Finite Group Databasebeta

← #45666993

Subgroups — #45666993

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×#22551577C211yesmaximal, characteristic
2C2×C4×D1028C211yesmaximal, characteristic
3C2×C4⋊C4×D257C211yesmaximal, characteristic, complemented
4#30656636C211yesmaximal, characteristic, complemented
5#30656634C211yesmaximal, characteristic, complemented
6#30656629C211yesmaximal, characteristic, complemented
7#30656626C211yesmaximal, characteristic, complemented
8#30656621C211yesmaximal, characteristic, complemented
9#30656615C211yesmaximal, characteristic
10#30656611C211yesmaximal, characteristic, complemented
11#30656611C211yesmaximal, characteristic, complemented
12#30656608C211yesmaximal, characteristic, complemented
13#30656599C211yesmaximal, characteristic, complemented
14#30656584C211yesmaximal, characteristic, complemented
15#387×C257C211yesmaximal, characteristic, complemented
16C22×D1028C2211yescharacteristic
17C22×C4×D257C2211yescharacteristic
18C22×C4×D257C2211yescharacteristic
19C22×C4×D257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21#22551577C418yescomplemented
22C4×D1028—24no—
23C2×C4⋊Dic257C2211yescharacteristic
24C2×C4⋊Dic257C2211yescharacteristic
25C2×C4⋊Dic257C2211yescharacteristic
26C2×#16861597C2211yescharacteristic
27C2×#16861597C2211yescharacteristic
28C2×#16861597C2211yescharacteristic
29C2×#16861597C2211yescharacteristic
30C4⋊C4×D257—24no—
31C2×#16861595C2211yescharacteristic
32C2×#16861595C2211yescharacteristic
33C2×#16861595C2211yescharacteristic
34C2×#16861595C2211yescharacteristic
35C2×#16861595C2211yescharacteristic
36C2×#16861595C2211yescharacteristic
37C2×#16861595C2211yescharacteristic
38C2×#16861595C2211yescharacteristic
39C2×C4×Dic257C2211yescharacteristic
40C2×C4×Dic257C2211yescharacteristic
41#22551499C2211yescharacteristic
42#22551499C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45#22551499C2211yescharacteristic
46#22551499C2211yescharacteristic
47C2×C4⋊C4×C257C2211yescharacteristic
48C2×C4⋊C4×C257C2211yescharacteristic
49C2×C42×C257C2211yescharacteristic
50#95×C257C2211yescharacteristic
51#95×C257C2211yescharacteristic
52#95×C257C2211yescharacteristic
53#95×C257C2211yescharacteristic
54C23×D257C2311yescharacteristic
55C23×D257C2311yescharacteristic
56C2×D1028—22no—
57C2×D1028—22no—
58C2×D1028C2×C414yes—
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—41no—
70C2×C4×D257—41no—
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—22no—
79C4⋊Dic257—41no—
80#16861597—22no—
81#16861597—22no—
82#16861597—41no—
83#16861597C2×C414yescomplemented
84#16861595—22no—
85#16861595—22no—
86#16861595—22no—
87#16861595—41no—
88#16861595—41no—
89#16861595C2×C414yescomplemented
90#16861595C2×C414yescomplemented
91#16861595C2×C414yescomplemented
92C4×Dic257—41no—
93C4×Dic257—41no—
94C22×C4×C257C2311yescharacteristic
95C22×C4×C257C2311yescharacteristic
96C22×C4×C257C2311yescharacteristic
97C22×C4×C257C2311yescharacteristic
98C22×C4×C257C2311yescharacteristic
99C22×C4×C257C2311yescharacteristic
