Finite Group Databasebeta

← #45667320

Subgroups — #45667320

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×#22551569C211yesmaximal, characteristic
2C2×#22551557C211yesmaximal, characteristic, complemented
3C22×#16861597C211yesmaximal, characteristic
4#30656641C211yesmaximal, characteristic, complemented
5#30656633C211yesmaximal, characteristic, complemented
6#30656632C211yesmaximal, characteristic, complemented
7#30656632C211yesmaximal, characteristic, complemented
8#30656622C211yesmaximal, characteristic, complemented
9#30656619C211yesmaximal, characteristic, complemented
10#30656609C211yesmaximal, characteristic
11#30656588C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic257C211yesmaximal, characteristic
13C22⋊C4×Dic257C211yesmaximal, characteristic
14C4×C22⋊Dic257C211yesmaximal, characteristic
15#385×C257C211yesmaximal, characteristic
16C22×Dic514C2211yescharacteristic
17C23×Dic257C2211yescharacteristic
18C23×Dic257C2211yescharacteristic
19#22551569C418yescomplemented
20#22551557—24no—
21C2×C4⋊Dic257C2211yescharacteristic
22C2×#16861597—24no—
23C2×#16861597C2211yescharacteristic
24C2×#16861597C2211yescharacteristic
25C2×#16861597C2211yescharacteristic
26C2×#16861597C2211yescharacteristic
27C2×#16861597C2211yescharacteristic
28C2×#16861597C2211yescharacteristic
29C2×#16861597C2211yescharacteristic
30C2×#16861597C2211yescharacteristic
31C2×C22⋊Dic257C2211yescharacteristic
32C2×C22⋊Dic257C2211yescharacteristic
33C2×C4×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
47C23×C4×C257C2211yescharacteristic
48C2×C4⋊C4×C257C2211yescharacteristic
49C2×C4⋊C4×C257C2211yescharacteristic
50C2×C22⋊C4×C257C2211yescharacteristic
51C2×C22⋊C4×C257C2211yescharacteristic
52#95×C257C2211yescharacteristic
53#95×C257C2211yescharacteristic
54C2×Dic514—22no—
55C2×Dic514—22no—
56C2×Dic514C2×C414yescomplemented
57C22×Dic257—22no—
58C22×Dic257—22no—
59C22×Dic257—22no—
60C22×Dic257—22no—
61C22×Dic257—41no—
62C22×Dic257C2×C414yes—
63C22×Dic257C2311yescharacteristic
64C22×Dic257C2311yescharacteristic
65C22×Dic257C2311yescharacteristic
66C22×Dic257C2311yescharacteristic
67C22×Dic257C2311yescharacteristic
68C22×Dic257C2311yescharacteristic
69C22×Dic257C2311yescharacteristic
70C22×Dic257C2311yescharacteristic
71C4⋊Dic257C2×C414yescomplemented
72#16861597—22no—
73#16861597—22no—
74#16861597—22no—
75#16861597—22no—
76#16861597—41no—
77#16861597C2×C414yescomplemented
78#16861597C2×C414yescomplemented
79#16861597D414yes—
80C22⋊Dic257C2×C414yes—
81C22⋊Dic257Q814yescomplemented
82C4×Dic257—22no—
83C4×Dic257—22no—
84C4×Dic257—22no—
85C4×Dic257—22no—
86C4×Dic257—41no—
87C4×Dic257—41no—
88C24×C257C2311yescharacteristic
89C22×C4×C257—22no—
90C22×C4×C257—22no—
91C22×C4×C257—41no—
92C22×C4×C257C2311yescharacteristic
93C22×C4×C257C2311yescharacteristic
94C22×C4×C257C2311yescharacteristic
95C22×C4×C257C2311yescharacteristic
96C22×C4×C257C2311yescharacteristic
97C22×C4×C257C2311yescharacteristic
98C4⋊C4×C257—22no—
99C4⋊C4×C257—41no—
100C22⋊C4×C257—22no—
101C22⋊C4×C257C2×C414yes—
102Dic514—28no—
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—21no—
114C2×Dic257—21no—
115C2×Dic257—28no—
116C2×Dic257—21no—
117C2×Dic257—21no—
118C2×Dic257—21no—
119C2×Dic257—44no—
120C2×Dic257C22×C412yes—
121C2×Dic257C22×C412yes—
122C2×Dic257C22×C412yes—
123C2×Dic257C22×C412yes—
124C2×Dic257C2×D412yes—
125C2×Dic257C2×D412yes—
126C2×Dic257C2×Q812yes—
127C2×Dic257C2×Q812yes—
128C2×Dic257#4112yes—
129C23×C257—21no—
130C23×C257—21no—
131C23×C257—21no—
132C23×C257—21no—
133C23×C257C22×C412yes—
134C23×C257C2×Q812yes—
135C23×C257#4112yes—
136C23×C257C2411yescharacteristic
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×C257—21no—
152C2×C4×C257—44no—
153C2×C4×C257C22×C412yes—
154C2×C4×C257C22×C412yes—
155C2×C4×C257C2×D412yes—
156C22×C257—22no—
157C22×C257—22no—
158C22×C257—22no—
159C22×C257—22no—
160C22×C257—22no—
161C22×C257—22no—
162C22×C257C4×Q814yescomplemented
163C22×C257C23×C411yescharacteristic
164C22×C257C22×D411yescharacteristic
165C22×C257C22×Q811yescharacteristic
166C22×C257C2×#4111yescharacteristic
167C22×C257C2×#4111yescharacteristic
168C22×C257H(5,𝔽2)11yescharacteristic
