Finite Group Databasebeta

← #45666995

Subgroups — #45666995

Order 32896.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551569C211yesmaximal, characteristic
2C2×#22551557C211yesmaximal, characteristic, complemented
3C2×#22551548C211yesmaximal, characteristic
4#30656641C211yesmaximal, characteristic, complemented
5#30656635C211yesmaximal, characteristic, complemented
6#30656634C211yesmaximal, characteristic, complemented
7#30656632C211yesmaximal, characteristic, complemented
8#30656631C211yesmaximal, characteristic
9#30656631C211yesmaximal, characteristic
10#30656630C211yesmaximal, characteristic
11#30656620C211yesmaximal, characteristic, complemented
12#30656619C211yesmaximal, characteristic, complemented
13#30656595C211yesmaximal, characteristic
14C4×#16861597C211yesmaximal, characteristic, complemented
15#388×C257C211yesmaximal, characteristic
16C22×Dic514C2211yescharacteristic
17C23×Dic257C2211yescharacteristic
18#22551569C418yescomplemented
19#22551557—24no—
20#22551548—24no—
21C2×C4⋊Dic257C2211yescharacteristic
22C2×C4⋊Dic257C2211yescharacteristic
23C2×#16861597C2211yescharacteristic
24C2×#16861597C2211yescharacteristic
25C2×#16861597C2211yescharacteristic
26C2×#16861597C2211yescharacteristic
27C2×#16861597C2211yescharacteristic
28C2×#16861597C2211yescharacteristic
29C2×#16861597C2211yescharacteristic
30C2×#16861597C2211yescharacteristic
31C2×C22⋊Dic257C2211yescharacteristic
32C2×C22⋊Dic257C2211yescharacteristic
33C2×C22⋊Dic257C2211yescharacteristic
34C2×C4×Dic257C2211yescharacteristic
35C2×C4×Dic257C2211yescharacteristic
36C2×C4×Dic257C2211yescharacteristic
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
54C2×Dic514—22no—
55C2×Dic514—22no—
56C2×Dic514C2×C414yes—
57C22×Dic257—22no—
58C22×Dic257—22no—
59C22×Dic257C2×C414yes—
60C22×Dic257C2311yescharacteristic
61C22×Dic257C2311yescharacteristic
62C22×Dic257C2311yescharacteristic
63C22×Dic257C2311yescharacteristic
64C22×Dic257C2311yescharacteristic
65C22×Dic257C2311yescharacteristic
66C22×Dic257C2311yescharacteristic
67C22×Dic257C2311yescharacteristic
68C4⋊Dic257—41no—
69C4⋊Dic257C2×C414yes—
70#16861597—22no—
71#16861597—22no—
72#16861597—22no—
73#16861597—22no—
74#16861597—22no—
75#16861597—41no—
76#16861597C2×C414yes—
77#16861597C2×C414yes—
78C22⋊Dic257—22no—
79C22⋊Dic257—41no—
80C22⋊Dic257C2×C414yes—
81C4×Dic257—22no—
82C4×Dic257—22no—
83C4×Dic257—22no—
84C4×Dic257—41no—
85C4×Dic257D414yes—
86C24×C257C2311yescharacteristic
87C22×C4×C257C2311yescharacteristic
88C22×C4×C257C2311yescharacteristic
89C22×C4×C257C2311yescharacteristic
90C22×C4×C257C2311yescharacteristic
91C22×C4×C257C2311yescharacteristic
92C22×C4×C257C2311yescharacteristic
93C4⋊C4×C257—22no—
94C22⋊C4×C257—22no—
95C22⋊C4×C257—41no—
96C22⋊C4×C257C2×C414yes—
97C42×C257—41no—
98Dic514—28no—
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—28no—
