Finite Group Databasebeta

← #45667487

Subgroups — #45667487

Order 32896.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551569C211yesmaximal, characteristic
2#30656643C211yesmaximal, characteristic, complemented
3#30656641C211yesmaximal, characteristic, complemented
4#30656634C211yesmaximal, characteristic, complemented
5#30656633C211yesmaximal, characteristic, complemented
6#30656631C211yesmaximal, characteristic
7#30656631C211yesmaximal, characteristic
8#30656630C211yesmaximal, characteristic
9#30656622C211yesmaximal, characteristic, complemented
10#30656621C211yesmaximal, characteristic, complemented
11#30656619C211yesmaximal, characteristic, complemented
12#30656618C211yesmaximal, characteristic, complemented
13#30656598C211yesmaximal, characteristic
14#30656595C211yesmaximal, characteristic
15#388×C257C211yesmaximal, characteristic
16C22×Dic514C2211yescharacteristic
17C23×Dic257C2211yescharacteristic
18#22551569C418yescomplemented
19C2×C4⋊Dic257C2211yescharacteristic
20C2×C4⋊Dic257C2211yescharacteristic
21C2×#16861597C2211yescharacteristic
22C2×#16861597C2211yescharacteristic
23C2×#16861597C2211yescharacteristic
24C2×#16861597C2211yescharacteristic
25C2×#16861597C2211yescharacteristic
26C2×#16861597C2211yescharacteristic
27C2×C22⋊Dic257C2211yescharacteristic
28C2×C22⋊Dic257C2211yescharacteristic
29C2×C22⋊Dic257C2211yescharacteristic
30C2×C4×Dic257C2211yescharacteristic
31C2×C4×Dic257C2211yescharacteristic
32C2×C4×Dic257C2211yescharacteristic
33#22551499C2211yescharacteristic
34#22551499C2211yescharacteristic
35#22551499C2211yescharacteristic
36#22551499C2211yescharacteristic
37#22551499C2211yescharacteristic
38#22551499C2211yescharacteristic
39#22551499C2211yescharacteristic
40#22551499C2211yescharacteristic
41#22551499C2211yescharacteristic
42#22551499C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45C2×C4⋊C4×C257C2211yescharacteristic
46C2×C22⋊C4×C257C2211yescharacteristic
47C2×C22⋊C4×C257C2211yescharacteristic
48C2×C22⋊C4×C257C2211yescharacteristic
49C2×C42×C257C2211yescharacteristic
50#95×C257C2211yescharacteristic
51#95×C257C2211yescharacteristic
52C2×Dic514—41no—
53C2×Dic514—41no—
54C2×Dic514C2×C414yes—
55C22×Dic257—41no—
56C22×Dic257—41no—
57C22×Dic257C2×C414yes—
58C22×Dic257C2311yescharacteristic
59C22×Dic257C2311yescharacteristic
60C22×Dic257C2311yescharacteristic
61C22×Dic257C2311yescharacteristic
62C22×Dic257C2311yescharacteristic
63C22×Dic257C2311yescharacteristic
64C22×Dic257C2311yescharacteristic
65C22×Dic257C2311yescharacteristic
66C4⋊Dic257—41no—
67C4⋊Dic257C2×C414yes—
68#16861597—41no—
69#16861597—41no—
70#16861597—41no—
71#16861597—41no—
72#16861597C2×C414yes—
73#16861597C2×C414yes—
74C22⋊Dic257—41no—
75C22⋊Dic257—41no—
76C22⋊Dic257C2×C414yes—
77C4×Dic257—41no—
78C4×Dic257—41no—
79C4×Dic257—41no—
80C24×C257C2311yescharacteristic
81C22×C4×C257C2311yescharacteristic
82C22×C4×C257C2311yescharacteristic
83C22×C4×C257C2311yescharacteristic
84C22×C4×C257C2311yescharacteristic
