Finite Group Databasebeta

← #44553129

Subgroups — #44553129

Order 32128.

596 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548690C211yesmaximal, characteristic
2#30631589C211yesmaximal, characteristic, complemented
3#30631588C211yesmaximal, characteristic, complemented
4#30631579C211yesmaximal, characteristic, complemented
5#30631577C211yesmaximal, characteristic, complemented
6#30631573C211yesmaximal, characteristic, complemented
7#30631565C211yesmaximal, characteristic, complemented
8#30631565C211yesmaximal, characteristic, complemented
9#30631565C211yesmaximal, characteristic, complemented
10#30631565C211yesmaximal, characteristic, complemented
11#30631549C211yesmaximal, characteristic, complemented
12C4×(C2×D251)⋊C4C211yesmaximal, characteristic, complemented
13#30631534C211yesmaximal, characteristic, complemented
14#30631534C211yesmaximal, characteristic, complemented
15#387×C251C211yesmaximal, characteristic, complemented
16C22×D1004C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C22×D1004C2211yescharacteristic
19C22×C4×D251C2211yescharacteristic
20C22×C4×D251C2211yescharacteristic
21C22×C4×D251C2211yescharacteristic
22#22548690C418yescomplemented
23C2×C4⋊Dic251C2211yescharacteristic
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×(C2×D251)⋊C4C2211yescharacteristic
31C2×(C2×D251)⋊C4C2211yescharacteristic
32C2×(C2×D251)⋊C4C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C2×C4×Dic251C2211yescharacteristic
37C2×C4×Dic251C2211yescharacteristic
38C2×C4×Dic251C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C2×C4⋊C4×C251C2211yescharacteristic
46C2×C4⋊C4×C251C2211yescharacteristic
47C2×C42×C251C2211yescharacteristic
48#95×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52C23×D251C2311yescharacteristic
53C23×D251C2311yescharacteristic
54C23×D251C2311yescharacteristic
55C2×D1004—41no—
56C2×D1004—41no—
57C2×D1004—41no—
58C2×D1004—41no—
59C2×D1004—41no—
60C2×D1004—41no—
61C2×D1004C2×C414yes—
62C2×D1004C2×C414yescomplemented
63C2×D1004C2×C414yescomplemented
64C22×Dic251C2311yescharacteristic
65C22×Dic251C2311yescharacteristic
66C22×Dic251C2311yescharacteristic
67C22×Dic251C2311yescharacteristic
68C22×Dic251C2311yescharacteristic
69C2×C4×D251—22no—
70C2×C4×D251—22no—
71C2×C4×D251—41no—
72C2×C4×D251—41no—
73C2×C4×D251—41no—
74C2×C4×D251—41no—
75C2×C4×D251—41no—
76C2×C4×D251—41no—
77C2×C4×D251C2×C414yescomplemented
78C4⋊Dic251—41no—
79(C2×D251)⋊C4—22no—
80(C2×D251)⋊C4—22no—
81(C2×D251)⋊C4—22no—
82(C2×D251)⋊C4—22no—
83(C2×D251)⋊C4—41no—
84(C2×D251)⋊C4—41no—
85(C2×D251)⋊C4—41no—
86(C2×D251)⋊C4—41no—
87(C2×D251)⋊C4—41no—
88(C2×D251)⋊C4—41no—
89(C2×D251)⋊C4C2×C414yescomplemented
90(C2×D251)⋊C4C2×C414yescomplemented
91C4×Dic251—41no—
92C4×Dic251—41no—
93C4×Dic251—41no—
94C22×C4×C251C2311yescharacteristic
95C22×C4×C251C2311yescharacteristic
96C22×C4×C251C2311yescharacteristic
97C22×C4×C251C2311yescharacteristic
98C22×C4×C251C2311yescharacteristic
99C22×C4×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C4⋊C4×C251—41no—
102C4⋊C4×C251C2×C414yescomplemented
103C42×C251—41no—
104C22×D251—21no—
105C22×D251—21no—
106C22×D251—21no—
107C22×D251—21no—
108C22×D251—21no—
109C22×D251—21no—
110C22×D251—21no—
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—21no—
115C22×D251—21no—
116C22×D251—21no—
117C22×D251—21no—
118C22×D251—21no—
119C22×D251—21no—
120C22×D251—21no—
121C22×D251—21no—
122C22×D251C22×C412yes—
123C22×D251C22×C412yes—
