Finite Group Databasebeta

← #44553116

Subgroups — #44553116

Order 32128.

484 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548693C211yesmaximal, characteristic
2#30631593C211yesmaximal, characteristic, complemented
3#30631591C211yesmaximal, characteristic, complemented
4#30631579C211yesmaximal, characteristic
5#30631578C211yesmaximal, characteristic
6#30631578C211yesmaximal, characteristic
7#30631577C211yesmaximal, characteristic
8#30631573C211yesmaximal, characteristic, complemented
9#30631571C211yesmaximal, characteristic, complemented
10#30631569C211yesmaximal, characteristic, complemented
11#30631568C211yesmaximal, characteristic, complemented
12#30631549C211yesmaximal, characteristic
13C4×Dic251⋊C4C211yesmaximal, characteristic, complemented
14C4×(C2×D251)⋊C4C211yesmaximal, characteristic
15C4×C4⋊C4×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic
17C22×C4×D251C2211yescharacteristic
18#22548693C418yescomplemented
19C2×C4⋊Dic251C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×(C2×D251)⋊C4C2211yescharacteristic
25C2×(C2×D251)⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2211yescharacteristic
27C2×(C2×D251)⋊C4C2211yescharacteristic
28C2×(C2×D251)⋊C4C2211yescharacteristic
29C2×(C2×D251)⋊C4C2211yescharacteristic
30C2×C4×Dic251C2211yescharacteristic
31C2×C4×Dic251C2211yescharacteristic
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
44C251⋊#95C2211yescharacteristic
45C2×C4⋊C4×C251C2211yescharacteristic
46C2×C4⋊C4×C251C2211yescharacteristic
47C2×C42×C251C2211yescharacteristic
48C2×C42×C251C2211yescharacteristic
49C2×C42×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52C23×D251C2311yescharacteristic
53C2×Dic502—41no—
54C2×Dic502—41no—
55C2×Dic502C2×C414yes—
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C22×Dic251C2311yescharacteristic
63C2×C4×D251—41no—
64C2×C4×D251—41no—
65C2×C4×D251C2×C414yes—
66C4⋊Dic251—41no—
67Dic251⋊C4—22no—
68Dic251⋊C4—41no—
69Dic251⋊C4C2×C414yes—
70Dic251⋊C4C2×C414yes—
71(C2×D251)⋊C4—22no—
72(C2×D251)⋊C4—22no—
73(C2×D251)⋊C4—41no—
74(C2×D251)⋊C4—41no—
75(C2×D251)⋊C4C2×C414yes—
76(C2×D251)⋊C4C2×C414yes—
77C4×Dic251—22no—
78C4×Dic251—41no—
79C4×Dic251—41no—
80C22×C4×C251C2311yescharacteristic
81C22×C4×C251C2311yescharacteristic
82C22×C4×C251C2311yescharacteristic
83C22×C4×C251C2311yescharacteristic
84C22×C4×C251C2311yescharacteristic
85C22×C4×C251C2311yescharacteristic
86C22×C4×C251C2311yescharacteristic
87C4⋊C4×C251—22no—
88C4⋊C4×C251C2×C414yes—
89C42×C251—22no—
90C42×C251—41no—
91C42×C251—41no—
92C22×D251—21no—
93C22×D251—21no—
94C22×D251—21no—
95C22×D251—21no—
96C22×D251—21no—
97C22×D251—21no—
98C22×D251C22×C412yes—
99Dic502—44no—
100C2×Dic251—21no—
101C2×Dic251—21no—
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×Dic251C22×C412yes—
119C2×Dic251C22×C412yes—
120C2×Dic251C22×C412yes—
121C4×D251—44no—
122C23×C251C2411yescharacteristic
123C2×C4×C251—21no—
124C2×C4×C251—21no—
125C2×C4×C251—21no—
126C2×C4×C251—21no—
127C2×C4×C251—21no—
128C2×C4×C251—21no—
129C2×C4×C251—21no—
130C2×C4×C251—21no—
131C2×C4×C251—21no—
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×C251C22×C412yes—
140C2×C4×C251C22×C412yes—
141C2×C4×C251C22×C412yes—
142C2×C4×C251#4112yes—
143C2×C4×C251#4112yes—
144C22×C251C23×C411yescharacteristic
145C22×C251C2×#4111yescharacteristic
146C22×C251C2×#4111yescharacteristic
147C22×C251H(5,𝔽2)11yescharacteristic
148C22×C251H(5,𝔽2)11yescharacteristic
149C22×C2512−1+411yescharacteristic
150C22×C2512−1+411yescharacteristic
151C2×D251—22no—
152C2×D251—22no—
153C2×D251—22no—
154C2×D251—41no—
155C2×D251—41no—
156C2×D251—41no—
157C2×D251—41no—
158C4×C251—22no—
159C4×C251—22no—
160C4×C251—22no—
161C4×C251—22no—
162C4×C251—22no—
163C4×C251—41no—
164C4×C251—41no—
165Dic251—22no—
166Dic251—22no—
167Dic251—22no—
168Dic251—41no—
169Dic251—41no—
170Dic251—41no—
171Dic251—41no—
172C2×C251C4×#4111yescharacteristic
173C2×C251#51811yescharacteristic
174C2×C251#51911yescharacteristic
175C2×C251#53911yescharacteristic
176C2×C251#53911yescharacteristic
177C2×C251#54211yescharacteristic
178C2×C251#54211yescharacteristic
179D251—42no—