85C22×C4×C257C2311yescharacteristic
86C22×C4×C257C2311yescharacteristic
87C4⋊C4×C257—41no—
88C22⋊C4×C257—41no—
89C22⋊C4×C257—41no—
90C22⋊C4×C257C2×C414yes—
91C42×C257—41no—
92Dic514—44no—
93C2×Dic257—21no—
94C2×Dic257—21no—
95C2×Dic257—21no—
96C2×Dic257—21no—
97C2×Dic257—21no—
98C2×Dic257—21no—
99C2×Dic257—21no—
100C2×Dic257—21no—
101C2×Dic257—21no—
102C2×Dic257—21no—
103C2×Dic257—21no—
104C2×Dic257—21no—
105C2×Dic257—21no—
106C2×Dic257—21no—
107C2×Dic257—21no—
108C2×Dic257—21no—
109C2×Dic257—21no—
110C2×Dic257—21no—
111C2×Dic257—21no—
112C2×Dic257—21no—
113C2×Dic257—44no—
114C2×Dic257C22×C412yes—
115C2×Dic257C22×C412yes—
116C2×Dic257C22×C412yes—
117C2×Dic257C22×C412yes—
118C23×C257—21no—
119C23×C257—21no—
120C23×C257—21no—
121C23×C257—21no—
122C23×C257—21no—
123C23×C257—21no—
124C23×C257C22×C412yes—
125C23×C257C2411yescharacteristic
126C2×C4×C257—21no—
127C2×C4×C257—21no—
128C2×C4×C257—21no—
129C2×C4×C257—21no—
130C2×C4×C257—21no—
131C2×C4×C257—21no—
132C2×C4×C257—21no—
133C2×C4×C257—21no—
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×C257C22×C412yes—
143C2×C4×C257C22×C412yes—
144C22×C257—22no—
145C22×C257—22no—
146C22×C257—22no—
147C22×C257—41no—
148C22×C257—41no—
149C22×C257—41no—
150C22×C257—41no—
151C22×C257C23×C411yescharacteristic
152C22×C257H(5,𝔽2)11yescharacteristic
153C22×C257H(5,𝔽2)11yescharacteristic
154C22×C257H(5,𝔽2)11yescharacteristic
155C22×C2572−1+411yescharacteristic
156C22×C2572−1+411yescharacteristic
157C22×C2572−1+411yescharacteristic
158C4×C257—22no—
159C4×C257—22no—
160C4×C257—41no—
161C4×C257—41no—
162C4×C257—41no—
163C4×C257—41no—
164Dic257—22no—
165Dic257—22no—
166Dic257—22no—
167Dic257—22no—
168Dic257—41no—
169Dic257—41no—
170Dic257—41no—
171Dic257—41no—
172C2×C257—42no—
173C2×C257#51811yescharacteristic
174C2×C257#51911yescharacteristic
175C2×C257#52011yescharacteristic
176C2×C257#56211yescharacteristic
177C2×C257#56211yescharacteristic
178C2×C257#56311yescharacteristic
179C2×C257#56311yescharacteristic
180C257#237811yescharacteristic, complemented, Sylow
181#2378—2571nomaximal, Sylow
182C2×C22⋊Q8—2571no—
183#391—2571no—
184#391—2571no—
185#389—2571no—
186#389—2571no—
187#388—2571no—
188#388—2571no—
189#388—2571no—
190#388D25711yescharacteristic
191#387—2571no—
192#387—2571no—
193#387—2571no—
194#387—2571no—
195#381—2571no—
196#380—2571no—
197C22×Q8—2571no—
198C23×C4—2571no—
199C22⋊Q8—2578no—
200C2×C4⋊C4—2571no—
201C2×C4⋊C4—2571no—
202C2×C4⋊C4—2571no—
203C2×C4⋊C4—2571no—
204C2×C4⋊C4—2571no—
205C2×C4⋊C4—2571no—
206C2×C4⋊C4—2571no—
207C2×C4⋊C4—2571no—
208C2×C4⋊C4C2×D25711yescharacteristic
209C2×C22⋊C4—2571no—
210C2×C22⋊C4—2571no—
211C2×C22⋊C4—2571no—
212C2×C22⋊C4C2×D25711yescharacteristic
213C2×C22⋊C4C2×D25711yescharacteristic
214C2×C22⋊C4C2×D25711yescharacteristic
215C2×C42—2571no—
216C2×C42—2571no—
217C2×C42—2571no—
218C2×C42C2×D25711yescharacteristic
219#95—2571no—
220#95—2571no—
221#95—2571no—
222#95—2571no—
223#95—2571no—
224#95—2571no—
225#95—2571no—
226#95—2571no—
227#95—2571no—
