Finite Group Databasebeta

← #45667255

Subgroups — #45667255

Order 32896.

588 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551552C211yesmaximal, characteristic
2#30656630C211yesmaximal, characteristic, complemented
3#30656624C211yesmaximal, characteristic, complemented
4#30656618C211yesmaximal, characteristic, complemented
5#30656613C211yesmaximal, characteristic
6#30656613C211yesmaximal, characteristic
7#30656604C211yesmaximal, characteristic, complemented
8#30656598C211yesmaximal, characteristic, complemented
9#30656594C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic257C211yesmaximal, characteristic, complemented
11C4×#16861595C211yesmaximal, characteristic, complemented
12#30656584C211yesmaximal, characteristic, complemented
13#30656583C211yesmaximal, characteristic, complemented
14#95×D257C211yesmaximal, characteristic, complemented
15#388×C257C211yesmaximal, characteristic, complemented
16C22×C22⋊D257C2211yescharacteristic
17C23×Dic257C2211yescharacteristic
18C22×C4×D257C2211yescharacteristic
19C22×C4×D257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21#22551552C418yescomplemented
22C2×C4⋊Dic257C2211yescharacteristic
23C2×#16861597C2211yescharacteristic
24C2×C22⋊Dic257C2211yescharacteristic
25C2×C22⋊Dic257C2211yescharacteristic
26C2×#16861595C2211yescharacteristic
27C2×#16861595C2211yescharacteristic
28C2×#16861595C2211yescharacteristic
29C2×C4×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
52C23×D257C2311yescharacteristic
53C2×C22⋊D257—22no—
54C2×C22⋊D257—22no—
55C2×C22⋊D257C2×C414yes—
56C22×Dic257—41no—
57C22×Dic257—41no—
58C22×Dic257C2×C414yescomplemented
59C22×Dic257C2311yescharacteristic
60C22×Dic257C2311yescharacteristic
61C22×Dic257C2311yescharacteristic
62C22×Dic257C2311yescharacteristic
63C22×Dic257C2311yescharacteristic
64C22×Dic257C2311yescharacteristic
65C22×Dic257C2311yescharacteristic
66C2×C4×D257—22no—
67C2×C4×D257—22no—
68C2×C4×D257—22no—
69C2×C4×D257—22no—
70C2×C4×D257—41no—
71C2×C4×D257—41no—
72C2×C4×D257—41no—
73C2×C4×D257—41no—
74C2×C4×D257C2×C414yes—
75C4⋊Dic257—41no—
76#16861597C2×C414yescomplemented
77C22⋊Dic257—22no—
78C22⋊Dic257—41no—
79#16861595—22no—
80#16861595—22no—
81#16861595C2×C414yes—
82C4×Dic257—22no—
83C4×Dic257—41no—
84C4×Dic257—41no—
85C4×Dic257C2×C414yescomplemented
86C24×C257C2311yescharacteristic
87C22×C4×C257C2311yescharacteristic
88C22×C4×C257C2311yescharacteristic
89C22×C4×C257C2311yescharacteristic
90C22×C4×C257C2311yescharacteristic
91C22×C4×C257C2311yescharacteristic
92C22×C4×C257C2311yescharacteristic
93C4⋊C4×C257—41no—
94C22⋊C4×C257—22no—
95C22⋊C4×C257—41no—
96C22⋊C4×C257C2×C414yescomplemented
97C42×C257—22no—
98C22×D257—21no—
99C22×D257—21no—
100C22×D257—21no—
101C22×D257—21no—
102C22×D257C22×C412yes—
103C22×D257#4112yes—
104C22×D257#4112yes—
105C22⋊D257—28no—
106C2×Dic257—21no—
107C2×Dic257—21no—
108C2×Dic257—21no—
109C2×Dic257—21no—
110C2×Dic257—21no—
111C2×Dic257—21no—
112C2×Dic257—21no—
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—44no—
124C2×Dic257C22×C412yes—