180C251#233211yescharacteristic, complemented, Sylow
181#2332—2511nomaximal, Sylow
182C2×C22⋊Q8—2511no—
183#391—2511no—
184#391—2511no—
185#388—2511no—
186#388—2511no—
187#388—2511no—
188#388—2511no—
189#387—2511no—
190#387—2511no—
191#387—2511no—
192#387—2511no—
193#381—2511no—
194C4×C4⋊C4—2511no—
195C4×C4⋊C4D25111yescharacteristic, complemented
196C4×C22⋊C4—2511no—
197C22×Q8—2511no—
198C23×C4—2511no—
199C22⋊Q8—2518no—
200C2×C4⋊C4—2511no—
201C2×C4⋊C4—2511no—
202C2×C4⋊C4—2511no—
203C2×C4⋊C4—2511no—
204C2×C4⋊C4—2511no—
205C2×C4⋊C4C2×D25111yescharacteristic
206C2×C4⋊C4C2×D25111yescharacteristic
207C2×C22⋊C4—2511no—
208C2×C22⋊C4—2511no—
209C2×C22⋊C4—2511no—
210C2×C22⋊C4—2511no—
211C2×C22⋊C4—2511no—
212C2×C22⋊C4—2511no—
213C2×C42—2511no—
214C2×C42—2511no—
215C2×C42—2511no—
216C2×C42C2×D25111yescharacteristic
217C2×C42C2×D25111yescharacteristic
218C2×C42C2×D25111yescharacteristic
219#95—2511no—
220#95—2511no—
221#95—2511no—
222#95—2511no—
223#95—2511no—
224#95—2511no—
225#95—2511no—
226#95—2511no—
227#95—2511no—
228#95—2511no—
229#95—2511no—
230#95—2511no—
231#95C2×D25111yescharacteristic
232#95C2×D25111yescharacteristic
233C24—2511no—
234C2×Q8—10041no—
235C2×Q8—10041no—
236C2×Q8—2514no—
237C22×C4—10041no—
238C22×C4—10041no—
239C22×C4—2514no—
240C22×C4—2511no—
241C22×C4—2511no—
242C22×C4—2511no—
243C22×C4—2511no—
244C22×C4—2511no—
245C22×C4—2511no—
246C22×C4—2511no—
247C22×C4C22×D25111yescharacteristic
248C22×C4C22×D25111yescharacteristic
249C22×C4C22×D25111yescharacteristic
250C22×C4C22×D25111yescharacteristic
251C22×C4C22×D25111yescharacteristic
252C22×C4C22×D25111yescharacteristic
253C22×C4C22×D25111yescharacteristic
254C4⋊C4—22no—
255C4⋊C4—10041no—
256C4⋊C4—10041no—
257C4⋊C4—5022no—
258C4⋊C4—2514no—
259C4⋊C4—2514no—
260C4⋊C4C4×D25114yes—
261C22⋊C4—10041no—
262C22⋊C4—10041no—
263C22⋊C4—5022no—
264C22⋊C4—5022no—
265C22⋊C4—2514no—
266C22⋊C4—2514no—
267C42—22no—
268C42—41no—
269C42—41no—
270C42—10041no—
271C42—10041no—
272C42—5022no—
273C23—5021no—
274C23—5021no—
275C23—5021no—
276C23—5021no—
277C23—5021no—
278C23—5021no—
279C23—2512no—
280C23C23×D25111yescharacteristic
281Q8—10044no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—10044no—
299C2×C4—5021no—
300C2×C4—5021no—
301C2×C4—5021no—
302C2×C4—5021no—
303C2×C4—5021no—
304C2×C4—5021no—
305C2×C4—5021no—
306C2×C4—5021no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—5021no—
311C2×C4—5021no—
312C2×C4—5021no—
313C2×C4—5021no—
314C2×C4—5021no—
315C2×C4—5021no—
316C2×C4—5021no—
317C2×C4—2512no—
318C2×C4—2512no—
319C2×C4—2512no—
320C2×C4C2×C4×D25112yes—
321C2×C4C2×C4×D25112yes—
322C2×C4C2×C4×D25112yes—
323C2×C4#1686081012yes—
324C2×C4#1686081012yes—
325C22—10041no—
326C22—10041no—
327C22—10041no—
328C22—10041no—
329C22—5022no—
330C22—5022no—
331C22—5022no—
332C22C22×C4×D25111yescharacteristic
333C22C2×#1686081011yescharacteristic
334C22C2×#1686081011yescharacteristic
335C22#2254876011yescharacteristic
336C22#2254876111yescharacteristic
337C22#2254876211yescharacteristic
338C22#2254876311yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—10041no—
347C4—10041no—
348C4—10041no—
349C4—10041no—
350C4—5022no—
351C4—5022no—
352C4—5022no—
353C2—10042no—
354C2C4×#1686081011yescharacteristic
355C2#3063215911yescharacteristic
356C2#3063216311yescharacteristic
357C2#3063227911yescharacteristic
358C2#3063228411yescharacteristic
359C2#3063230811yescharacteristic
360C2#3063230911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
FittingC4×C4⋊C4×C251C2
socle, cosocleC23×C251C24
prime radicalC23×D251C23
hypercenterC4×C4⋊C4D251
hypocenterC251#2332
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #30631593C2semidirect
3. #30631591C2semidirect
8. #30631573C2semidirect
9. #30631571C2semidirect
10. #30631569C2semidirect
11. #30631568C2semidirect
13. C4×Dic251⋊C4C2semidirect
15. C4×C4⋊C4×C251C2semidirect
18. #22548693C4semidirect
180. C251#2332semidirect
195. C4×C4⋊C4D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486932
#306315932
#306315912
#306315792
#306315782
#306315772
#306315732
#306315712
#306315692
#306315682
#306315492
C4×Dic251⋊C42
C4×(C2×D251)⋊C42
C4×C4⋊C4×C2512
#2332251