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×Dic257C22×C412yes—
118C2×Dic257C22×C412yes—
119C2×Dic257C22×C412yes—
120C2×Dic257C22×C412yes—
121C2×Dic257C2×D412yes—
122C2×Dic257C2×D412yes—
123C2×Dic257#4112yes—
124C23×C257—21no—
125C23×C257—21no—
126C23×C257—21no—
127C23×C257—21no—
128C23×C257—21no—
129C23×C257—21no—
130C23×C257C22×C412yes—
131C23×C257C2411yescharacteristic
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×C257—21no—
143C2×C4×C257—21no—
144C2×C4×C257—21no—
145C2×C4×C257—21no—
146C2×C4×C257—21no—
147C2×C4×C257C22×C412yes—
148C2×C4×C257C22×C412yes—
149C2×C4×C257C2×D412yes—
150C22×C257—22no—
151C22×C257—22no—
152C22×C257—22no—
153C22×C257—41no—
154C22×C257—41no—
155C22×C257—41no—
156C22×C257—41no—
157C22×C257C23×C411yescharacteristic
158C22×C257C22×D411yescharacteristic
159C22×C257C2×#4111yescharacteristic
160C22×C257H(5,𝔽2)11yescharacteristic
161C22×C2572−1+411yescharacteristic
162C22×C2572−1+411yescharacteristic
163C22×C2572−1+411yescharacteristic
164C4×C257—22no—
165C4×C257—22no—
166C4×C257—22no—
167C4×C257—41no—
168C4×C257—41no—
169C4×C257—41no—
170Dic257—22no—
171Dic257—22no—
172Dic257—22no—
173Dic257—22no—
174Dic257—41no—
175Dic257—41no—
176Dic257—41no—
177Dic257C4×D414yes—
178C2×C257—42no—
179C2×C257C2×C4×D411yescharacteristic
180C2×C257#51911yescharacteristic
181C2×C257#52011yescharacteristic
182C2×C257#53611yescharacteristic
183C2×C257#53711yescharacteristic
184C2×C257#54111yescharacteristic
185C2×C257#54211yescharacteristic
186C257#232311yescharacteristic, complemented, Sylow
187#2323—2571nomaximal, Sylow
188C2×C22⋊Q8—2571no—
189C2×C4×Q8—2571no—
190C2×#117—2571no—
191#391—2571no—
192#389—2571no—
193#389—2571no—
194#389—2571no—
195#388—2571no—
196#388—2571no—
197#388—2571no—
198#388D25711yescharacteristic
199#387—2571no—
200#387—2571no—
201#380—2571no—
202C4×C4⋊C4—2571no—
203C22×Q8—2571no—
204C23×C4—2571no—
205C22⋊Q8—2578no—
206C4×Q8—5144no—
207#117—5144no—
208C2×C4⋊C4—2571no—
209C2×C4⋊C4—2571no—
210C2×C4⋊C4—2571no—
211C2×C4⋊C4—2571no—
212C2×C4⋊C4—2571no—
213C2×C4⋊C4—2571no—
214C2×C4⋊C4—2571no—
215C2×C4⋊C4—2571no—
216C2×C4⋊C4—2571no—
217C2×C4⋊C4—2571no—
218C2×C4⋊C4C2×D25711yescharacteristic
219C2×C22⋊C4—2571no—
220C2×C22⋊C4—2571no—
221C2×C22⋊C4—2571no—
222C2×C22⋊C4C2×D25711yescharacteristic
223C2×C22⋊C4C2×D25711yescharacteristic
224C2×C22⋊C4C2×D25711yescharacteristic
225C2×C42—2571no—
226C2×C42—2571no—
227C2×C42—2571no—
228C2×C42—2571no—
229C2×C42—2571no—
230C2×C42C2×D25711yescharacteristic
231#95—2571no—
232#95—2571no—
233#95—2571no—
234#95—2571no—
235#95—2571no—
236#95—2571no—
237#95—2571no—
238#95—2571no—
239#95C2×D25711yescharacteristic
240#95C2×D25711yescharacteristic
241C24C22×D25711yescharacteristic
242C2×Q8—2574no—
243C2×Q8—5142no—
244C2×Q8—5142no—
245C22×C4—2571no—
246C22×C4—2574no—
247C22×C4—2571no—
248C22×C4—2571no—
249C22×C4—2571no—
