Finite Group Databasebeta

← #45666990

Subgroups — #45666990

Order 32896.

632 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551574C211yesmaximal, characteristic
2C2×#22551552C211yesmaximal, characteristic
3C2×C4⋊C4×D257C211yesmaximal, characteristic, complemented
4#30656635C211yesmaximal, characteristic, complemented
5#30656632C211yesmaximal, characteristic, complemented
6#30656631C211yesmaximal, characteristic, complemented
7#30656624C211yesmaximal, characteristic, complemented
8#30656620C211yesmaximal, characteristic, complemented
9#30656613C211yesmaximal, characteristic
10#30656612C211yesmaximal, characteristic, complemented
11#30656611C211yesmaximal, characteristic, complemented
12#30656609C211yesmaximal, characteristic, complemented
13#30656599C211yesmaximal, characteristic, complemented
14C4×#16861595C211yesmaximal, characteristic, complemented
15#388×C257C211yesmaximal, characteristic, complemented
16C22×C22⋊D257C2211yescharacteristic
17C23×Dic257C2211yescharacteristic
18C22×C4×D257C2211yescharacteristic
19C22×C4×D257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21#22551574C418yescomplemented
22#22551552—24no—
23C2×C4⋊Dic257C2211yescharacteristic
24C2×C4⋊Dic257C2211yescharacteristic
25C2×C4⋊Dic257C2211yescharacteristic
26C2×#16861597C2211yescharacteristic
27C2×#16861597C2211yescharacteristic
28C2×#16861597C2211yescharacteristic
29C2×#16861597C2211yescharacteristic
30C2×#16861597C2211yescharacteristic
31C4⋊C4×D257—24no—
32C2×C22⋊Dic257C2211yescharacteristic
33C2×C22⋊Dic257C2211yescharacteristic
34C2×#16861595C2211yescharacteristic
35C2×#16861595C2211yescharacteristic
36C2×#16861595C2211yescharacteristic
37C2×C4×Dic257C2211yescharacteristic
38C2×C4×Dic257C2211yescharacteristic
39#22551499C2211yescharacteristic
40#22551499C2211yescharacteristic
41#22551499C2211yescharacteristic
42#22551499C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45#22551499C2211yescharacteristic
46#22551499C2211yescharacteristic
47C2×C4⋊C4×C257C2211yescharacteristic
48C2×C22⋊C4×C257C2211yescharacteristic
49C2×C22⋊C4×C257C2211yescharacteristic
50C2×C22⋊C4×C257C2211yescharacteristic
51C2×C42×C257C2211yescharacteristic
52#95×C257C2211yescharacteristic
53#95×C257C2211yescharacteristic
54C23×D257C2311yescharacteristic
55C2×C22⋊D257—22no—
56C2×C22⋊D257—22no—
57C2×C22⋊D257C2×C414yes—
58C22×Dic257—22no—
59C22×Dic257—41no—
60C22×Dic257—41no—
61C22×Dic257C2311yescharacteristic
62C22×Dic257C2311yescharacteristic
63C22×Dic257C2311yescharacteristic
64C22×Dic257C2311yescharacteristic
65C22×Dic257C2311yescharacteristic
66C22×Dic257C2311yescharacteristic
67C22×Dic257C2311yescharacteristic
68C2×C4×D257—22no—
69C2×C4×D257—22no—
70C2×C4×D257—22no—
71C2×C4×D257—41no—
72C2×C4×D257—41no—
73C2×C4×D257—41no—
74C2×C4×D257—41no—
75C2×C4×D257C2×C414yescomplemented
76C2×C4×D257D414yes—
77C4⋊Dic257—22no—
78C4⋊Dic257—41no—
79C4⋊Dic257C2×C414yescomplemented
80#16861597—22no—
81#16861597—22no—
82#16861597—22no—
83#16861597—41no—
84#16861597C2×C414yescomplemented
85C22⋊Dic257—41no—
86C22⋊Dic257C2×C414yescomplemented
87#16861595—22no—
88#16861595—22no—
89#16861595C2×C414yescomplemented
90C4×Dic257—22no—
91C4×Dic257—41no—
92C24×C257C2311yescharacteristic
93C22×C4×C257C2311yescharacteristic
94C22×C4×C257C2311yescharacteristic
95C22×C4×C257C2311yescharacteristic
96C22×C4×C257C2311yescharacteristic
97C22×C4×C257C2311yescharacteristic
98C22×C4×C257C2311yescharacteristic
99C4⋊C4×C257—22no—