100C22×C4×C257C2311yescharacteristic
101C4⋊C4×C257—22no—
102C4⋊C4×C257C2×C414yescomplemented
103C42×C257—22no—
104C22×D257—21no—
105C22×D257—21no—
106C22×D257—21no—
107C22×D257—21no—
108C22×D257—21no—
109C22×D257—21no—
110C22×D257—21no—
111C22×D257—21no—
112C22×D257—21no—
113C22×D257—21no—
114C22×D257C22×C412yes—
115C22×D257C22×C412yes—
116C22×D257C2×D412yes—
117C22×D257#4112yes—
118D1028—28no—
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—21no—
130C2×Dic257—21no—
131C2×Dic257—21no—
132C2×Dic257—21no—
133C2×Dic257—21no—
134C2×Dic257C22×C412yes—
135C2×Dic257C22×C412yes—
136C2×Dic257C2×D412yes—
137C4×D257—28no—
138C4×D257—44no—
139C4×D257—44no—
140C4×D257—44no—
141C23×C257C2411yescharacteristic
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×C257—21no—
155C2×C4×C257—21no—
156C2×C4×C257—21no—
157C2×C4×C257—21no—
158C2×C4×C257—21no—
159C2×C4×C257C22×C412yes—
160C2×C4×C257C22×C412yes—
161C2×C4×C257C22×C412yes—
162C2×C4×C257C2×D412yes—
163C22×C257C23×C411yescharacteristic
164C22×C257C22×D411yescharacteristic
165C22×C257C2×#4111yescharacteristic
166C22×C257H(5,𝔽2)11yescharacteristic
167C22×C257H(5,𝔽2)11yescharacteristic
168C22×C2572−1+411yescharacteristic
169C22×C2572−1+411yescharacteristic
170C2×D257—22no—
171C2×D257—22no—
172C2×D257—22no—
173C2×D257—22no—
174C2×D257—22no—
175C2×D257—22no—
176C2×D257—22no—
177C2×D257—22no—
178C2×D257—22no—
179C2×D257—41no—
180C2×D257—41no—
181C2×D257—41no—
182C2×D257—41no—
183C2×D257C4×D414yes—
184C4×C257—22no—
185C4×C257—22no—
186C4×C257—22no—
187C4×C257—22no—
188C4×C257—41no—
189C4×C257—41no—
190C4×C257—41no—
191Dic257—22no—
192Dic257—22no—
193Dic257—22no—
194Dic257—41no—
195Dic257—41no—
196Dic257—41no—
197C2×C257C2×C4×D411yescharacteristic
198C2×C257#52011yescharacteristic
199C2×C257#52011yescharacteristic
200C2×C257#53411yescharacteristic
201C2×C257#53611yescharacteristic
202C2×C257#53911yescharacteristic
203C2×C257#53911yescharacteristic
204D257—24no—
205D257—42no—
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#388—2571no—
215#387—2571no—
216#387D25711yescharacteristic, complemented
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—5144no—
238C2×C4⋊C4C2×D25711yescharacteristic
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⋊C4—2571no—
246C2×C22⋊C4—2571no—
247C2×C22⋊C4—2571no—
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#95C2×D25711yescharacteristic
258#95C2×D25711yescharacteristic
259#95C2×D25711yescharacteristic
260#95C2×D25711yescharacteristic
261C24—2571no—
262C24—2571no—
263C2×D4—2574no—
264C2×D4—5142no—
265C2×D4—5142no—
266C22×C4—2571no—
267C22×C4—2571no—
268C22×C4—2571no—
269C22×C4—2574no—
270C22×C4—2574no—
271C22×C4—2571no—
272C22×C4—2571no—
273C22×C4—2571no—
274C22×C4—5142no—
275C22×C4—5142no—
276C22×C4—5142no—
277C22×C4—5142no—
278C22×C4—10281no—
279C22×C4—10281no—
280C22×C4—10281no—
281C22×C4—10281no—
282C22×C4—10281no—
283C22×C4—10281no—
284C22×C4C22×D25711yescharacteristic
285C22×C4C22×D25711yescharacteristic
286C22×C4C22×D25711yescharacteristic