250C22×C4—2571no—
251C22×C4—2571no—
252C22×C4—2571no—
253C22×C4—2571no—
254C22×C4—5142no—
255C22×C4—5142no—
256C22×C4C22×D25711yescharacteristic
257C22×C4C22×D25711yescharacteristic
258C22×C4C22×D25711yescharacteristic
259C22×C4C22×D25711yescharacteristic
260C22×C4C22×D25711yescharacteristic
261C22×C4C22×D25711yescharacteristic
262C4⋊C4—2574no—
263C4⋊C4—2574no—
264C4⋊C4—2574no—
265C4⋊C4—22no—
266C4⋊C4—5142no—
267C4⋊C4—5142no—
268C4⋊C4—5142no—
269C4⋊C4—5142no—
270C4⋊C4—5142no—
271C4⋊C4—10281no—
272C4⋊C4—10281no—
273C22⋊C4—2574no—
274C22⋊C4—22no—
275C22⋊C4—5142no—
276C22⋊C4—41no—
277C22⋊C4—10281no—
278C22⋊C4C4×D25714yes—
279C42—2574no—
280C42—5142no—
281C42—5142no—
282C42—5142no—
283C42—41no—
284C42—10281no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23C2×C4×D25712yes—
292C23C23×D25711yescharacteristic
293Q8—5148no—
294C2×C4—2572no—
295C2×C4—2572no—
296C2×C4—2572no—
297C2×C4—2572no—
298C2×C4—2572no—
299C2×C4—2572no—
300C2×C4—2572no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—5148no—
317C2×C4—5141no—
318C2×C4—5141no—
319C2×C4—5141no—
320C2×C4—5141no—
321C2×C4—5141no—
322C2×C4—5141no—
323C2×C4—5141no—
324C2×C4—5141no—
325C2×C4—5141no—
326C2×C4—5141no—
327C2×C4—5141no—
328C2×C4—5141no—
329C2×C4—5141no—
330C2×C4—5141no—
331C2×C4—5141no—
332C2×C4—5141no—
333C2×C4—5141no—
334C2×C4C2×C4×D25712yes—
335C2×C4C2×C4×D25712yes—
336C2×C4D4×D25712yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22C22×C4×D25711yescharacteristic
345C22C2×D4×D25711yescharacteristic
346C22#41×D25711yescharacteristic
347C22#2255163711yescharacteristic
348C22#2255163911yescharacteristic
349C22#2255163911yescharacteristic
350C22#2255163911yescharacteristic
351C4—2574no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—5142no—
356C4—5142no—
357C4—5142no—
358C4—5142no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—10281no—
363C4—10281no—
364C4—10281no—
365C2—42no—
366C2C4×D4×D25711yescharacteristic
367C2#3065721411yescharacteristic
368C2#3065721611yescharacteristic
369C2#3065730711yescharacteristic
370C2#3065731411yescharacteristic
371C2#3065734411yescharacteristic
372C2#3065735511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#388×C257C2
socle, cosocleC23×C257C24
prime radicalC24×C257C23
hypercenter#388D257
hypocenterC257#2323
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#22551557C2semidirect
4. #30656641C2semidirect
5. #30656635C2semidirect
6. #30656634C2semidirect
7. #30656632C2semidirect
11. #30656620C2semidirect
12. #30656619C2semidirect
14. C4×#16861597C2semidirect
18. #22551569C4semidirect
186. C257#2323semidirect

Maximal subgroups

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

subgroupindex
C2×#225515692
C2×#225515572
C2×#225515482
#306566412
#306566352
#306566342
#306566322
#306566312
#306566302
#306566202
#306566192
#306565952
C4×#168615972
#388×C2572
#2323257