Finite Group Databasebeta

← #45667449

Subgroups — #45667449

Order 32896.

552 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551577C211yesmaximal, characteristic
2#30656639C211yesmaximal, characteristic
3#30656628C211yesmaximal, characteristic, complemented
4#30656628C211yesmaximal, characteristic, complemented
5#30656627C211yesmaximal, characteristic, complemented
6#30656627C211yesmaximal, characteristic, complemented
7#30656624C211yesmaximal, characteristic, complemented
8#30656623C211yesmaximal, characteristic, complemented
9#30656615C211yesmaximal, characteristic
10C4×#16861597C211yesmaximal, characteristic, complemented
11C4×#16861595C211yesmaximal, characteristic, complemented
12C4×#16861595C211yesmaximal, characteristic, complemented
13#30656584C211yesmaximal, characteristic, complemented
14#30656584C211yesmaximal, characteristic, complemented
15C4×C4⋊C4×C257C211yesmaximal, characteristic, complemented
16C22×D1028C2211yescharacteristic
17C22×C4×D257C2211yescharacteristic
18C22×C4×D257C2211yescharacteristic
19#22551577C418yescomplemented
20C2×C4⋊Dic257C2211yescharacteristic
21C2×#16861597C2211yescharacteristic
22C2×#16861597C2211yescharacteristic
23C2×#16861595C2211yescharacteristic
24C2×#16861595C2211yescharacteristic
25C2×#16861595C2211yescharacteristic
26C2×#16861595C2211yescharacteristic
27C2×#16861595C2211yescharacteristic
28C2×#16861595C2211yescharacteristic
29C2×#16861595C2211yescharacteristic
30C2×#16861595C2211yescharacteristic
31C2×#16861595C2211yescharacteristic
32C2×#16861595C2211yescharacteristic
33C2×C4×Dic257C2211yescharacteristic
34C2×C4×Dic257C2211yescharacteristic
35C2×C4×Dic257C2211yescharacteristic
36C2×C4×Dic257C2211yescharacteristic
37#22551499C2211yescharacteristic
38#22551499C2211yescharacteristic
39#22551499C2211yescharacteristic
40#22551499C2211yescharacteristic
41#22551499C2211yescharacteristic
42#22551499C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45C2×C4⋊C4×C257C2211yescharacteristic
46C2×C4⋊C4×C257C2211yescharacteristic
47C2×C42×C257C2211yescharacteristic
48C2×C42×C257C2211yescharacteristic
49C2×C42×C257C2211yescharacteristic
50#95×C257C2211yescharacteristic
51#95×C257C2211yescharacteristic
52C23×D257C2311yescharacteristic
53C23×D257C2311yescharacteristic
54C2×D1028—41no—
55C2×D1028—41no—
56C2×D1028C2×C414yes—
57C22×Dic257C2311yescharacteristic
58C22×Dic257C2311yescharacteristic
59C22×Dic257C2311yescharacteristic
60C22×Dic257C2311yescharacteristic
61C22×Dic257C2311yescharacteristic
62C22×Dic257C2311yescharacteristic
63C2×C4×D257—22no—
64C2×C4×D257—22no—
65C2×C4×D257—41no—
66C2×C4×D257—41no—
67C2×C4×D257—41no—
68C2×C4×D257C2×C414yescomplemented
69C4⋊Dic257—41no—
70#16861597—22no—
71#16861597C2×C414yescomplemented
72#16861595—22no—
73#16861595—22no—
74#16861595—22no—
75#16861595—22no—
76#16861595—22no—
77#16861595—22no—
78#16861595—41no—
79#16861595C2×C414yes—
80#16861595C2×C414yes—
81#16861595C2×C414yescomplemented
82C4×Dic257—22no—
83C4×Dic257—41no—
84C4×Dic257—41no—
85C4×Dic257—41no—
86C22×C4×C257C2311yescharacteristic
87C22×C4×C257C2311yescharacteristic
88C22×C4×C257C2311yescharacteristic
89C22×C4×C257C2311yescharacteristic
90C22×C4×C257C2311yescharacteristic
91C22×C4×C257C2311yescharacteristic
92C22×C4×C257C2311yescharacteristic
93C4⋊C4×C257—22no—
94C4⋊C4×C257C2×C414yescomplemented
95C42×C257—22no—
96C42×C257—41no—
97C42×C257—41no—
98C22×D257—21no—
99C22×D257—21no—