124C22×D251C22×C412yes—
125D1004—44no—
126D1004—44no—
127D1004—44no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—21no—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—21no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C2×Dic251—21no—
140C2×Dic251—21no—
141C2×Dic251C22×C412yes—
142C2×Dic251#4112yes—
143C4×D251—44no—
144C4×D251—44no—
145C4×D251—44no—
146C23×C251C2411yescharacteristic
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×C251—21no—
155C2×C4×C251—21no—
156C2×C4×C251—21no—
157C2×C4×C251—21no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251C22×C412yes—
165C2×C4×C251C22×C412yes—
166C2×C4×C251C22×C412yes—
167C2×C4×C251#4112yes—
168C22×C251C23×C411yescharacteristic
169C22×C251C2×#4111yescharacteristic
170C22×C251C2×#4111yescharacteristic
171C22×C251H(5,𝔽2)11yescharacteristic
172C22×C251H(5,𝔽2)11yescharacteristic
173C22×C251H(5,𝔽2)11yescharacteristic
174C22×C251H(5,𝔽2)11yescharacteristic
175C2×D251—22no—
176C2×D251—22no—
177C2×D251—22no—
178C2×D251—22no—
179C2×D251—22no—
180C2×D251—22no—
181C2×D251—22no—
182C2×D251—22no—
183C2×D251—22no—
184C2×D251—41no—
185C2×D251—41no—
186C2×D251—41no—
187C2×D251—41no—
188C2×D251—41no—
189C2×D251—41no—
190C2×D251—41no—
191C2×D251—41no—
192C2×D251—41no—
193C2×D251—41no—
194C2×D251—41no—
195C2×D251—41no—
196C4×C251—22no—
197C4×C251—22no—
198C4×C251—22no—
199C4×C251—22no—
200C4×C251—41no—
201C4×C251—41no—
202C4×C251—41no—
203Dic251—22no—
204Dic251—22no—
205Dic251—41no—
206Dic251—41no—
207Dic251—41no—
208C2×C251C4×#4111yescharacteristic
209C2×C251#51811yescharacteristic
210C2×C251#51811yescharacteristic
211C2×C251#53811yescharacteristic
212C2×C251#53811yescharacteristic
213C2×C251#54011yescharacteristic
214C2×C251#54011yescharacteristic
215D251—42no—
216D251—42no—
217D251—42no—
218C251#233311yescharacteristic, complemented, Sylow
219#2333—2511nomaximal, Sylow
220C2×#121—2511no—
221#390—2511no—
222#390—2511no—
223#388—2511no—
224#388—2511no—
225#387—2511no—
226#387D25111yescharacteristic, complemented
227#386—2511no—
228#386—2511no—
229#386—2511no—
230#386—2511no—
231#381—2511no—
232C4×C22⋊C4—2511no—
233C4×C22⋊C4—2511no—
234C4×C22⋊C4—2511no—
235C22×D4—2511no—
236C22×D4—2511no—
237C22×D4—2511no—
238C23×C4—2511no—
239C23×C4—2511no—
240C23×C4—2511no—
241#121—2518no—
242C2×C4⋊C4—2511no—
243C2×C4⋊C4C2×D25111yescharacteristic
244C2×C4⋊C4C2×D25111yescharacteristic
245C2×C22⋊C4—2511no—
246C2×C22⋊C4—2511no—
247C2×C22⋊C4—2511no—
248C2×C22⋊C4—2511no—
249C2×C22⋊C4—2511no—
250C2×C22⋊C4—2511no—
251C2×C22⋊C4—2511no—
252C2×C22⋊C4—2511no—
253C2×C22⋊C4—2511no—
254C2×C22⋊C4—2511no—
255C2×C22⋊C4—2511no—
256C2×C22⋊C4—2511no—
257C2×C42—2511no—
258C2×C42—2511no—
259C2×C42—2511no—
260C2×C42C2×D25111yescharacteristic
261#95—2511no—
262#95—2511no—
263#95—2511no—
264#95—2511no—
265#95—2511no—
266#95—2511no—
267#95C2×D25111yescharacteristic
268#95C2×D25111yescharacteristic
269#95C2×D25111yescharacteristic
270#95C2×D25111yescharacteristic
271C24—2511no—
272C24—2511no—
273C24—2511no—
274C2×D4—10041no—
275C2×D4—10041no—
276C2×D4—10041no—
277C2×D4—10041no—
278C2×D4—10041no—
279C2×D4—10041no—
280C2×D4—2514no—
281C2×D4—2514no—
282C2×D4—2514no—
283C22×C4—10041no—
284C22×C4—10041no—
285C22×C4—10041no—
286C22×C4—10041no—
287C22×C4—10041no—
288C22×C4—10041no—
289C22×C4—5022no—
290C22×C4—5022no—
291C22×C4—2514no—
292C22×C4—2511no—
293C22×C4—2511no—
294C22×C4—2511no—
295C22×C4—2511no—
296C22×C4—2511no—