169C22×C2572−1+411yescharacteristic
170C4×C257—22no—
171C4×C257—22no—
172C4×C257—22no—
173C4×C257—41no—
174C4×C257—41no—
175C4×C257—41no—
176Dic257—22no—
177Dic257—22no—
178Dic257—22no—
179Dic257—22no—
180Dic257—22no—
181Dic257—22no—
182Dic257—41no—
183Dic257C4×D414yes—
184C2×C257—24no—
185C2×C257C2×C4×D411yescharacteristic
186C2×C257C2×C4×Q811yescharacteristic
187C2×C257#52011yescharacteristic
188C2×C257#54611yescharacteristic
189C2×C257D4×Q811yescharacteristic
190C2×C257#55211yescharacteristic
191C2×C257#55411yescharacteristic
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#389—2571no—
201#387—2571no—
202#387—2571no—
203#385—2571no—
204#385D25711yescharacteristic
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×C4C2×D25711yescharacteristic
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—2571no—
223C2×C4⋊C4—2571no—
224C2×C4⋊C4—5144no—
225C2×C4⋊C4C2×D25711yescharacteristic
226C2×C4⋊C4C2×D25711yescharacteristic
227C2×C22⋊C4—2571no—
228C2×C22⋊C4—2571no—
229C2×C22⋊C4C2×D25711yescharacteristic
230C2×C22⋊C4C2×D25711yescharacteristic
231C2×C42—2571no—
232C2×C42—2571no—
233C2×C42—2571no—
234C2×C42—2571no—
235C2×C42—2571no—
236C2×C42—2571no—
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
247C24C22×D25711yescharacteristic
248C2×Q8—2574no—
249C2×Q8—5142no—
250C2×Q8—5142no—
251C22×C4—2571no—
252C22×C4—2574no—
253C22×C4—2571no—
254C22×C4—2571no—
255C22×C4—2571no—
256C22×C4—2571no—
257C22×C4—2571no—
258C22×C4—2571no—
259C22×C4—2571no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—5142no—
263C22×C4—5142no—
264C22×C4—5142no—
265C22×C4—5142no—
266C22×C4—41no—
267C22×C4—10281no—
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—2574no—
278C4⋊C4—22no—
279C4⋊C4—5142no—
280C4⋊C4—5142no—
281C4⋊C4—5142no—
282C4⋊C4—5142no—
283C4⋊C4—41no—
284C4⋊C4—10281no—
285C22⋊C4—2574no—
286C22⋊C4—2574no—
287C22⋊C4—22no—
288C22⋊C4C4×D25714yes—
289C42—5142no—
290C42—5142no—
291C42—5142no—
292C42—5142no—
293C42—10281no—
294C42—10281no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23C2×C4×D25712yes—
300C23Q8×D25712yes—
301C23#1686161712yes—
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—2572no—
312C2×C4—2572no—
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—21no—
328C2×C4—5148no—
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—5141no—
343C2×C4—5141no—
344C2×C4—44no—
345C2×C4—10284no—
346C2×C4C2×C4×D25712yes—
347C2×C4C2×C4×D25712yes—
348C2×C4D4×D25712yes—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22#2255155714yescomplemented
356C22C22×C4×D25711yescharacteristic
357C22C2×D4×D25711yescharacteristic
358C22C2×Q8×D25711yescharacteristic
359C22#41×D25711yescharacteristic
360C22C2×#1686161711yescharacteristic
361C22#2255163711yescharacteristic
362C22#2255163911yescharacteristic
363C4—2574no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—5142no—
368C4—5142no—
369C4—5142no—
370C4—5142no—
371C4—5142no—
372C4—5142no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—10281no—
377C2—24no—
378C2C4×D4×D25711yescharacteristic
379C2C2×#2255155711yescharacteristic
380C2#3065721611yescharacteristic
381C2#3065737411yescharacteristic
382C2#3065740011yescharacteristic
383C2#3065742411yescharacteristic
384C2#3065744111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#385×C257C2
socle, cosocleC23×C257C24
prime radicalC24×C257C23
hypercenter#385D257
hypocenterC257#2360
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#22551557C2semidirect
4. #30656641C2semidirect
5. #30656633C2semidirect
6. #30656632C2semidirect
7. #30656632C2semidirect
8. #30656622C2semidirect
9. #30656619C2semidirect
11. #30656588C2semidirect
19. #22551569C4semidirect
56. C2×Dic514C2×C4semidirect
71. C4⋊Dic257C2×C4semidirect
77. #16861597C2×C4semidirect
78. #16861597C2×C4semidirect
81. C22⋊Dic257Q8semidirect
162. C22×C257C4×Q8semidirect
192. C257#2360semidirect
355. C22#22551557semidirect

Maximal subgroups

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

subgroupindex
C2×#225515692
C2×#225515572
C22×#168615972
#306566412
#306566332
#306566322
#306566222
#306566192
#306566092
#306565882
C22⋊C4×Dic2572
C4×C22⋊Dic2572
#385×C2572
#2360257