100C22×D257—21no—
101C22×D257—21no—
102C22×D257—21no—
103C22×D257—21no—
104C22×D257—21no—
105C22×D257—21no—
106C22×D257—21no—
107C22×D257—21no—
108C22×D257—21no—
109C22×D257—21no—
110C22×D257C22×C412yes—
111C22×D257C22×C412yes—
112D1028—44no—
113C2×Dic257—21no—
114C2×Dic257—21no—
115C2×Dic257—21no—
116C2×Dic257—21no—
117C2×Dic257—21no—
118C2×Dic257—21no—
119C2×Dic257—21no—
120C2×Dic257—21no—
121C2×Dic257—21no—
122C2×Dic257—21no—
123C2×Dic257—21no—
124C2×Dic257—21no—
125C2×Dic257—21no—
126C2×Dic257—21no—
127C2×Dic257—21no—
128C2×Dic257C22×C412yes—
129C2×Dic257C22×C412yes—
130C2×Dic257#4112yes—
131C4×D257—44no—
132C4×D257—44no—
133C23×C257C2411yescharacteristic
134C2×C4×C257—21no—
135C2×C4×C257—21no—
136C2×C4×C257—21no—
137C2×C4×C257—21no—
138C2×C4×C257—21no—
139C2×C4×C257—21no—
140C2×C4×C257—21no—
141C2×C4×C257—21no—
142C2×C4×C257—21no—
143C2×C4×C257—21no—
144C2×C4×C257—21no—
145C2×C4×C257—21no—
146C2×C4×C257—21no—
147C2×C4×C257—21no—
148C2×C4×C257—21no—
149C2×C4×C257C22×C412yes—
150C2×C4×C257C22×C412yes—
151C2×C4×C257C22×C412yes—
152C2×C4×C257#4112yes—
153C2×C4×C257#4112yes—
154C2×C4×C257#4112yes—
155C22×C257C23×C411yescharacteristic
156C22×C257C2×#4111yescharacteristic
157C22×C257C2×#4111yescharacteristic
158C22×C257C2×#4111yescharacteristic
159C22×C257C2×#4111yescharacteristic
160C22×C257H(5,𝔽2)11yescharacteristic
161C22×C257H(5,𝔽2)11yescharacteristic
162C2×D257—22no—
163C2×D257—22no—
164C2×D257—22no—
165C2×D257—22no—
166C2×D257—22no—
167C2×D257—22no—
168C2×D257—41no—
169C2×D257—41no—
170C2×D257—41no—
171C2×D257—41no—
172C2×D257—41no—
173C2×D257—41no—
174C2×D257—41no—
175C2×D257—41no—
176C4×C257—22no—
177C4×C257—22no—
178C4×C257—22no—
179C4×C257—22no—
180C4×C257—22no—
181C4×C257—22no—
182C4×C257—41no—
183Dic257—22no—
184Dic257—22no—
185Dic257—22no—
186Dic257—41no—
187Dic257—41no—
188Dic257—41no—
189C2×C257C4×#4111yescharacteristic
190C2×C257C4×#4111yescharacteristic
191C2×C257#51811yescharacteristic
192C2×C257#55111yescharacteristic
193C2×C257#55311yescharacteristic
194C2×C257#55311yescharacteristic
195C2×C257#55911yescharacteristic
196D257—42no—
197D257—42no—
198C257#237411yescharacteristic, complemented, Sylow
199#2374—2571nomaximal, Sylow
200C2×#123—2571no—
201#390—2571no—
202#388—2571no—
203#388—2571no—
204#388—2571no—
205#388—2571no—
206#387—2571no—
207#387—2571no—
208#386—2571no—
209C4×C4⋊C4—2571no—
210C4×C4⋊C4D25711yescharacteristic, complemented
211C4×C22⋊C4—2571no—
212C4×C22⋊C4—2571no—
213C4×C22⋊C4—2571no—
214C4×C22⋊C4—2571no—
215C22×D4—2571no—
216C23×C4—2571no—
217C23×C4—2571no—
218#123—2578no—
219C2×C4⋊C4—2571no—
220C2×C4⋊C4—2571no—
221C2×C4⋊C4—2571no—
222C2×C4⋊C4C2×D25711yescharacteristic
223C2×C4⋊C4C2×D25711yescharacteristic
224C2×C22⋊C4—2571no—
225C2×C22⋊C4—2571no—
226C2×C22⋊C4—2571no—
227C2×C22⋊C4—2571no—
228C2×C22⋊C4—2571no—
229C2×C22⋊C4—2571no—
230C2×C22⋊C4—2571no—
231C2×C22⋊C4—2571no—
232C2×C22⋊C4—2571no—
233C2×C22⋊C4—2571no—
234C2×C42—2571no—
235C2×C42—2571no—
236C2×C42—2571no—
237C2×C42—2571no—
238C2×C42C2×D25711yescharacteristic
239C2×C42C2×D25711yescharacteristic
240C2×C42C2×D25711yescharacteristic
241#95—2571no—
242#95—2571no—
243#95—2571no—
244#95—2571no—
245#95—2571no—
246#95—2571no—
247#95—2571no—
248#95—2571no—
249#95C2×D25711yescharacteristic
250#95C2×D25711yescharacteristic
251C24—2571no—
252C24—2571no—
253C2×D4—2574no—
254C2×D4—10281no—
255C2×D4—10281no—
256C22×C4—2571no—
257C22×C4—2571no—
258C22×C4—2571no—
