Finite Group Databasebeta

← #44558535

Subgroups — #44558535

Order 32128.

416 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30631629C211yesmaximal, characteristic, complemented
2#30631627C211yesmaximal, characteristic, complemented
3#30631615C211yesmaximal, characteristic, complemented
4#30631614C211yesmaximal, characteristic, complemented
5#30631608C211yesmaximal, characteristic, complemented
6#30631603C211yesmaximal, characteristic
7#30631603C211yesmaximal, characteristic
8#30631602C211yesmaximal, characteristic
9#30631600C211yesmaximal, characteristic, complemented
10#30631599C211yesmaximal, characteristic, complemented
11#30631598C211yesmaximal, characteristic, complemented
12#30631594C211yesmaximal, characteristic, complemented
13#30631580C211yesmaximal, characteristic, complemented
14#30631576C211yesmaximal, characteristic, complemented
15#393×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic
17C22×C22⋊D251C2211yescharacteristic
18C2×C4⋊Dic251C2211yescharacteristic
19C2×C4⋊Dic251C2211yescharacteristic
20C2×C4⋊Dic251C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×C22⋊Dic251C2211yescharacteristic
24C2×C22⋊Dic251C2211yescharacteristic
25C2×C22⋊Dic251C2211yescharacteristic
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×C4×Dic251C2211yescharacteristic
33C251⋊#95C2211yescharacteristic
34C251⋊#95C2211yescharacteristic
35C251⋊#95C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C22×Q8×C251C2211yescharacteristic
45C2×C22⋊C4×C251C2211yescharacteristic
46C2×C22⋊C4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48#95×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51C23×D251C2311yescharacteristic
52C2×Dic502—41no—
53C2×Dic502—41no—
54C2×Dic502—41no—
55C2×C22⋊D251—41no—
56C2×C22⋊D251—41no—
57C2×C22⋊D251—41no—
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C22×Dic251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C4⋊Dic251—41no—
66C4⋊Dic251—41no—
67C4⋊Dic251—41no—
68Dic251⋊C4—41no—
69Dic251⋊C4—41no—
70C22⋊Dic251—41no—
71C22⋊Dic251—41no—
72C22⋊Dic251—41no—
73(C2×D251)⋊C4—22no—
74(C2×D251)⋊C4—41no—
75(C2×D251)⋊C4—41no—
76(C2×D251)⋊C4—41no—
77(C2×D251)⋊C4—41no—
78(C2×D251)⋊C4—41no—
79C4×Dic251—41no—
80C24×C251C2311yescharacteristic
81C2×Q8×C251—22no—
82C2×Q8×C251—41no—
83C2×Q8×C251—41no—
84C22×C4×C251C2311yescharacteristic
85C22×C4×C251C2311yescharacteristic
86C22×C4×C251C2311yescharacteristic
87C22×C4×C251C2311yescharacteristic
88C22×C4×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C22⋊C4×C251—22no—
91C22⋊C4×C251—41no—
92C22⋊C4×C251—41no—
93C22×D251—21no—
94C22×D251—21no—
95C22×D251—21no—
96C22×D251—21no—
97C22×D251—21no—
98C22×D251—21no—
99C22×D251—21no—
100Dic502—44no—
101C22⋊D251—44no—
102C2×Dic251—21no—
103C2×Dic251—21no—
104C2×Dic251—21no—
105C2×Dic251—21no—
106C2×Dic251—21no—
107C2×Dic251—21no—
108C2×Dic251—21no—
109C2×Dic251—21no—
110C2×Dic251—21no—
111C2×Dic251—21no—
112C2×Dic251—21no—
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—
123C23×C251—21no—
124C23×C251—21no—
125C23×C251—21no—
126C23×C251—21no—
127C23×C251—21no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251C2411yescharacteristic
131Q8×C251—44no—
132C2×C4×C251—21no—
133C2×C4×C251—21no—
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#4112yes—
150C22×C251—41no—
151C22×C251—41no—
152C22×C251—41no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251C2×#4111yescharacteristic
158C22×C251H(5,𝔽2)11yescharacteristic
159C22×C251H(5,𝔽2)11yescharacteristic
160C22×C251H(5,𝔽2)11yescharacteristic
161C22×C251H(5,𝔽2)11yescharacteristic
162C22×C2512−1+411yescharacteristic
163C22×C2512−1+411yescharacteristic
164C2×D251—41no—
165C2×D251—41no—
166C2×D251—41no—
167C2×D251—41no—
168C2×D251—41no—
169C2×D251—41no—
170C2×D251—41no—
171C4×C251—22no—
172C4×C251—41no—
173C4×C251—41no—
174C4×C251—41no—
175C4×C251—41no—
176C4×C251—41no—
177Dic251—41no—
178Dic251—41no—
179Dic251—41no—
180Dic251—41no—
181Dic251—41no—
182Dic251—41no—
183Dic251—41no—
184C2×C251—81no—
185C2×C251#53811yescharacteristic
186C2×C251#54211yescharacteristic
187C2×C251#54211yescharacteristic
188C2×C251H(3,𝔽4)11yescharacteristic
189C2×C251#56211yescharacteristic
190C2×C251#56211yescharacteristic
191C2×C251#56311yescharacteristic
