Finite Group Databasebeta

← #45667322

Subgroups — #45667322

Order 32896.

620 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551570C211yesmaximal, characteristic
2C2×#22551557C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D257C211yesmaximal, characteristic
4#30656641C211yesmaximal, characteristic, complemented
5#30656634C211yesmaximal, characteristic, complemented
6#30656632C211yesmaximal, characteristic, complemented
7#30656623C211yesmaximal, characteristic, complemented
8#30656618C211yesmaximal, characteristic, complemented
9#30656608C211yesmaximal, characteristic
10#30656608C211yesmaximal, characteristic
11C4×#16861597C211yesmaximal, characteristic, complemented
12C4×#16861595C211yesmaximal, characteristic
13#30656584C211yesmaximal, characteristic
14#30656584C211yesmaximal, characteristic
15#389×C257C211yesmaximal, characteristic, complemented
16C22×Dic514C2211yescharacteristic
17C22×C4×D257C2211yescharacteristic
18C22×C4×D257C2211yescharacteristic
19C22×C4×D257C2211yescharacteristic
20#22551570C418yescomplemented
21#22551557—24no—
22C2×C4⋊Dic257C2211yescharacteristic
23C2×#16861597C2211yescharacteristic
24C2×#16861597C2211yescharacteristic
25C2×#16861597C2211yescharacteristic
26C2×#16861597C2211yescharacteristic
27C2×#16861597C2211yescharacteristic
28C2×#16861597C2211yescharacteristic
29C4⋊C4×D257—24no—
30C2×#16861595C2211yescharacteristic
31C2×#16861595C2211yescharacteristic
32C2×#16861595C2211yescharacteristic
33C2×#16861595C2211yescharacteristic
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×C4⋊C4×C257C2211yescharacteristic
49C2×C4⋊C4×C257C2211yescharacteristic
50C2×C4⋊C4×C257C2211yescharacteristic
51C2×C42×C257C2211yescharacteristic
52#95×C257C2211yescharacteristic
53#95×C257C2211yescharacteristic
54C23×D257C2311yescharacteristic
55C2×Dic514—22no—
56C2×Dic514—22no—
57C2×Dic514C2×C414yescomplemented
58C22×Dic257C2311yescharacteristic
59C22×Dic257C2311yescharacteristic
60C22×Dic257C2311yescharacteristic
61C22×Dic257C2311yescharacteristic
62C22×Dic257C2311yescharacteristic
63C22×Dic257C2311yescharacteristic
64C22×Dic257C2311yescharacteristic
65C2×C4×D257—22no—
66C2×C4×D257—22no—
67C2×C4×D257—22no—
68C2×C4×D257—22no—
69C2×C4×D257—22no—
70C2×C4×D257—22no—
71C2×C4×D257—41no—
72C2×C4×D257—41no—
73C2×C4×D257C2×C414yes—
74C4⋊Dic257—22no—
75#16861597—22no—
76#16861597—22no—
77#16861597—41no—
78#16861597C2×C414yescomplemented
79#16861597C2×C414yescomplemented
80#16861597D414yes—
81#16861595—22no—
82#16861595C2×C414yes—
83#16861595C2×C414yes—
84#16861595Q814yescomplemented
85C4×Dic257—22no—
86C4×Dic257—22no—
87C4×Dic257—22no—
88C4×Dic257—22no—
89C4×Dic257—41no—
90C22×C4×C257C2311yescharacteristic
91C22×C4×C257C2311yescharacteristic
92C22×C4×C257C2311yescharacteristic
93C22×C4×C257C2311yescharacteristic
94C22×C4×C257C2311yescharacteristic
95C22×C4×C257C2311yescharacteristic
96C22×C4×C257C2311yescharacteristic
97C4⋊C4×C257—22no—
98C4⋊C4×C257—22no—
99C4⋊C4×C257—41no—
100C4⋊C4×C257C2×C414yescomplemented
101C42×C257—41no—
102C22×D257—21no—
