Finite Group Databasebeta

← #44553174

Subgroups — #44553174

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×#22548702C211yesmaximal, characteristic
2#30631591C211yesmaximal, characteristic, complemented
3#30631591C211yesmaximal, characteristic, complemented
4#30631581C211yesmaximal, characteristic, complemented
5#30631581C211yesmaximal, characteristic, complemented
6#30631575C211yesmaximal, characteristic, complemented
7#30631575C211yesmaximal, characteristic, complemented
8#30631567C211yesmaximal, characteristic
9#30631560C211yesmaximal, characteristic, complemented
10#30631559C211yesmaximal, characteristic, complemented
11#30631554C211yesmaximal, characteristic, complemented
12#30631538C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic251C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic251C211yesmaximal, characteristic, complemented
15#386×C251C211yesmaximal, characteristic
16C22×Dic502C2211yescharacteristic
17C23×Dic251C2211yescharacteristic
18#22548702C418yescomplemented
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×C22⋊Dic251C2211yescharacteristic
24C2×C22⋊Dic251C2211yescharacteristic
25C2×C22⋊Dic251C2211yescharacteristic
26C2×C22⋊Dic251C2211yescharacteristic
27C2×C22⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C4×Dic251C2211yescharacteristic
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
45C22×D4×C251C2211yescharacteristic
46C23×C4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52C2×Dic502—41no—
53C2×Dic502—41no—
54C2×Dic502C2×C414yescomplemented
55C22×Dic251—22no—
56C22×Dic251—41no—
57C22×Dic251—41no—
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C22×Dic251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C22×Dic251C2311yescharacteristic
66Dic251⋊C4—22no—
67Dic251⋊C4—22no—
68Dic251⋊C4—41no—
69Dic251⋊C4—41no—
70C22⋊Dic251—22no—
71C22⋊Dic251—41no—
72C22⋊Dic251—41no—
73C22⋊Dic251C2×C414yescomplemented
74C22⋊Dic251C2×C414yescomplemented
75C22⋊Dic251C2×C414yescomplemented
76C22⋊Dic251C2×C414yescomplemented
77C4×Dic251—41no—
78C4×Dic251—41no—
79C4×Dic251—41no—
80C4×Dic251—41no—
81C4×Dic251C2×C414yescomplemented
82C24×C251C2311yescharacteristic
83C24×C251C2311yescharacteristic
84C2×D4×C251—41no—
85C2×D4×C251—41no—
86C2×D4×C251C2×C414yes—
87C22×C4×C251—22no—
88C22×C4×C251—41no—
89C22×C4×C251—41no—
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C22×C4×C251C2311yescharacteristic
95C22⋊C4×C251—22no—
96C22⋊C4×C251—41no—
97C22⋊C4×C251—41no—
98Dic502—44no—
99C2×Dic251—21no—
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×Dic251—44no—
119C2×Dic251C22×C412yes—
120C2×Dic251C22×C412yes—
121C2×Dic251C22×C412yes—
122C2×Dic251C22×C412yes—
123C2×Dic251#4112yes—
124C23×C251—21no—
125C23×C251—21no—
126C23×C251—21no—
127C23×C251—21no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—21no—
134C23×C251—21no—
135C23×C251—21no—
136C23×C251C22×C412yes—
137C23×C251C22×C412yes—
138C23×C251C2411yescharacteristic
139D4×C251—44no—
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—44no—
154C2×C4×C251C22×C412yes—
155C2×C4×C251#4112yes—
156C22×C251—22no—
157C22×C251—22no—
158C22×C251—22no—
159C22×C251—22no—
160C22×C251—22no—
161C22×C251—22no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251—41no—
166C22×C251—41no—
167C22×C251—41no—
168C22×C251—41no—
169C22×C251—41no—
170C22×C251C23×C411yescharacteristic
171C22×C251C2×#4111yescharacteristic
172C22×C251C2×#4111yescharacteristic
173C22×C251H(5,𝔽2)11yescharacteristic
174C22×C251H(5,𝔽2)11yescharacteristic
175C22×C251H(5,𝔽2)11yescharacteristic
176C22×C2512−1+411yescharacteristic
177C4×C251—22no—
178C4×C251—22no—
179C4×C251—41no—
180C4×C251—41no—
181C4×C251—41no—
182Dic251—22no—
183Dic251—22no—
184Dic251—22no—
185Dic251—22no—