100C22⋊C4×C257—22no—
101C22⋊C4×C257—41no—
102C22⋊C4×C257C2×C414yescomplemented
103C42×C257—41no—
104C22×D257—21no—
105C22×D257—21no—
106C22×D257—21no—
107C22×D257—21no—
108C22×D257C22×C412yes—
109C22×D257C2×D412yes—
110C22×D257#4112yes—
111C22⋊D257—28no—
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—21no—
124C2×Dic257—21no—
125C2×Dic257—21no—
126C2×Dic257—21no—
127C2×Dic257—21no—
128C2×Dic257—21no—
129C2×Dic257—44no—
130C2×Dic257C22×C412yes—
131C2×Dic257C22×C412yes—
132C2×Dic257C22×C412yes—
133C2×Dic257C2×D412yes—
134C4×D257—28no—
135C4×D257—44no—
136C4×D257—44no—
137C23×C257—21no—
138C23×C257—21no—
139C23×C257—21no—
140C23×C257—21no—
141C23×C257—21no—
142C23×C257—21no—
143C23×C257C22×C412yes—
144C23×C257C2411yescharacteristic
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×C257—21no—
155C2×C4×C257—21no—
156C2×C4×C257—21no—
157C2×C4×C257—21no—
158C2×C4×C257—21no—
159C2×C4×C257—21no—
160C2×C4×C257C22×C412yes—
161C2×C4×C257C22×C412yes—
162C2×C4×C257C2×D412yes—
163C22×C257—22no—
164C22×C257—22no—
165C22×C257—22no—
166C22×C257—41no—
167C22×C257—41no—
168C22×C257—41no—
169C22×C257—41no—
170C22×C257C23×C411yescharacteristic
171C22×C257C22×D411yescharacteristic
172C22×C257C2×#4111yescharacteristic
173C22×C257H(5,𝔽2)11yescharacteristic
174C22×C257H(5,𝔽2)11yescharacteristic
175C22×C2572−1+411yescharacteristic
176C22×C2572−1+411yescharacteristic
177C2×D257—22no—
178C2×D257—22no—
179C2×D257—22no—
180C2×D257—22no—
181C2×D257—22no—
182C2×D257—22no—
183C2×D257C4×D414yes—
184C4×C257—22no—
185C4×C257—22no—
186C4×C257—22no—
187C4×C257—41no—
188C4×C257—41no—
189C4×C257—41no—
190Dic257—22no—
191Dic257—22no—
192Dic257—22no—
193Dic257—22no—
194Dic257—41no—
195Dic257—41no—
196Dic257—41no—
197C2×C257—42no—
198C2×C257C2×C4×D411yescharacteristic
199C2×C257#52011yescharacteristic
200C2×C257#52011yescharacteristic
201C2×C257#53411yescharacteristic
202C2×C257#53611yescharacteristic
203C2×C257#53911yescharacteristic
204C2×C257#53911yescharacteristic
205D257—24no—
206C257#232211yescharacteristic, complemented, Sylow
207#2322—2571nomaximal, Sylow
208C2×#123—2571no—
209C2×C4×D4—2571no—
210C22×C4⋊C4—2571no—
211#389—2571no—
212#389—2571no—
213#388—2571no—
214#388D25711yescharacteristic, complemented
215#387—2571no—
216#387—2571no—
217#386—2571no—
218#385—2571no—
219#385—2571no—
220#385—2571no—
221#381—2571no—
222C4×C22⋊C4—2571no—
223C22×D4—2571no—
224C23×C4—2571no—
225C23×C4—2571no—
226C23×C4—2571no—
227C23×C4—2571no—
228#123—2578no—
229C4×D4—5144no—
230C2×C4⋊C4—2571no—
231C2×C4⋊C4—2571no—
232C2×C4⋊C4—2571no—
233C2×C4⋊C4—2571no—
234C2×C4⋊C4—2571no—
235C2×C4⋊C4—2571no—
236C2×C4⋊C4—2571no—
237C2×C4⋊C4—2571no—
238C2×C4⋊C4—5144no—
239C2×C4⋊C4C2×D25711yescharacteristic
240C2×C22⋊C4—2571no—
241C2×C22⋊C4—2571no—
242C2×C22⋊C4—2571no—
243C2×C22⋊C4—2571no—
244C2×C22⋊C4—2571no—
245C2×C22⋊C4C2×D25711yescharacteristic
246C2×C22⋊C4C2×D25711yescharacteristic
247C2×C22⋊C4C2×D25711yescharacteristic
248C2×C42—2571no—
249C2×C42—2571no—
250C2×C42C2×D25711yescharacteristic
251#95—2571no—
252#95—2571no—
253#95—2571no—
254#95—2571no—
255#95—2571no—
256#95—2571no—
257#95—2571no—
258#95—2571no—
259#95C2×D25711yescharacteristic
260#95C2×D25711yescharacteristic
261C24—2571no—
262C24C22×D25711yescharacteristic
263C2×D4—2574no—
264C2×D4—5142no—
265C2×D4—5142no—
266C22×C4—2571no—
267C22×C4—2571no—
268C22×C4—2571no—
269C22×C4—2571no—
270C22×C4—2574no—
271C22×C4—2574no—
272C22×C4—2571no—
273C22×C4—2571no—
