Finite Group Databasebeta

← #44553411

Subgroups — #44553411

Order 32128.

716 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548690C211yesmaximal, characteristic
2C2×C4×D1004C211yesmaximal, characteristic, complemented
3C2×#22548677C211yesmaximal, 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
11C4×(C2×D251)⋊C4C211yesmaximal, characteristic, complemented
12#30631534C211yesmaximal, characteristic, complemented
13#30631534C211yesmaximal, characteristic, complemented
14#95×D251C211yesmaximal, characteristic, complemented
15#387×C251C211yesmaximal, characteristic, complemented
16C22×D1004C2211yescharacteristic
17C22×D1004C2211yescharacteristic
18C22×D1004C2211yescharacteristic
19C22×C4×D251C2211yescharacteristic
20C22×C4×D251C2211yescharacteristic
21C22×C4×D251C2211yescharacteristic
22C22×C4×D251C2211yescharacteristic
23C22×C4×D251C2211yescharacteristic
24#22548690C418yescomplemented
25C4×D1004—24no—
26#22548677—24no—
27C2×C4⋊Dic251C2211yescharacteristic
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×(C2×D251)⋊C4C2211yescharacteristic
37C2×(C2×D251)⋊C4C2211yescharacteristic
38C2×C4×Dic251C2211yescharacteristic
39C2×C4×Dic251C2211yescharacteristic
40C2×C4×Dic251C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C251⋊#95C2211yescharacteristic
47C2×C4⋊C4×C251C2211yescharacteristic
48C2×C4⋊C4×C251C2211yescharacteristic
49C2×C42×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52#95×C251C2211yescharacteristic
53#95×C251C2211yescharacteristic
54C23×D251C2311yescharacteristic
55C23×D251C2311yescharacteristic
56C23×D251C2311yescharacteristic
57C2×D1004—22no—
58C2×D1004—22no—
59C2×D1004—22no—
60C2×D1004—22no—
61C2×D1004—41no—
62C2×D1004—41no—
63C2×D1004C2×C414yescomplemented
64C2×D1004C2×C414yescomplemented
65C2×D1004C2×C414yescomplemented
66C22×Dic251C2311yescharacteristic
67C22×Dic251C2311yescharacteristic
68C22×Dic251C2311yescharacteristic
69C22×Dic251C2311yescharacteristic
70C22×Dic251C2311yescharacteristic
71C2×C4×D251—22no—
72C2×C4×D251—22no—
73C2×C4×D251—22no—
74C2×C4×D251—22no—
75C2×C4×D251—22no—
76C2×C4×D251—22no—
77C2×C4×D251—22no—
78C2×C4×D251—22no—
79C2×C4×D251—41no—
80C2×C4×D251—41no—
81C2×C4×D251—41no—
82C2×C4×D251—41no—
83C2×C4×D251—41no—
84C2×C4×D251—41no—
85C2×C4×D251C2×C414yescomplemented
86C4⋊Dic251—22no—
87(C2×D251)⋊C4—22no—
88(C2×D251)⋊C4—22no—
89(C2×D251)⋊C4—22no—
90(C2×D251)⋊C4—22no—
91(C2×D251)⋊C4—22no—
92(C2×D251)⋊C4—41no—
93(C2×D251)⋊C4—41no—
94(C2×D251)⋊C4C2×C414yescomplemented
95(C2×D251)⋊C4C2×C414yescomplemented
96(C2×D251)⋊C4D414yescomplemented
97C4×Dic251—22no—
98C4×Dic251—41no—
99C4×Dic251—41no—
100C22×C4×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102C22×C4×C251C2311yescharacteristic
103C22×C4×C251C2311yescharacteristic
104C22×C4×C251C2311yescharacteristic
105C22×C4×C251C2311yescharacteristic
106C22×C4×C251C2311yescharacteristic
107C4⋊C4×C251—22no—
108C4⋊C4×C251C2×C414yescomplemented
109C42×C251—22no—
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×D251—21no—
123C22×D251—21no—
124C22×D251—21no—
125C22×D251—21no—
126C22×D251C22×C412yes—
127C22×D251C22×C412yes—
128C22×D251C22×C412yes—
129C22×D251C2×D412yes—
130C22×D251#4112yes—
131D1004—28no—
132D1004—28no—
133D1004—44no—
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—21no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C2×Dic251—21no—
140C2×Dic251—21no—
141C2×Dic251—21no—
142C2×Dic251—21no—
143C2×Dic251—21no—
144C2×Dic251—21no—