259C22×C4—2574no—
260C22×C4—2571no—
261C22×C4—2571no—
262C22×C4—2571no—
263C22×C4—5142no—
264C22×C4—5142no—
265C22×C4—10281no—
266C22×C4—10281no—
267C22×C4—10281no—
268C22×C4C22×D25711yescharacteristic
269C22×C4C22×D25711yescharacteristic
270C22×C4C22×D25711yescharacteristic
271C22×C4C22×D25711yescharacteristic
272C22×C4C22×D25711yescharacteristic
273C22×C4C22×D25711yescharacteristic
274C22×C4C22×D25711yescharacteristic
275C4⋊C4—2574no—
276C4⋊C4—22no—
277C4⋊C4—5142no—
278C4⋊C4—10281no—
279C4⋊C4C4×D25714yescomplemented
280C22⋊C4—2574no—
281C22⋊C4—2574no—
282C22⋊C4—2574no—
283C22⋊C4—5142no—
284C22⋊C4—5142no—
285C22⋊C4—5142no—
286C22⋊C4—5142no—
287C22⋊C4—5142no—
288C22⋊C4—5142no—
289C22⋊C4—10281no—
290C42—22no—
291C42—5142no—
292C42—41no—
293C42—41no—
294C42—10281no—
295C42—10281no—
296C42—10281no—
297C23—2572no—
298C23—2572no—
299C23—5141no—
300C23—5141no—
301C23—5141no—
302C23—5141no—
303C23—5141no—
304C23—5141no—
305C23—5141no—
306C23—5141no—
307C23—5141no—
308C23—5141no—
309C23—5141no—
310C23—5141no—
311C23C23×D25711yescharacteristic
312D4—10284no—
313C2×C4—2572no—
314C2×C4—2572no—
315C2×C4—2572no—
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—5141no—
332C2×C4—5141no—
333C2×C4—5141no—
334C2×C4—5141no—
335C2×C4—5141no—
336C2×C4—5141no—
337C2×C4—5141no—
338C2×C4—5141no—
339C2×C4—5141no—
340C2×C4—5141no—
341C2×C4—5141no—
342C2×C4—5141no—
343C2×C4—5141no—
344C2×C4—5141no—
345C2×C4—5141no—
346C2×C4—10284no—
347C2×C4—10284no—
348C2×C4C2×C4×D25712yes—
349C2×C4C2×C4×D25712yes—
350C2×C4C2×C4×D25712yes—
351C2×C4#1686161812yes—
352C2×C4#1686161812yes—
353C2×C4#1686161912yes—
354C22—5142no—
355C22—5142no—
356C22—5142no—
357C22—5142no—
358C22—5142no—
359C22—5142no—
360C22—10281no—
361C22—10281no—
362C22—10281no—
363C22—10281no—
364C22—10281no—
365C22—10281no—
366C22—10281no—
367C22—10281no—
368C22C22×C4×D25711yescharacteristic
369C22#41×D25711yescharacteristic
370C22C2×#1686161811yescharacteristic
371C22C2×#1686161811yescharacteristic
372C22C2×#1686161911yescharacteristic
373C22#2255163711yescharacteristic
374C22#2255163811yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—5142no—
382C4—5142no—
383C4—5142no—
384C4—41no—
385C4—10281no—
386C4—10281no—
387C4—10281no—
388C2—10282no—
389C2—10282no—
390C2C4×#1686161811yescharacteristic
391C2C4×#1686161911yescharacteristic
392C2#3065720911yescharacteristic
393C2#3065741111yescharacteristic
394C2#3065743511yescharacteristic
395C2#3065743611yescharacteristic
396C2#3065747211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
FittingC4×C4⋊C4×C257C2
socle, cosocleC23×C257C24
prime radicalC22×D1028C22
hypercenterC4×C4⋊C4D257
hypocenterC257#2374
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #30656628C2semidirect
4. #30656628C2semidirect
5. #30656627C2semidirect
6. #30656627C2semidirect
7. #30656624C2semidirect
8. #30656623C2semidirect
10. C4×#16861597C2semidirect
11. C4×#16861595C2semidirect
12. C4×#16861595C2semidirect
13. #30656584C2semidirect
14. #30656584C2semidirect
15. C4×C4⋊C4×C257C2semidirect
19. #22551577C4semidirect
68. C2×C4×D257C2×C4semidirect
71. #16861597C2×C4semidirect
81. #16861595C2×C4semidirect
94. C4⋊C4×C257C2×C4semidirect
198. C257#2374semidirect
210. C4×C4⋊C4D257semidirect
279. C4⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515772
#306566392
#306566282
#306566272
#306566242
#306566232
#306566152
C4×#168615972
C4×#168615952
#306565842
C4×C4⋊C4×C2572
#2374257