274C22×C4—2571no—
275C22×C4—5142no—
276C22×C4—5142no—
277C22×C4—5142no—
278C22×C4—5142no—
279C22×C4—10281no—
280C22×C4—10281no—
281C22×C4—10281no—
282C22×C4—10281no—
283C22×C4—10281no—
284C22×C4—10281no—
285C22×C4C22×D25711yescharacteristic
286C22×C4C22×D25711yescharacteristic
287C22×C4C22×D25711yescharacteristic
288C22×C4C22×D25711yescharacteristic
289C22×C4C22×D25711yescharacteristic
290C22×C4C22×D25711yescharacteristic
291C4⋊C4—2574no—
292C4⋊C4—2574no—
293C4⋊C4—22no—
294C4⋊C4—5142no—
295C4⋊C4—5142no—
296C4⋊C4—5142no—
297C4⋊C4—5142no—
298C4⋊C4—10281no—
299C4⋊C4—10281no—
300C22⋊C4—2574no—
301C22⋊C4—2574no—
302C22⋊C4—22no—
303C22⋊C4—5142no—
304C22⋊C4—5142no—
305C22⋊C4—41no—
306C22⋊C4—10281no—
307C22⋊C4C4×D25714yescomplemented
308C42—5142no—
309C42—41no—
310C42—10281no—
311C23—2572no—
312C23—2572no—
313C23—2572no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—5141no—
321C23—5141no—
322C23—5141no—
323C23—5141no—
324C23C2×C4×D25712yes—
325C23C23×D25711yescharacteristic
326D4—5148no—
327C2×C4—2572no—
328C2×C4—2572no—
329C2×C4—2572no—
330C2×C4—2572no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—5148no—
347C2×C4—5141no—
348C2×C4—5141no—
349C2×C4—5141no—
350C2×C4—5141no—
351C2×C4—5141no—
352C2×C4—5141no—
353C2×C4—5141no—
354C2×C4—5141no—
355C2×C4—5141no—
356C2×C4—5141no—
357C2×C4—5141no—
358C2×C4—5141no—
359C2×C4—5141no—
360C2×C4—5141no—
361C2×C4—5141no—
362C2×C4—5141no—
363C2×C4—5141no—
364C2×C4—10284no—
365C2×C4—10284no—
366C2×C4—10284no—
367C2×C4C2×C4×D25712yes—
368C2×C4C2×C4×D25712yes—
369C2×C4D4×D25712yes—
370C22—2574no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—5142no—
375C22—5142no—
376C22—5142no—
377C22—5142no—
378C22—5142no—
379C22—5142no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22C22×C4×D25711yescharacteristic
385C22C2×D4×D25711yescharacteristic
386C22#41×D25711yescharacteristic
387C22#2255163711yescharacteristic
388C22#2255163811yescharacteristic
389C22#2255163911yescharacteristic
390C22#2255163911yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—5142no—
395C4—5142no—
396C4—5142no—
397C4—5142no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—10281no—
402C4—10281no—
403C4—10281no—
404C2—5144no—
405C2—42no—
406C2C4×D4×D25711yescharacteristic
407C2#3065721611yescharacteristic
408C2#3065721711yescharacteristic
409C2#3065729511yescharacteristic
410C2#3065730711yescharacteristic
411C2#3065732811yescharacteristic
412C2#3065733011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#388×C257C2
socle, cosocleC23×C257C24
prime radicalC22×C22⋊D257C22
hypercenter#388D257
hypocenterC257#2322
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×D257C2semidirect
4. #30656635C2semidirect
5. #30656632C2semidirect
6. #30656631C2semidirect
7. #30656624C2semidirect
8. #30656620C2semidirect
10. #30656612C2semidirect
11. #30656611C2semidirect
12. #30656609C2semidirect
13. #30656599C2semidirect
14. C4×#16861595C2semidirect
15. #388×C257C2semidirect
21. #22551574C4semidirect
75. C2×C4×D257C2×C4semidirect
79. C4⋊Dic257C2×C4semidirect
84. #16861597C2×C4semidirect
86. C22⋊Dic257C2×C4semidirect
89. #16861595C2×C4semidirect
102. C22⋊C4×C257C2×C4semidirect
206. C257#2322semidirect
214. #388D257semidirect
307. C22⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515742
C2×#225515522
C2×C4⋊C4×D2572
#306566352
#306566322
#306566312
#306566242
#306566202
#306566132
#306566122
#306566112
#306566092
#306565992
C4×#168615952
#388×C2572
#2322257