145C2×Dic251—21no—
146C2×Dic251C22×C412yes—
147C2×Dic251C2×D412yes—
148C2×Dic251#4112yes—
149C4×D251—44no—
150C4×D251—44no—
151C4×D251—44no—
152C4×D251—44no—
153C4×D251—44no—
154C23×C251C2411yescharacteristic
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×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—21no—
171C2×C4×C251C22×C412yes—
172C2×C4×C251C22×C412yes—
173C2×C4×C251C22×C412yes—
174C2×C4×C251C2×D412yes—
175C2×C4×C251#4112yes—
176C22×C251C23×C411yescharacteristic
177C22×C251C22×D411yescharacteristic
178C22×C251C2×#4111yescharacteristic
179C22×C251C2×#4111yescharacteristic
180C22×C251C2×#4111yescharacteristic
181C22×C251H(5,𝔽2)11yescharacteristic
182C22×C251H(5,𝔽2)11yescharacteristic
183C2×D251—22no—
184C2×D251—22no—
185C2×D251—22no—
186C2×D251—22no—
187C2×D251—22no—
188C2×D251—22no—
189C2×D251—22no—
190C2×D251—22no—
191C2×D251—22no—
192C2×D251—22no—
193C2×D251—22no—
194C2×D251—22no—
195C2×D251—41no—
196C2×D251—41no—
197C2×D251—41no—
198C2×D251—41no—
199C2×D251—41no—
200C2×D251—41no—
201C2×D251—41no—
202C2×D251—41no—
203C2×D251C4×D414yes—
204C4×C251—22no—
205C4×C251—22no—
206C4×C251—22no—
207C4×C251—22no—
208C4×C251—22no—
209C4×C251—41no—
210C4×C251—41no—
211Dic251—22no—
212Dic251—22no—
213Dic251—22no—
214Dic251—41no—
215Dic251—41no—
216C2×C251C2×C4×D411yescharacteristic
217C2×C251C4×#4111yescharacteristic
218C2×C251#51811yescharacteristic
219C2×C251#54611yescharacteristic
220C2×C251#54611yescharacteristic
221C2×C251#55311yescharacteristic
222C2×C251#55311yescharacteristic
223D251—24no—
224D251—42no—
225D251—42no—
226C251#236411yescharacteristic, complemented, Sylow
227#2364—2511nomaximal, Sylow
228C2×#121—2511no—
229C2×C4×D4—2511no—
230C2×C4×D4—2511no—
231#388—2511no—
232#388—2511no—
233#387—2511no—
234#387D25111yescharacteristic, complemented
235#386—2511no—
236#386—2511no—
237#386—2511no—
238#386—2511no—
239C4×C22⋊C4—2511no—
240C4×C22⋊C4—2511no—
241C4×C22⋊C4—2511no—
242C2×#95—2511no—
243C22×D4—2511no—
244C22×D4—2511no—
245C22×D4—2511no—
246C23×C4—2511no—
247C23×C4—2511no—
248C23×C4—2511no—
249C23×C4—2511no—
250C23×C4—2511no—
251#121—2518no—
252C4×D4—5024no—
253C4×D4—5024no—
254C2×C4⋊C4—2511no—
255C2×C4⋊C4C2×D25111yescharacteristic
256C2×C4⋊C4C2×D25111yescharacteristic
257C2×C22⋊C4—2511no—
258C2×C22⋊C4—2511no—
259C2×C22⋊C4—2511no—
260C2×C22⋊C4—2511no—
261C2×C22⋊C4—2511no—
262C2×C22⋊C4—2511no—
263C2×C22⋊C4—2511no—
264C2×C22⋊C4—2511no—
265C2×C22⋊C4—2511no—
266C2×C22⋊C4—2511no—
267C2×C42—2511no—
268C2×C42—2511no—
269C2×C42—2511no—
270C2×C42C2×D25111yescharacteristic
271#95—2511no—
272#95—2511no—
273#95—2511no—
274#95—2511no—
275#95—2511no—
276#95—2511no—
277#95C2×D25111yescharacteristic
278#95C2×D25111yescharacteristic
279#95C2×D25111yescharacteristic
280#95C2×D25111yescharacteristic
281C24—2511no—
282C24—2511no—
283C24—2511no—
284C2×D4—10041no—
285C2×D4—10041no—
286C2×D4—5022no—
287C2×D4—5022no—
288C2×D4—5022no—
289C2×D4—5022no—
290C2×D4—2514no—
291C2×D4—2514no—
292C2×D4—2514no—
293C22×C4—10041no—
294C22×C4—10041no—
295C22×C4—10041no—
296C22×C4—10041no—
297C22×C4—10041no—
298C22×C4—10041no—
299C22×C4—5022no—
300C22×C4—5022no—
301C22×C4—5022no—
302C22×C4—5022no—
303C22×C4—5022no—
304C22×C4—5022no—
305C22×C4—5022no—
306C22×C4—5022no—
307C22×C4—2514no—
308C22×C4—2511no—
309C22×C4—2511no—
310C22×C4—2511no—
311C22×C4—2511no—
312C22×C4—2511no—
313C22×C4C22×D25111yescharacteristic
314C22×C4C22×D25111yescharacteristic
