Finite Group Databasebeta

← #45667366

Subgroups — #45667366

Order 32896.

716 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551567C211yesmaximal, characteristic
2C2×C4×D1028C211yesmaximal, characteristic, complemented
3C2×#22551554C211yesmaximal, characteristic, complemented
4#30656629C211yesmaximal, characteristic, complemented
5#30656627C211yesmaximal, characteristic, complemented
6#30656623C211yesmaximal, characteristic, complemented
7#30656615C211yesmaximal, characteristic, complemented
8#30656615C211yesmaximal, characteristic, complemented
9#30656615C211yesmaximal, characteristic, complemented
10#30656615C211yesmaximal, characteristic, complemented
11C4×#16861595C211yesmaximal, characteristic, complemented
12#30656584C211yesmaximal, characteristic, complemented
13#30656584C211yesmaximal, characteristic, complemented
14#95×D257C211yesmaximal, characteristic, complemented
15#387×C257C211yesmaximal, characteristic, complemented
16C22×D1028C2211yescharacteristic
17C22×D1028C2211yescharacteristic
18C22×D1028C2211yescharacteristic
19C22×C4×D257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21C22×C4×D257C2211yescharacteristic
22C22×C4×D257C2211yescharacteristic
23C22×C4×D257C2211yescharacteristic
24#22551567C418yescomplemented
25C4×D1028—24no—
26#22551554—24no—
27C2×C4⋊Dic257C2211yescharacteristic
28C2×#16861595C2211yescharacteristic
29C2×#16861595C2211yescharacteristic
30C2×#16861595C2211yescharacteristic
31C2×#16861595C2211yescharacteristic
32C2×#16861595C2211yescharacteristic
33C2×#16861595C2211yescharacteristic
34C2×#16861595C2211yescharacteristic
35C2×#16861595C2211yescharacteristic
36C2×#16861595C2211yescharacteristic
37C2×#16861595C2211yescharacteristic
38C2×C4×Dic257C2211yescharacteristic
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
56C23×D257C2311yescharacteristic
57C2×D1028—22no—
58C2×D1028—22no—
59C2×D1028—22no—
60C2×D1028—22no—
61C2×D1028—41no—
62C2×D1028—41no—
63C2×D1028C2×C414yescomplemented
64C2×D1028C2×C414yescomplemented
65C2×D1028C2×C414yescomplemented
66C22×Dic257C2311yescharacteristic
67C22×Dic257C2311yescharacteristic
68C22×Dic257C2311yescharacteristic
69C22×Dic257C2311yescharacteristic
70C22×Dic257C2311yescharacteristic
71C2×C4×D257—22no—
72C2×C4×D257—22no—
73C2×C4×D257—22no—
74C2×C4×D257—22no—
75C2×C4×D257—22no—
76C2×C4×D257—22no—
77C2×C4×D257—22no—
78C2×C4×D257—22no—
79C2×C4×D257—41no—
80C2×C4×D257—41no—
81C2×C4×D257—41no—
82C2×C4×D257—41no—
83C2×C4×D257—41no—
84C2×C4×D257—41no—
85C2×C4×D257C2×C414yescomplemented
86C4⋊Dic257—22no—
87#16861595—22no—
88#16861595—22no—
89#16861595—22no—
90#16861595—22no—
91#16861595—22no—
92#16861595—41no—
93#16861595—41no—
94#16861595C2×C414yescomplemented
95#16861595C2×C414yescomplemented
96#16861595D414yescomplemented
97C4×Dic257—22no—
98C4×Dic257—41no—
99C4×Dic257—41no—
100C22×C4×C257C2311yescharacteristic