297C22×C4C22×D25111yescharacteristic
298C22×C4C22×D25111yescharacteristic
299C22×C4C22×D25111yescharacteristic
300C22×C4C22×D25111yescharacteristic
301C22×C4C22×D25111yescharacteristic
302C22×C4C22×D25111yescharacteristic
303C22×C4C22×D25111yescharacteristic
304C4⋊C4—41no—
305C4⋊C4—10041no—
306C4⋊C4C4×D25114yescomplemented
307C22⋊C4—10041no—
308C22⋊C4—10041no—
309C22⋊C4—10041no—
310C22⋊C4—10041no—
311C22⋊C4—10041no—
312C22⋊C4—10041no—
313C22⋊C4—5022no—
314C22⋊C4—5022no—
315C22⋊C4—5022no—
316C22⋊C4—5022no—
317C22⋊C4—2514no—
318C22⋊C4—2514no—
319C42—41no—
320C42—10041no—
321C42—10041no—
322C42—10041no—
323C23—5021no—
324C23—5021no—
325C23—5021no—
326C23—5021no—
327C23—5021no—
328C23—5021no—
329C23—5021no—
330C23—5021no—
331C23—5021no—
332C23—5021no—
333C23—5021no—
334C23—5021no—
335C23—5021no—
336C23—5021no—
337C23—5021no—
338C23—5021no—
339C23—5021no—
340C23—5021no—
341C23—2512no—
342C23—2512no—
343C23—2512no—
344C23C23×D25111yescharacteristic
345D4—10044no—
346D4—10044no—
347D4—10044no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—10044no—
366C2×C4—10044no—
367C2×C4—10044no—
368C2×C4—5021no—
369C2×C4—5021no—
370C2×C4—5021no—
371C2×C4—5021no—
372C2×C4—5021no—
373C2×C4—5021no—
374C2×C4—5021no—
375C2×C4—5021no—
376C2×C4—5021no—
377C2×C4—5021no—
378C2×C4—5021no—
379C2×C4—5021no—
380C2×C4—5021no—
381C2×C4—2512no—
382C2×C4—2512no—
383C2×C4C2×C4×D25112yes—
384C2×C4C2×C4×D25112yes—
385C2×C4C2×C4×D25112yes—
386C2×C4#1686081112yes—
387C22—10041no—
388C22—10041no—
389C22—10041no—
390C22—10041no—
391C22—10041no—
392C22—10041no—
393C22—10041no—
394C22—10041no—
395C22—10041no—
396C22—10041no—
397C22—10041no—
398C22—10041no—
399C22—5022no—
400C22—5022no—
401C22—5022no—
402C22—5022no—
403C22—5022no—
404C22—5022no—
405C22—5022no—
406C22—5022no—
407C22—5022no—
408C22C22×C4×D25111yescharacteristic
409C22#41×D25111yescharacteristic
410C22C2×#1686081111yescharacteristic
411C22#2254876011yescharacteristic
412C22#2254876111yescharacteristic
413C22#2254876111yescharacteristic
414C22#2254876111yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—10041no—
423C4—10041no—
424C4—10041no—
425C4—5022no—
426C4—5022no—
427C2—10042no—
428C2—10042no—
429C2—10042no—
430C2C4×#1686081111yescharacteristic
431C2#3063215911yescharacteristic
432C2#3063216111yescharacteristic
433C2#3063227611yescharacteristic
434C2#3063227711yescharacteristic
435C2#3063228911yescharacteristic
436C2#3063229311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#387×C251C2
socle, cosocleC23×C251C24
prime radicalC2×#22548690C2
hypercenter#387D251
hypocenterC251#2333
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #30631589C2semidirect
3. #30631588C2semidirect
4. #30631579C2semidirect
5. #30631577C2semidirect
6. #30631573C2semidirect
7. #30631565C2semidirect
8. #30631565C2semidirect
9. #30631565C2semidirect
10. #30631565C2semidirect
11. #30631549C2semidirect
12. C4×(C2×D251)⋊C4C2semidirect
13. #30631534C2semidirect
14. #30631534C2semidirect
15. #387×C251C2semidirect
22. #22548690C4semidirect
62. C2×D1004C2×C4semidirect
63. C2×D1004C2×C4semidirect
77. C2×C4×D251C2×C4semidirect
89. (C2×D251)⋊C4C2×C4semidirect
90. (C2×D251)⋊C4C2×C4semidirect
102. C4⋊C4×C251C2×C4semidirect
218. C251#2333semidirect
226. #387D251semidirect
306. C4⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486902
#306315892
#306315882
#306315792
#306315772
#306315732
#306315652
#306315492
C4×(C2×D251)⋊C42
#306315342
#387×C2512
#2333251