125C2×Dic257C22×C412yes—
126C2×Dic257C22×C412yes—
127C2×Dic257#4112yes—
128C4×D257—44no—
129C4×D257—44no—
130C4×D257—44no—
131C23×C257—21no—
132C23×C257—21no—
133C23×C257—21no—
134C23×C257—21no—
135C23×C257—21no—
136C23×C257—21no—
137C23×C257C22×C412yes—
138C23×C257C2411yescharacteristic
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×C257—21no—
150C2×C4×C257—21no—
151C2×C4×C257—21no—
152C2×C4×C257—21no—
153C2×C4×C257—21no—
154C2×C4×C257C22×C412yes—
155C2×C4×C257C22×C412yes—
156C2×C4×C257#4112yes—
157C22×C257—22no—
158C22×C257—22no—
159C22×C257—22no—
160C22×C257—41no—
161C22×C257—41no—
162C22×C257—41no—
163C22×C257—41no—
164C22×C257C23×C411yescharacteristic
165C22×C257C2×#4111yescharacteristic
166C22×C257C2×#4111yescharacteristic
167C22×C257C2×#4111yescharacteristic
168C22×C257C2×#4111yescharacteristic
169C22×C257H(5,𝔽2)11yescharacteristic
170C22×C257H(5,𝔽2)11yescharacteristic
171C2×D257—22no—
172C2×D257—22no—
173C2×D257—22no—
174C2×D257—22no—
175C2×D257—22no—
176C2×D257—22no—
177C2×D257#11714yes—
178C4×C257—22no—
179C4×C257—22no—
180C4×C257—22no—
181C4×C257—41no—
182C4×C257—41no—
183C4×C257—41no—
184Dic257—22no—
185Dic257—22no—
186Dic257—22no—
187Dic257—22no—
188Dic257—41no—
189Dic257—41no—
190Dic257—41no—
191C2×C257—42no—
192C2×C257C2×#11711yescharacteristic
193C2×C257C4×#4111yescharacteristic
194C2×C257#51811yescharacteristic
195C2×C257#55111yescharacteristic
196C2×C257#55111yescharacteristic
197C2×C257#55311yescharacteristic
198C2×C257#55311yescharacteristic
199D257—24no—
200C257#235311yescharacteristic, complemented, Sylow
201#2353—2571nomaximal, Sylow
202C2×C4×D4—2571no—
203#388—2571no—
204#388D25711yescharacteristic, complemented
205#387—2571no—
206#387—2571no—
207#386—2571no—
208#386—2571no—
209#383—2571no—
210#381—2571no—
211#380—2571no—
212C4×C22⋊C4—2571no—
213C4×C22⋊C4—2571no—
214C4×C22⋊C4—2571no—
215#376—2571no—
216C2×#95—2571no—
217C22×D4—2571no—
218C23×C4—2571no—
219C23×C4—2571no—
220C23×C4—2571no—
221C23×C4—2571no—
222C4×D4—2578no—
223C2×C4⋊C4—2571no—
224C2×C4⋊C4—2571no—
225C2×C4⋊C4C2×D25711yescharacteristic
226C2×C22⋊C4—2571no—
227C2×C22⋊C4—2571no—
228C2×C22⋊C4—2571no—
229C2×C22⋊C4—2571no—
230C2×C22⋊C4—2571no—
231C2×C22⋊C4C2×D25711yescharacteristic
232C2×C22⋊C4C2×D25711yescharacteristic
233C2×C22⋊C4C2×D25711yescharacteristic
234C2×C42—2571no—
235C2×C42—2571no—
236C2×C42—2571no—
237C2×C42—2571no—
238C2×C42C2×D25711yescharacteristic
239#95—2571no—
240#95—2571no—
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#95—2571no—
250#95—2571no—
251#95C2×D25711yescharacteristic
252#95C2×D25711yescharacteristic
253C24—2571no—
254C24C22×D25711yescharacteristic
255C2×D4—2574no—
256C2×D4—5142no—
257C2×D4—5142no—
258C22×C4—2571no—
259C22×C4—2571no—
260C22×C4—2571no—
261C22×C4—2571no—
262C22×C4—2574no—
263C22×C4—2574no—
264C22×C4—2571no—
265C22×C4—2571no—
266C22×C4—2571no—
267C22×C4—5142no—
268C22×C4—5142no—
269C22×C4—5142no—
270C22×C4—5142no—
271C22×C4—10281no—
272C22×C4—10281no—
273C22×C4—10281no—
274C22×C4—10281no—
