Finite Group Databasebeta

← #44553176

Subgroups — #44553176

Order 32128.

540 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548701C211yesmaximal, characteristic
2#30631592C211yesmaximal, characteristic, complemented
3#30631591C211yesmaximal, characteristic, complemented
4#30631578C211yesmaximal, characteristic, complemented
5#30631576C211yesmaximal, characteristic, complemented
6#30631575C211yesmaximal, characteristic, complemented
7#30631564C211yesmaximal, characteristic
8#30631563C211yesmaximal, characteristic
9#30631559C211yesmaximal, characteristic, complemented
10#30631558C211yesmaximal, characteristic, complemented
11#30631554C211yesmaximal, characteristic, complemented
12C4×Dic251⋊C4C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic251C211yesmaximal, characteristic, complemented
14#30631534C211yesmaximal, characteristic, complemented
15#388×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic
17C22×C22⋊D251C2211yescharacteristic
18C23×Dic251C2211yescharacteristic
19C22×C4×D251C2211yescharacteristic
20#22548701C418yescomplemented
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
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×C4×Dic251C2211yescharacteristic
32C2×C4×Dic251C2211yescharacteristic
33C2×C4×Dic251C2211yescharacteristic
34C2×C4×Dic251C2211yescharacteristic
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×C22⋊C4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C42×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52C23×D251C2311yescharacteristic
53C2×Dic502—41no—
54C2×Dic502—41no—
55C2×Dic502C2×C414yescomplemented
56C2×C22⋊D251—41no—
57C2×C22⋊D251—41no—
58C2×C22⋊D251C2×C414yes—
59C22×Dic251—22no—
60C22×Dic251—41no—
61C22×Dic251—41no—
62C22×Dic251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C22×Dic251C2311yescharacteristic
66C22×Dic251C2311yescharacteristic
67C22×Dic251C2311yescharacteristic
68C22×Dic251C2311yescharacteristic
69C2×C4×D251—22no—
70C2×C4×D251—41no—
71C2×C4×D251—41no—
72Dic251⋊C4—22no—
73Dic251⋊C4—22no—
74Dic251⋊C4—41no—
75C22⋊Dic251—22no—
76C22⋊Dic251C2×C414yescomplemented
77(C2×D251)⋊C4—22no—
78(C2×D251)⋊C4—41no—
79(C2×D251)⋊C4—41no—
80(C2×D251)⋊C4C2×C414yescomplemented
81(C2×D251)⋊C4C2×C414yescomplemented
82C4×Dic251—41no—
83C4×Dic251—41no—
84C4×Dic251—41no—
85C4×Dic251C2×C414yescomplemented
86C24×C251C2311yescharacteristic
87C22×C4×C251C2311yescharacteristic
88C22×C4×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C4⋊C4×C251—41no—
94C22⋊C4×C251—41no—
95C22⋊C4×C251—41no—
96C22⋊C4×C251C2×C414yescomplemented
97C42×C251—41no—
98C22×D251—21no—
99C22×D251—21no—
100C22×D251—21no—
101C22×D251—21no—
102C22×D251—21no—
103C22×D251—21no—
104C22×D251C22×C412yes—
105Dic502—44no—
106C22⋊D251—44no—
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—
123C2×Dic251—44no—
124C2×Dic251C22×C412yes—
125C2×Dic251C22×C412yes—
126C2×Dic251C22×C412yes—
127C2×Dic251#4112yes—
128C2×Dic251#4112yes—
129C4×D251—44no—
130C23×C251—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—21no—
134C23×C251—21no—
135C23×C251—21no—
136C23×C251C22×C412yes—
137C23×C251C2411yescharacteristic
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×C251—21no—
154C2×C4×C251C22×C412yes—
155C2×C4×C251C22×C412yes—
156C22×C251—22no—
157C22×C251—22no—
158C22×C251—22no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251C23×C411yescharacteristic
164C22×C251C2×#4111yescharacteristic
165C22×C251C2×#4111yescharacteristic
166C22×C251H(5,𝔽2)11yescharacteristic
167C22×C251H(5,𝔽2)11yescharacteristic
168C22×C251H(5,𝔽2)11yescharacteristic
169C22×C2512−1+411yescharacteristic
170C2×D251—22no—
171C2×D251—22no—
172C2×D251—22no—
173C2×D251—41no—
174C2×D251—41no—
175C2×D251—41no—
176C2×D251—41no—
177C4×C251—22no—
178C4×C251—22no—
179C4×C251—41no—
180C4×C251—41no—
181C4×C251—41no—
182C4×C251—41no—
183Dic251—22no—
184Dic251—22no—
185Dic251—22no—
186Dic251—22no—
187Dic251—22no—
188Dic251—41no—
189Dic251—41no—
190C2×C251—42no—
191C2×C251C4×#4111yescharacteristic
192C2×C251#51811yescharacteristic
193C2×C251#52011yescharacteristic
194C2×C251#53811yescharacteristic
195C2×C251#53811yescharacteristic
196C2×C251#54211yescharacteristic
197C2×C251#54211yescharacteristic
198D251—42no—
199C251#233711yescharacteristic, complemented, Sylow
200#2337—2511nomaximal, Sylow
201C2×#124—2511no—
202#391—2511no—
203#391—2511no—
204#388—2511no—
205#388—2511no—