101C22×C4×C257C2311yescharacteristic
102C22×C4×C257C2311yescharacteristic
103C22×C4×C257C2311yescharacteristic
104C22×C4×C257C2311yescharacteristic
105C22×C4×C257C2311yescharacteristic
106C22×C4×C257C2311yescharacteristic
107C4⋊C4×C257—22no—
108C4⋊C4×C257C2×C414yescomplemented
109C42×C257—22no—
110C22×D257—21no—
111C22×D257—21no—
112C22×D257—21no—
113C22×D257—21no—
114C22×D257—21no—
115C22×D257—21no—
116C22×D257—21no—
117C22×D257—21no—
118C22×D257—21no—
119C22×D257—21no—
120C22×D257—21no—
121C22×D257—21no—
122C22×D257—21no—
123C22×D257—21no—
124C22×D257—21no—
125C22×D257—21no—
126C22×D257C22×C412yes—
127C22×D257C22×C412yes—
128C22×D257C22×C412yes—
129C22×D257C2×D412yes—
130C22×D257#4112yes—
131D1028—28no—
132D1028—28no—
133D1028—44no—
134C2×Dic257—21no—
135C2×Dic257—21no—
136C2×Dic257—21no—
137C2×Dic257—21no—
138C2×Dic257—21no—
139C2×Dic257—21no—
140C2×Dic257—21no—
141C2×Dic257—21no—
142C2×Dic257—21no—
143C2×Dic257—21no—
144C2×Dic257—21no—
145C2×Dic257—21no—
146C2×Dic257C22×C412yes—
147C2×Dic257C2×D412yes—
148C2×Dic257#4112yes—
149C4×D257—44no—
150C4×D257—44no—
151C4×D257—44no—
152C4×D257—44no—
153C4×D257—44no—
154C23×C257C2411yescharacteristic
155C2×C4×C257—21no—
156C2×C4×C257—21no—
157C2×C4×C257—21no—
158C2×C4×C257—21no—
159C2×C4×C257—21no—
160C2×C4×C257—21no—
161C2×C4×C257—21no—
162C2×C4×C257—21no—
163C2×C4×C257—21no—
164C2×C4×C257—21no—
165C2×C4×C257—21no—
166C2×C4×C257—21no—
167C2×C4×C257—21no—
168C2×C4×C257—21no—
169C2×C4×C257—21no—
170C2×C4×C257—21no—
171C2×C4×C257C22×C412yes—
172C2×C4×C257C22×C412yes—
173C2×C4×C257C22×C412yes—
174C2×C4×C257C2×D412yes—
175C2×C4×C257#4112yes—
176C22×C257C23×C411yescharacteristic
177C22×C257C22×D411yescharacteristic
178C22×C257C2×#4111yescharacteristic
179C22×C257C2×#4111yescharacteristic
180C22×C257C2×#4111yescharacteristic
181C22×C257H(5,𝔽2)11yescharacteristic
182C22×C257H(5,𝔽2)11yescharacteristic
183C2×D257—22no—
184C2×D257—22no—
185C2×D257—22no—
186C2×D257—22no—
187C2×D257—22no—
188C2×D257—22no—
189C2×D257—22no—
190C2×D257—22no—
191C2×D257—22no—
192C2×D257—22no—
193C2×D257—22no—
194C2×D257—22no—
195C2×D257—41no—
196C2×D257—41no—
197C2×D257—41no—
198C2×D257—41no—
199C2×D257—41no—
200C2×D257—41no—
201C2×D257—41no—
202C2×D257—41no—
203C2×D257C4×D414yes—
204C4×C257—22no—
205C4×C257—22no—
206C4×C257—22no—
207C4×C257—22no—
208C4×C257—22no—
209C4×C257—41no—
210C4×C257—41no—
211Dic257—22no—
212Dic257—22no—
213Dic257—22no—
214Dic257—41no—
215Dic257—41no—
216C2×C257C2×C4×D411yescharacteristic
217C2×C257C4×#4111yescharacteristic
218C2×C257#51811yescharacteristic
219C2×C257#54611yescharacteristic
220C2×C257#54611yescharacteristic
221C2×C257#55311yescharacteristic
222C2×C257#55311yescharacteristic
223D257—24no—
224D257—42no—
225D257—42no—
226C257#236411yescharacteristic, complemented, Sylow
227#2364—2571nomaximal, Sylow
228C2×#121—2571no—