228#95—2571no—
229#95—2571no—
230#95—2571no—
231#95C2×D25711yescharacteristic
232#95C2×D25711yescharacteristic
233C24C22×D25711yescharacteristic
234C2×Q8—2574no—
235C2×Q8—10281no—
236C2×Q8—10281no—
237C22×C4—2571no—
238C22×C4—2571no—
239C22×C4—2571no—
240C22×C4—2571no—
241C22×C4—2574no—
242C22×C4—2571no—
243C22×C4—2571no—
244C22×C4—2571no—
245C22×C4—2571no—
246C22×C4—10281no—
247C22×C4—10281no—
248C22×C4C22×D25711yescharacteristic
249C22×C4C22×D25711yescharacteristic
250C22×C4C22×D25711yescharacteristic
251C22×C4C22×D25711yescharacteristic
252C22×C4C22×D25711yescharacteristic
253C22×C4C22×D25711yescharacteristic
254C4⋊C4—2574no—
255C4⋊C4—2574no—
256C4⋊C4—2574no—
257C4⋊C4—41no—
258C4⋊C4—10281no—
259C4⋊C4—10281no—
260C4⋊C4—10281no—
261C4⋊C4—10281no—
262C4⋊C4—10281no—
263C22⋊C4—2574no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—10281no—
267C22⋊C4—10281no—
268C22⋊C4C4×D25714yes—
269C42—41no—
270C42—10281no—
271C42—10281no—
272C42—10281no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23C2×C4×D25712yes—
280C23C23×D25711yescharacteristic
281Q8—10284no—
282C2×C4—2572no—
283C2×C4—2572no—
284C2×C4—2572no—
285C2×C4—2572no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—5141no—
303C2×C4—5141no—
304C2×C4—5141no—
305C2×C4—5141no—
306C2×C4—5141no—
307C2×C4—5141no—
308C2×C4—5141no—
309C2×C4—5141no—
310C2×C4—5141no—
311C2×C4—5141no—
312C2×C4—5141no—
313C2×C4—5141no—
314C2×C4—5141no—
315C2×C4—5141no—
316C2×C4—5141no—
317C2×C4—5141no—
318C2×C4—5141no—
319C2×C4—5141no—
320C2×C4—5141no—
321C2×C4—5141no—
322C2×C4—10284no—
323C2×C4C2×C4×D25712yes—
324C2×C4C2×C4×D25712yes—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C22×C4×D25711yescharacteristic
333C22#2255163711yescharacteristic
334C22#2255163711yescharacteristic
335C22#2255163811yescharacteristic
336C22#2255163911yescharacteristic
337C22#2255163911yescharacteristic
338C22#2255163911yescharacteristic
339C4—22no—
340C4—22no—
341C4—5142no—
342C4—5142no—
343C4—5142no—
344C4—5142no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—10281no—
350C4—10281no—
351C4—10281no—
352C4—10281no—
353C2—42no—
354C2#3065720911yescharacteristic
355C2#3065721411yescharacteristic
356C2#3065721611yescharacteristic
357C2#3065748111yescharacteristic
358C2#3065748211yescharacteristic
359C2#3065748911yescharacteristic
360C2#3065749011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#388×C257C2
socle, cosocleC23×C257C24
prime radicalC24×C257C23
hypercenter#388D257
hypocenterC257#2378
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #30656643C2semidirect
3. #30656641C2semidirect
4. #30656634C2semidirect
5. #30656633C2semidirect
9. #30656622C2semidirect
10. #30656621C2semidirect
11. #30656619C2semidirect
12. #30656618C2semidirect
18. #22551569C4semidirect
180. C257#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#225515692
#306566432
#306566412
#306566342
#306566332
#306566312
#306566302
#306566222
#306566212
#306566192
#306566182
#306565982
#306565952
#388×C2572
#2378257