192D251—81no—
193C251#283211yescharacteristic, complemented, Sylow
194#2832—2511nomaximal, Sylow
195#400—2511no—
196#400—2511no—
197#397—2511no—
198#397—2511no—
199#395—2511no—
200#394—2511no—
201#394—2511no—
202#394—2511no—
203#393—2511no—
204#393—2511no—
205#393—2511no—
206#393D25111yescharacteristic, complemented
207#391—2511no—
208#388—2511no—
209#388—2511no—
210C22×Q8—2511no—
211C22×Q8C2×D25111yescharacteristic
212C22×D4—2511no—
213C2×C4⋊C4—2511no—
214C2×C4⋊C4—2511no—
215C2×C4⋊C4—2511no—
216C2×C4⋊C4—2511no—
217C2×C4⋊C4—2511no—
218C2×C22⋊C4—2511no—
219C2×C22⋊C4—2511no—
220C2×C22⋊C4—2511no—
221C2×C22⋊C4—2511no—
222C2×C22⋊C4—2511no—
223C2×C22⋊C4—2511no—
224C2×C22⋊C4—2511no—
225C2×C22⋊C4—2511no—
226C2×C22⋊C4—2511no—
227C2×C22⋊C4C2×D25111yescharacteristic
228C2×C22⋊C4C2×D25111yescharacteristic
229C2×C22⋊C4C2×D25111yescharacteristic
230C2×C42—2511no—
231#95—2511no—
232#95—2511no—
233#95—2511no—
234#95—2511no—
235#95—2511no—
236#95—2511no—
237#95—2511no—
238#95—2511no—
239#95—2511no—
240#95—2511no—
241#95—2511no—
242#95C2×D25111yescharacteristic
243#95C2×D25111yescharacteristic
244#95C2×D25111yescharacteristic
245C24—2511no—
246C24C22×D25111yescharacteristic
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—10041no—
251C2×Q8—10041no—
252C2×Q8—10041no—
253C2×D4—10041no—
254C2×D4—10041no—
255C2×D4—10041no—
256C22×C4—2511no—
257C22×C4—2511no—
258C22×C4—2511no—
259C22×C4—2511no—
260C22×C4—2511no—
261C22×C4—2511no—
262C22×C4—2511no—
263C22×C4C22×D25111yescharacteristic
264C22×C4C22×D25111yescharacteristic
265C22×C4C22×D25111yescharacteristic
266C22×C4C22×D25111yescharacteristic
267C22×C4C22×D25111yescharacteristic
268C22×C4C22×D25111yescharacteristic
269C4⋊C4—10041no—
270C4⋊C4—10041no—
271C4⋊C4—10041no—
272C4⋊C4—10041no—
273C4⋊C4—10041no—
274C22⋊C4—22no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—10041no—
278C22⋊C4—10041no—
279C22⋊C4—10041no—
280C22⋊C4—10041no—
281C22⋊C4—10041no—
282C22⋊C4—10041no—
283C22⋊C4—10041no—
284C22⋊C4—10041no—
285C22⋊C4—5022no—
286C42—10041no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—5021no—
295C23—5021no—
296C23—5021no—
297C23—5021no—
298C23—5021no—
299C23—5021no—
300C23—5021no—
301C23C23×D25111yescharacteristic
302Q8—44no—
303Q8—10044no—
304D4—10044no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
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—5021no—
323C2×C4—5021no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4—5021no—
330C2×C4—5021no—
331C2×C4—5021no—
332C2×C4—5021no—
333C2×C4—5021no—
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#1686080912yes—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—10041no—
352C22—10041no—
353C22—10041no—
354C22—10041no—
355C22—10041no—
356C22—10041no—
357C22—10041no—
358C22C2×#1686080911yescharacteristic
359C22#2254876011yescharacteristic
360C22#2254876011yescharacteristic
361C22#2254876111yescharacteristic
362C22#2254876111yescharacteristic
363C22#2254876211yescharacteristic
364C22#2254876311yescharacteristic
365C4—22no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—10041no—
372C4—10041no—
373C4—10041no—
374C4—10041no—
375C4—10041no—
376C4—10041no—
377C4—10041no—
378C2—81no—
379C2—20081no—
380C2#3063227211yescharacteristic
381C2#3063230611yescharacteristic
382C2#3063231011yescharacteristic
383C2#3063242911yescharacteristic
384C2#3063243111yescharacteristic
385C2#3063243311yescharacteristic
386C2#3063243811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#393×C251C2
prime radicalC22×C22⋊D251C22
hypercenter#393D251
hypocenterC251#2832
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. #30631629C2semidirect
2. #30631627C2semidirect
3. #30631615C2semidirect
4. #30631614C2semidirect
5. #30631608C2semidirect
9. #30631600C2semidirect
10. #30631599C2semidirect
11. #30631598C2semidirect
12. #30631594C2semidirect
13. #30631580C2semidirect
14. #30631576C2semidirect
15. #393×C251C2semidirect
193. C251#2832semidirect
206. #393D251semidirect

Maximal subgroups

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

subgroupindex
#306316292
#306316272
#306316152
#306316142
#306316082
#306316032
#306316022
#306316002
#306315992
#306315982
#306315942
#306315802
#306315762
#393×C2512
#2832251