—5021no—
346C2×C4—5021no—
347C2×C4—5021no—
348C2×C4—5021no—
349C2×C4—5021no—
350C2×C4—5021no—
351C2×C4—2512no—
352C2×C4Q8×D25112yes—
353C2×C4#1686081012yes—
354C22—10041no—
355C22—10041no—
356C22—10041no—
357C22—10041no—
358C22—10041no—
359C22—10041no—
360C22—10041no—
361C22—10041no—
362C22—5022no—
363C22—5022no—
364C22—5022no—
365C22—5022no—
366C22—5022no—
367C22—5022no—
368C22C2×Q8×D25111yescharacteristic
369C22#41×D25111yescharacteristic
370C22C2×#1686081011yescharacteristic
371C22#2254876011yescharacteristic
372C22#2254876111yescharacteristic
373C22#2254876111yescharacteristic
374C22#2254876111yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—10041no—
383C4—10041no—
384C4—10041no—
385C4—10041no—
386C4—10041no—
387C4—5022no—
388C2—10042no—
389C2—10042no—
390C2#3063227011yescharacteristic
391C2#3063227311yescharacteristic
392C2#3063231211yescharacteristic
393C2#3063239311yescharacteristic
394C2#3063239711yescharacteristic
395C2#3063240011yescharacteristic
396C2#3063242811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#387×C251C2
prime radicalC22×D1004C22
hypercenter#387D251
hypocenterC251#2784
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. #30631629C2semidirect
2. #30631611C2semidirect
3. #30631609C2semidirect
4. #30631608C2semidirect
5. #30631600C2semidirect
6. #30631600C2semidirect
8. #30631571C2semidirect
11. #30631558C2semidirect
12. #30631558C2semidirect
13. #30631546C2semidirect
14. #30631544C2semidirect
15. #387×C251C2semidirect
58. C2×D1004Q8semidirect
198. C251#2784semidirect
208. #387D251semidirect

Maximal subgroups

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

subgroupindex
#306316292
#306316112
#306316092
#306316082
#306316002
#306315892
#306315712
#306315652
#306315582
#306315462
#306315442
#387×C2512
#2784251