206#388—2511no—
207#388D25111yescharacteristic, complemented
208#386—2511no—
209#386—2511no—
210#385—2511no—
211#385—2511no—
212#383—2511no—
213C4×C4⋊C4—2511no—
214C4×C22⋊C4—2511no—
215C4×C22⋊C4—2511no—
216C22×Q8—2511no—
217C22×D4—2511no—
218C23×C4—2511no—
219C23×C4—2511no—
220#124—2518no—
221C2×C4⋊C4—2511no—
222C2×C4⋊C4—2511no—
223C2×C4⋊C4—2511no—
224C2×C4⋊C4C2×D25111yescharacteristic
225C2×C22⋊C4—2511no—
226C2×C22⋊C4—2511no—
227C2×C22⋊C4—2511no—
228C2×C22⋊C4—2511no—
229C2×C22⋊C4—2511no—
230C2×C22⋊C4—2511no—
231C2×C22⋊C4—2511no—
232C2×C22⋊C4C2×D25111yescharacteristic
233C2×C22⋊C4C2×D25111yescharacteristic
234C2×C22⋊C4C2×D25111yescharacteristic
235C2×C42—2511no—
236C2×C42—2511no—
237C2×C42—2511no—
238C2×C42—2511no—
239C2×C42C2×D25111yescharacteristic
240#95—2511no—
241#95—2511no—
242#95—2511no—
243#95—2511no—
244#95—2511no—
245#95—2511no—
246#95—2511no—
247#95—2511no—
248#95—2511no—
249#95—2511no—
250#95C2×D25111yescharacteristic
251#95C2×D25111yescharacteristic
252C24—2511no—
253C24C22×D25111yescharacteristic
254C2×Q8—10041no—
255C2×Q8—10041no—
256C2×Q8—2514no—
257C2×D4—10041no—
258C2×D4—10041no—
259C2×D4—2514no—
260C22×C4—10041no—
261C22×C4—10041no—
262C22×C4—10041no—
263C22×C4—10041no—
264C22×C4—5022no—
265C22×C4—5022no—
266C22×C4—2511no—
267C22×C4—2511no—
268C22×C4—2511no—
269C22×C4—2511no—
270C22×C4—2511no—
271C22×C4—2511no—
272C22×C4—2511no—
273C22×C4C22×D25111yescharacteristic
274C22×C4C22×D25111yescharacteristic
275C22×C4C22×D25111yescharacteristic
276C22×C4C22×D25111yescharacteristic
277C22×C4C22×D25111yescharacteristic
278C22×C4C22×D25111yescharacteristic
279C4⋊C4—41no—
280C4⋊C4—10041no—
281C4⋊C4—5022no—
282C4⋊C4—5022no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—10041no—
286C22⋊C4—10041no—
287C22⋊C4—5022no—
288C22⋊C4—5022no—
289C22⋊C4—2514no—
290C22⋊C4—2514no—
291C22⋊C4—2514no—
292C22⋊C4C4×D25114yescomplemented
293C42—41no—
294C42—10041no—
295C42—10041no—
296C42—10041no—
297C42—2514no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—5021no—
305C23—5021no—
306C23—5021no—
307C23—5021no—
308C23—5021no—
309C23—5021no—
310C23—2512no—
311C23C2×C4×D25112yes—
312C23C23×D25111yescharacteristic
313Q8—10044no—
314D4—10044no—
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—10044no—
332C2×C4—10044no—
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—5021no—
344C2×C4—5021no—
345C2×C4—5021no—
346C2×C4—5021no—
347C2×C4—5021no—
348C2×C4—5021no—
349C2×C4—2512no—
350C2×C4—2512no—
351C2×C4—2512no—
352C2×C4—2512no—
353C2×C4—2512no—
354C2×C4C2×C4×D25112yes—
355C2×C4C2×C4×D25112yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—10041no—
364C22—10041no—
365C22—10041no—
366C22—10041no—
367C22—5022no—
368C22—5022no—
369C22—5022no—
370C22C22×C4×D25111yescharacteristic
371C22#41×D25111yescharacteristic
372C22#41×D25111yescharacteristic
373C22#2254876011yescharacteristic
374C22#2254876011yescharacteristic
375C22#2254876111yescharacteristic
376C22#2254876211yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—10041no—
384C4—10041no—
385C4—5022no—
386C4—5022no—
387C4—5022no—
388C4—5022no—
389C4—5022no—
390C2—42no—
391C2—10042no—
392C2#3063215511yescharacteristic
393C2#3063215911yescharacteristic
394C2#3063216611yescharacteristic
395C2#3063226911yescharacteristic
396C2#3063227311yescharacteristic
397C2#3063230111yescharacteristic
398C2#3063230311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#388×C251C2
socle, cosocleC23×C251C24
prime radicalC22×C22⋊D251C22
hypercenter#388D251
hypocenterC251#2337
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #30631592C2semidirect
3. #30631591C2semidirect
4. #30631578C2semidirect
5. #30631576C2semidirect
6. #30631575C2semidirect
9. #30631559C2semidirect
10. #30631558C2semidirect
11. #30631554C2semidirect
12. C4×Dic251⋊C4C2semidirect
13. C22⋊C4×Dic251C2semidirect
14. #30631534C2semidirect
15. #388×C251C2semidirect
20. #22548701C4semidirect
55. C2×Dic502C2×C4semidirect
76. C22⋊Dic251C2×C4semidirect
80. (C2×D251)⋊C4C2×C4semidirect
81. (C2×D251)⋊C4C2×C4semidirect
85. C4×Dic251C2×C4semidirect
96. C22⋊C4×C251C2×C4semidirect
199. C251#2337semidirect
207. #388D251semidirect
292. C22⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487012
#306315922
#306315912
#306315782
#306315762
#306315752
#306315642
#306315632
#306315592
#306315582
#306315542
C4×Dic251⋊C42
C22⋊C4×Dic2512
#306315342
#388×C2512
#2337251