186Dic251—22no—
187Dic251—41no—
188Dic251—41no—
189Dic251—41no—
190C2×C251—42no—
191C2×C251—42no—
192C2×C251C4×#4111yescharacteristic
193C2×C251#51811yescharacteristic
194C2×C251#52011yescharacteristic
195C2×C251#53811yescharacteristic
196C2×C251#53811yescharacteristic
197C2×C251#54211yescharacteristic
198C2×C251#54211yescharacteristic
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#388—2511no—
208#386—2511no—
209#386D25111yescharacteristic
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×D4C2×D25111yescharacteristic
218C23×C4—2511no—
219C23×C4C2×D25111yescharacteristic
220#124—2518no—
221C2×C4⋊C4—2511no—
222C2×C4⋊C4—2511no—
223C2×C4⋊C4—2511no—
224C2×C4⋊C4—2511no—
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×C42—2511no—
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
252C24C22×D25111yescharacteristic
253C24C22×D25111yescharacteristic
254C2×Q8—10041no—
255C2×Q8—10041no—
256C2×Q8—2514no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4C4×D25114yes—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—10041no—
264C22×C4—10041no—
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×C4—2511no—
274C22×C4C22×D25111yescharacteristic
275C22×C4C22×D25111yescharacteristic
276C22×C4C22×D25111yescharacteristic
277C22×C4C22×D25111yescharacteristic
278C22×C4C22×D25111yescharacteristic
279C4⋊C4—10041no—
280C4⋊C4—10041no—
281C4⋊C4—5022no—
282C4⋊C4—5022no—
283C22⋊C4—22no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—10041no—
287C22⋊C4—10041no—
288C22⋊C4—5022no—
289C22⋊C4—2514no—
290C22⋊C4—2514no—
291C22⋊C4—2514no—
292C22⋊C4—2514no—
293C42—10041no—
294C42—10041no—
295C42—10041no—
296C42—10041no—
297C42—2514no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23C2×C4×D25112yes—
311C23C2×C4×D25112yes—
312C23C23×D25111yescharacteristic
313Q8—10044no—
314D4—44no—
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—44no—
329C2×C4—10044no—
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—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×C4#1686080912yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C22×C4×D25111yescharacteristic
371C22#41×D25111yescharacteristic
372C22C2×#1686080911yescharacteristic
373C22#2254876011yescharacteristic
374C22#2254876011yescharacteristic
375C22#2254876011yescharacteristic
376C22#2254876211yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—10041no—
383C4—10041no—
384C4—10041no—
385C4—5022no—
386C4—5022no—
387C4—5022no—
388C4—5022no—
389C4—5022no—
390C2—42no—
391C2—42no—
392C2C4×#1686080911yescharacteristic
393C2#3063216011yescharacteristic
394C2#3063216611yescharacteristic
395C2#3063226911yescharacteristic
396C2#3063227211yescharacteristic
397C2#3063230411yescharacteristic
398C2#3063230511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#386×C251C2
socle, cosocleC23×C251C24
prime radicalC22×D4×C251C22
hypercenter#386D251
hypocenterC251#2337
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #30631591C2semidirect
3. #30631591C2semidirect
4. #30631581C2semidirect
5. #30631581C2semidirect
6. #30631575C2semidirect
7. #30631575C2semidirect
9. #30631560C2semidirect
10. #30631559C2semidirect
11. #30631554C2semidirect
12. #30631538C2semidirect
13. C22⋊C4×Dic251C2semidirect
14. C4×C22⋊Dic251C2semidirect
18. #22548702C4semidirect
54. C2×Dic502C2×C4semidirect
73. C22⋊Dic251C2×C4semidirect
74. C22⋊Dic251C2×C4semidirect
75. C22⋊Dic251C2×C4semidirect
76. C22⋊Dic251C2×C4semidirect
81. C4×Dic251C2×C4semidirect
199. C251#2337semidirect

Maximal subgroups

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

subgroupindex
C2×#225487022
#306315912
#306315812
#306315752
#306315672
#306315602
#306315592
#306315542
#306315382
C22⋊C4×Dic2512
C4×C22⋊Dic2512
#386×C2512
#2337251