315C22×C4C22×D25111yescharacteristic
316C22×C4C22×D25111yescharacteristic
317C22×C4C22×D25111yescharacteristic
318C22×C4C22×D25111yescharacteristic
319C22×C4C22×D25111yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—5022no—
322C4⋊C4C4×D25114yescomplemented
323C22⋊C4—10041no—
324C22⋊C4—10041no—
325C22⋊C4—5022no—
326C22⋊C4—5022no—
327C22⋊C4—5022no—
328C22⋊C4—5022no—
329C22⋊C4—5022no—
330C22⋊C4—2514no—
331C22⋊C4—2514no—
332C22⋊C4—2514no—
333C42—22no—
334C42—10041no—
335C42—10041no—
336C42—5022no—
337C23—5021no—
338C23—5021no—
339C23—5021no—
340C23—5021no—
341C23—5021no—
342C23—5021no—
343C23—5021no—
344C23—5021no—
345C23—5021no—
346C23—5021no—
347C23—5021no—
348C23—5021no—
349C23—5021no—
350C23—5021no—
351C23—5021no—
352C23—5021no—
353C23—2512no—
354C23—2512no—
355C23—2512no—
356C23—2512no—
357C23—2512no—
358C23C23×D25111yescharacteristic
359D4—10044no—
360D4—5028no—
361D4—5028no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—10044no—
379C2×C4—10044no—
380C2×C4—10044no—
381C2×C4—10044no—
382C2×C4—10044no—
383C2×C4—5021no—
384C2×C4—5021no—
385C2×C4—5021no—
386C2×C4—5021no—
387C2×C4—5021no—
388C2×C4—5021no—
389C2×C4—5021no—
390C2×C4—5021no—
391C2×C4—5021no—
392C2×C4—5021no—
393C2×C4—5021no—
394C2×C4—5021no—
395C2×C4—2512no—
396C2×C4—2512no—
397C2×C4—2512no—
398C2×C4C2×C4×D25112yes—
399C2×C4C2×C4×D25112yes—
400C2×C4C2×C4×D25112yes—
401C2×C4D4×D25112yes—
402C2×C4#1686081112yes—
403C22—10041no—
404C22—10041no—
405C22—10041no—
406C22—10041no—
407C22—10041no—
408C22—10041no—
409C22—10041no—
410C22—10041no—
411C22—5022no—
412C22—5022no—
413C22—5022no—
414C22—5022no—
415C22—5022no—
416C22—5022no—
417C22—5022no—
418C22—5022no—
419C22—5022no—
420C22—5022no—
421C22—5022no—
422C22—5022no—
423C22—2514no—
424C22C22×C4×D25111yescharacteristic
425C22C2×D4×D25111yescharacteristic
426C22#41×D25111yescharacteristic
427C22#41×D25111yescharacteristic
428C22C2×#1686081111yescharacteristic
429C22#2254876111yescharacteristic
430C22#2254876111yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—10041no—
439C4—10041no—
440C4—5022no—
441C4—5022no—
442C4—5022no—
443C2—10042no—
444C2—10042no—
445C2—5024no—
446C2C4×D4×D25111yescharacteristic
447C2C4×#1686081111yescharacteristic
448C2#3063216111yescharacteristic
449C2#3063232811yescharacteristic
450C2#3063232911yescharacteristic
451C2#3063238811yescharacteristic
452C2#3063239011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#387×C251C2
socle, cosocleC23×C251C24
prime radicalC2×#22548690C2
hypercenter#387D251
hypocenterC251#2364
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×D1004C2semidirect
3. C2×#22548677C2semidirect
4. #30631579C2semidirect
5. #30631577C2semidirect
6. #30631573C2semidirect
7. #30631565C2semidirect
8. #30631565C2semidirect
9. #30631565C2semidirect
10. #30631565C2semidirect
11. C4×(C2×D251)⋊C4C2semidirect
12. #30631534C2semidirect
13. #30631534C2semidirect
14. #95×D251C2semidirect
15. #387×C251C2semidirect
24. #22548690C4semidirect
63. C2×D1004C2×C4semidirect
64. C2×D1004C2×C4semidirect
65. C2×D1004C2×C4semidirect
85. C2×C4×D251C2×C4semidirect
94. (C2×D251)⋊C4C2×C4semidirect
95. (C2×D251)⋊C4C2×C4semidirect
96. (C2×D251)⋊C4D4semidirect
108. C4⋊C4×C251C2×C4semidirect
226. C251#2364semidirect
234. #387D251semidirect
322. 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
C2×C4×D10042
C2×#225486772
#306315792
#306315772
#306315732
#306315652
C4×(C2×D251)⋊C42
#306315342
#95×D2512
#387×C2512
#2364251