103C22×D257—21no—
104C22×D257—21no—
105C22×D257—21no—
106C22×D257C22×C412yes—
107C22×D257C2×Q812yes—
108C22×D257#4112yes—
109Dic514—28no—
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—21no—
124C2×Dic257C22×C412yes—
125C2×Dic257C22×C412yes—
126C2×Dic257C22×C412yes—
127C2×Dic257C2×D412yes—
128C2×Dic257C2×D412yes—
129C2×Dic257C2×Q812yes—
130C2×Dic257#4112yes—
131C4×D257—28no—
132C4×D257—44no—
133C4×D257—44no—
134C23×C257C2411yescharacteristic
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×C257—21no—
150C2×C4×C257—21no—
151C2×C4×C257C22×C412yes—
152C2×C4×C257C22×C412yes—
153C2×C4×C257C22×C412yes—
154C2×C4×C257C2×D412yes—
155C2×C4×C257C2×Q812yes—
156C22×C257C23×C411yescharacteristic
157C22×C257C22×D411yescharacteristic
158C22×C257C22×Q811yescharacteristic
159C22×C257C2×#4111yescharacteristic
160C22×C257C2×#4111yescharacteristic
161C22×C257H(5,𝔽2)11yescharacteristic
162C22×C2572−1+411yescharacteristic
163C2×D257—22no—
164C2×D257—22no—
165C2×D257—22no—
166C2×D257—22no—
167C2×D257—22no—
168C2×D257—22no—
169C2×D257C4×Q814yescomplemented
170C4×C257—22no—
171C4×C257—22no—
172C4×C257—22no—
173C4×C257—22no—
174C4×C257—22no—
175C4×C257—41no—
176C4×C257—41no—
177Dic257—22no—
178Dic257—22no—
179Dic257—22no—
180Dic257—22no—
181Dic257—41no—
182Dic257—41no—
183Dic257C4×D414yes—
184C2×C257C2×C4×D411yescharacteristic
185C2×C257C2×C4×Q811yescharacteristic
186C2×C257#52011yescharacteristic
187C2×C257#54611yescharacteristic
188C2×C257D4×Q811yescharacteristic
189C2×C257#55211yescharacteristic
190C2×C257#55411yescharacteristic
191D257—24no—
192C257#236011yescharacteristic, complemented, Sylow
193#2360—2571nomaximal, Sylow
194C2×C22⋊Q8—2571no—
195C2×C4×Q8—2571no—
196C22×C4⋊C4—2571no—
197#391—2571no—
198#389—2571no—
199#389—2571no—
200#389D25711yescharacteristic, complemented
201#387—2571no—
202#387—2571no—
203#385—2571no—
204#385—2571no—
205C4×C4⋊C4—2571no—
206C4×C22⋊C4—2571no—
207C4×C22⋊C4—2571no—
208C4×C22⋊C4—2571no—
209C22×Q8—2571no—
210C23×C4—2571no—
211C23×C4—2571no—
212C23×C4—2571no—
213C22⋊Q8—2578no—
214C4×Q8—5144no—
215C2×C4⋊C4—2571no—
216C2×C4⋊C4—2571no—
217C2×C4⋊C4—2571no—
218C2×C4⋊C4—2571no—
219C2×C4⋊C4—2571no—
220C2×C4⋊C4—2571no—
221C2×C4⋊C4—2571no—
222C2×C4⋊C4—5144no—
223C2×C4⋊C4C2×D25711yescharacteristic
224C2×C4⋊C4C2×D25711yescharacteristic
225C2×C4⋊C4C2×D25711yescharacteristic
226C2×C4⋊C4C2×D25711yescharacteristic
227C2×C22⋊C4—2571no—
228C2×C22⋊C4—2571no—
229C2×C22⋊C4—2571no—
230C2×C22⋊C4—2571no—
231C2×C42—2571no—
232C2×C42—2571no—
233C2×C42—2571no—
234C2×C42—2571no—
235C2×C42—2571no—
236C2×C42C2×D25711yescharacteristic
237#95—2571no—
238#95—2571no—
239#95—2571no—
240#95—2571no—
241#95—2571no—
242#95—2571no—
243#95—2571no—
244#95—2571no—
245#95C2×D25711yescharacteristic
246#95C2×D25711yescharacteristic
247C24—2571no—
248C2×Q8—2574no—
249C2×Q8—5142no—
250C2×Q8—5142no—
251C22×C4—2571no—
252C22×C4—2571no—
253C22×C4—2571no—
254C22×C4—2571no—
255C22×C4—2571no—
256C22×C4—2574no—
257C22×C4—2571no—
258C22×C4—2571no—
259C22×C4—5142no—
260C22×C4—5142no—
261C22×C4—5142no—