287C22×C4C22×D25711yescharacteristic
288C22×C4C22×D25711yescharacteristic
289C22×C4C22×D25711yescharacteristic
290C22×C4C22×D25711yescharacteristic
291C4⋊C4—2574no—
292C4⋊C4—22no—
293C4⋊C4—5142no—
294C4⋊C4—5142no—
295C4⋊C4—5142no—
296C4⋊C4—5142no—
297C4⋊C4—10281no—
298C4⋊C4—10281no—
299C4⋊C4C4×D25714yescomplemented
300C22⋊C4—2574no—
301C22⋊C4—2574no—
302C22⋊C4—2574no—
303C22⋊C4—5142no—
304C22⋊C4—5142no—
305C22⋊C4—5142no—
306C22⋊C4—10281no—
307C22⋊C4—10281no—
308C42—22no—
309C42—10281no—
310C42—10281no—
311C23—2572no—
312C23—2572no—
313C23—2572no—
314C23—2572no—
315C23—5141no—
316C23—5141no—
317C23—5141no—
318C23—5141no—
319C23—5141no—
320C23—5141no—
321C23—5141no—
322C23—5141no—
323C23—5141no—
324C23—5141no—
325C23C23×D25711yescharacteristic
326D4—5148no—
327C2×C4—2572no—
328C2×C4—2572no—
329C2×C4—2572no—
330C2×C4—21no—
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—21no—
347C2×C4—5148no—
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—10284no—
364C2×C4—10284no—
365C2×C4—10284no—
366C2×C4C2×C4×D25712yes—
367C2×C4C2×C4×D25712yes—
368C2×C4C2×C4×D25712yes—
369C2×C4D4×D25712yes—
370C22—2574no—
371C22—5142no—
372C22—5142no—
373C22—5142no—
374C22—5142no—
375C22—5142no—
376C22—5142no—
377C22—5142no—
378C22—5142no—
379C22—5142no—
380C22—10281no—
381C22—10281no—
382C22—10281no—
383C22—10281no—
384C22C22×C4×D25711yescharacteristic
385C22C2×D4×D25711yescharacteristic
386C22#41×D25711yescharacteristic
387C22#2255163811yescharacteristic
388C22#2255163811yescharacteristic
389C22#2255163911yescharacteristic
390C22#2255164011yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—5142no—
396C4—5142no—
397C4—5142no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—10281no—
402C4—10281no—
403C4—10281no—
404C2—5144no—
405C2—10282no—
406C2C4×D4×D25711yescharacteristic
407C2#3065721711yescharacteristic
408C2#3065721911yescharacteristic
409C2#3065729811yescharacteristic
410C2#3065731011yescharacteristic
411C2#3065732811yescharacteristic
412C2#3065733511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#387×C257C2
socle, cosocleC23×C257C24
prime radicalC22×D1028C22
hypercenter#387D257
hypocenterC257#2322
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×D257C2semidirect
4. #30656636C2semidirect
5. #30656634C2semidirect
6. #30656629C2semidirect
7. #30656626C2semidirect
8. #30656621C2semidirect
10. #30656611C2semidirect
11. #30656611C2semidirect
12. #30656608C2semidirect
13. #30656599C2semidirect
14. #30656584C2semidirect
15. #387×C257C2semidirect
21. #22551577C4semidirect
75. C2×C4×D257C2×C4semidirect
83. #16861597C2×C4semidirect
89. #16861595C2×C4semidirect
90. #16861595C2×C4semidirect
91. #16861595C2×C4semidirect
102. C4⋊C4×C257C2×C4semidirect
206. C257#2322semidirect
216. #387D257semidirect
299. C4⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515772
C2×C4×D10282
C2×C4⋊C4×D2572
#306566362
#306566342
#306566292
#306566262
#306566212
#306566152
#306566112
#306566082
#306565992
#306565842
#387×C2572
#2322257