275C22×C4—10281no—
276C22×C4—10281no—
277C22×C4C22×D25711yescharacteristic
278C22×C4C22×D25711yescharacteristic
279C22×C4C22×D25711yescharacteristic
280C22×C4C22×D25711yescharacteristic
281C22×C4C22×D25711yescharacteristic
282C22×C4C22×D25711yescharacteristic
283C4⋊C4—2574no—
284C4⋊C4—41no—
285C4⋊C4—10281no—
286C22⋊C4—2574no—
287C22⋊C4—22no—
288C22⋊C4—5142no—
289C22⋊C4—5142no—
290C22⋊C4—5142no—
291C22⋊C4—41no—
292C22⋊C4—10281no—
293C22⋊C4C4×D25714yescomplemented
294C42—2574no—
295C42—22no—
296C42—5142no—
297C42—10281no—
298C42—10281no—
299C23—2572no—
300C23—2572no—
301C23—2572no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—5141no—
309C23—5141no—
310C23—5141no—
311C23—5141no—
312C23C2×C4×D25712yes—
313C23C23×D25711yescharacteristic
314D4—5148no—
315C2×C4—2572no—
316C2×C4—2572no—
317C2×C4—2572no—
318C2×C4—2572no—
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—21no—
332C2×C4—21no—
333C2×C4—21no—
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—5141no—
347C2×C4—5141no—
348C2×C4—5141no—
349C2×C4—5141no—
350C2×C4—5141no—
351C2×C4—10284no—
352C2×C4—10284no—
353C2×C4—10284no—
354C2×C4—10284no—
355C2×C4C2×C4×D25712yes—
356C2×C4C2×C4×D25712yes—
357C2×C4#1686161712yes—
358C22—2574no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—5142no—
363C22—5142no—
364C22—5142no—
365C22—5142no—
366C22—5142no—
367C22—5142no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22C22×C4×D25711yescharacteristic
373C22#41×D25711yescharacteristic
374C22#41×D25711yescharacteristic
375C22#41×D25711yescharacteristic
376C22C2×#1686161711yescharacteristic
377C22#2255163711yescharacteristic
378C22#2255163811yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—5142no—
383C4—5142no—
384C4—5142no—
385C4—5142no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—10281no—
390C4—10281no—
391C4—10281no—
392C2—5144no—
393C2—42no—
394C2#117×D25711yescharacteristic
395C2C4×#1686161711yescharacteristic
396C2#3065720911yescharacteristic
397C2#3065741011yescharacteristic
398C2#3065741711yescharacteristic
399C2#3065743411yescharacteristic
400C2#3065743811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#388×C257C2
socle, cosocleC23×C257C24
prime radicalC22×C22⋊D257C22
hypercenter#388D257
hypocenterC257#2353
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #30656630C2semidirect
3. #30656624C2semidirect
4. #30656618C2semidirect
7. #30656604C2semidirect
8. #30656598C2semidirect
9. #30656594C2semidirect
10. C22⋊C4×Dic257C2semidirect
11. C4×#16861595C2semidirect
12. #30656584C2semidirect
13. #30656583C2semidirect
14. #95×D257C2semidirect
15. #388×C257C2semidirect
21. #22551552C4semidirect
58. C22×Dic257C2×C4semidirect
76. #16861597C2×C4semidirect
85. C4×Dic257C2×C4semidirect
96. C22⋊C4×C257C2×C4semidirect
200. C257#2353semidirect
204. #388D257semidirect
293. C22⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515522
#306566302
#306566242
#306566182
#306566132
#306566042
#306565982
#306565942
C22⋊C4×Dic2572
C4×#168615952
#306565842
#306565832
#95×D2572
#388×C2572
#2353257