262C22×C4—5142no—
263C22×C4—5142no—
264C22×C4—5142no—
265C22×C4—10281no—
266C22×C4—10281no—
267C22×C4C22×D25711yescharacteristic
268C22×C4C22×D25711yescharacteristic
269C22×C4C22×D25711yescharacteristic
270C22×C4C22×D25711yescharacteristic
271C22×C4C22×D25711yescharacteristic
272C22×C4C22×D25711yescharacteristic
273C22×C4C22×D25711yescharacteristic
274C4⋊C4—2574no—
275C4⋊C4—2574no—
276C4⋊C4—2574no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—5142no—
280C4⋊C4—5142no—
281C4⋊C4—5142no—
282C4⋊C4—41no—
283C4⋊C4—10281no—
284C4⋊C4C4×D25714yescomplemented
285C22⋊C4—2574no—
286C22⋊C4—2574no—
287C22⋊C4—2574no—
288C22⋊C4—5142no—
289C42—5142no—
290C42—5142no—
291C42—5142no—
292C42—5142no—
293C42—41no—
294C42—10281no—
295C23—2572no—
296C23—2572no—
297C23—2572no—
298C23—5141no—
299C23—5141no—
300C23—5141no—
301C23—5141no—
302C23C23×D25711yescharacteristic
303Q8—5148no—
304C2×C4—2572no—
305C2×C4—2572no—
306C2×C4—2572no—
307C2×C4—2572no—
308C2×C4—2572no—
309C2×C4—2572no—
310C2×C4—2572no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
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—5148no—
328C2×C4—5141no—
329C2×C4—5141no—
330C2×C4—5141no—
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—10284no—
343C2×C4—10284no—
344C2×C4C2×C4×D25712yes—
345C2×C4C2×C4×D25712yes—
346C2×C4C2×C4×D25712yes—
347C2×C4D4×D25712yes—
348C2×C4Q8×D25712yes—
349C22—2574no—
350C22—5142no—
351C22—5142no—
352C22—5142no—
353C22—5142no—
354C22—5142no—
355C22—5142no—
356C22C22×C4×D25711yescharacteristic
357C22C2×D4×D25711yescharacteristic
358C22C2×Q8×D25711yescharacteristic
359C22#41×D25711yescharacteristic
360C22#41×D25711yescharacteristic
361C22#2255163711yescharacteristic
362C22#2255164011yescharacteristic
363C4—2574no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—22no—
369C4—5142no—
370C4—5142no—
371C4—5142no—
372C4—5142no—
373C4—41no—
374C4—41no—
375C4—10281no—
376C4—10281no—
377C2—5144no—
378C2C4×D4×D25711yescharacteristic
379C2C4×Q8×D25711yescharacteristic
380C2#3065721811yescharacteristic
381C2#3065737611yescharacteristic
382C2#3065740211yescharacteristic
383C2#3065741911yescharacteristic
384C2#3065744111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#389×C257C2
socle, cosocleC23×C257C24
prime radicalC23×D257C23
hypercenter#389D257
hypocenterC257#2360
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×#22551557C2semidirect
4. #30656641C2semidirect
5. #30656634C2semidirect
6. #30656632C2semidirect
7. #30656623C2semidirect
8. #30656618C2semidirect
11. C4×#16861597C2semidirect
15. #389×C257C2semidirect
20. #22551570C4semidirect
57. C2×Dic514C2×C4semidirect
78. #16861597C2×C4semidirect
79. #16861597C2×C4semidirect
84. #16861595Q8semidirect
100. C4⋊C4×C257C2×C4semidirect
169. C2×D257C4×Q8semidirect
192. C257#2360semidirect
200. #389D257semidirect
284. C4⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515702
C2×#225515572
C2×C4⋊C4×D2572
#306566412
#306566342
#306566322
#306566232
#306566182
#306566082
C4×#168615972
C4×#168615952
#306565842
#389×C2572
#2360257