229C2×C4×D4—2571no—
230C2×C4×D4—2571no—
231#388—2571no—
232#388—2571no—
233#387—2571no—
234#387D25711yescharacteristic, complemented
235#386—2571no—
236#386—2571no—
237#386—2571no—
238#386—2571no—
239C4×C22⋊C4—2571no—
240C4×C22⋊C4—2571no—
241C4×C22⋊C4—2571no—
242C2×#95—2571no—
243C22×D4—2571no—
244C22×D4—2571no—
245C22×D4—2571no—
246C23×C4—2571no—
247C23×C4—2571no—
248C23×C4—2571no—
249C23×C4—2571no—
250C23×C4—2571no—
251#121—2578no—
252C4×D4—5144no—
253C4×D4—5144no—
254C2×C4⋊C4—2571no—
255C2×C4⋊C4C2×D25711yescharacteristic
256C2×C4⋊C4C2×D25711yescharacteristic
257C2×C22⋊C4—2571no—
258C2×C22⋊C4—2571no—
259C2×C22⋊C4—2571no—
260C2×C22⋊C4—2571no—
261C2×C22⋊C4—2571no—
262C2×C22⋊C4—2571no—
263C2×C22⋊C4—2571no—
264C2×C22⋊C4—2571no—
265C2×C22⋊C4—2571no—
266C2×C22⋊C4—2571no—
267C2×C42—2571no—
268C2×C42—2571no—
269C2×C42—2571no—
270C2×C42C2×D25711yescharacteristic
271#95—2571no—
272#95—2571no—
273#95—2571no—
274#95—2571no—
275#95—2571no—
276#95—2571no—
277#95C2×D25711yescharacteristic
278#95C2×D25711yescharacteristic
279#95C2×D25711yescharacteristic
280#95C2×D25711yescharacteristic
281C24—2571no—
282C24—2571no—
283C24—2571no—
284C2×D4—2574no—
285C2×D4—2574no—
286C2×D4—2574no—
287C2×D4—5142no—
288C2×D4—5142no—
289C2×D4—5142no—
290C2×D4—5142no—
291C2×D4—10281no—
292C2×D4—10281no—
293C22×C4—2571no—
294C22×C4—2571no—
295C22×C4—2571no—
296C22×C4—2574no—
297C22×C4—2571no—
298C22×C4—2571no—
299C22×C4—5142no—
300C22×C4—5142no—
301C22×C4—5142no—
302C22×C4—5142no—
303C22×C4—5142no—
304C22×C4—5142no—
305C22×C4—5142no—
306C22×C4—5142no—
307C22×C4—10281no—
308C22×C4—10281no—
309C22×C4—10281no—
310C22×C4—10281no—
311C22×C4—10281no—
312C22×C4—10281no—
313C22×C4C22×D25711yescharacteristic
314C22×C4C22×D25711yescharacteristic
315C22×C4C22×D25711yescharacteristic
316C22×C4C22×D25711yescharacteristic
317C22×C4C22×D25711yescharacteristic
318C22×C4C22×D25711yescharacteristic
319C22×C4C22×D25711yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—5142no—
322C4⋊C4C4×D25714yescomplemented
323C22⋊C4—2574no—
324C22⋊C4—2574no—
325C22⋊C4—2574no—
326C22⋊C4—5142no—
327C22⋊C4—5142no—
328C22⋊C4—5142no—
329C22⋊C4—5142no—
330C22⋊C4—5142no—
331C22⋊C4—10281no—
332C22⋊C4—10281no—
333C42—22no—
334C42—5142no—
335C42—10281no—
336C42—10281no—
337C23—2572no—
338C23—2572no—
339C23—2572no—
340C23—2572no—
341C23—2572no—
342C23—5141no—
343C23—5141no—
344C23—5141no—
345C23—5141no—
346C23—5141no—
347C23—5141no—
348C23—5141no—
349C23—5141no—
350C23—5141no—
351C23—5141no—
352C23—5141no—
353C23—5141no—
354C23—5141no—
355C23—5141no—
356C23—5141no—
357C23—5141no—
358C23C23×D25711yescharacteristic
359D4—5148no—
360D4—5148no—
361D4—10284no—
362C2×C4—2572no—
363C2×C4—2572no—
364C2×C4—2572no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—5141no—
382C2×C4—5141no—
383C2×C4—5141no—
384C2×C4—5141no—
385C2×C4—5141no—
386C2×C4—5141no—
387C2×C4—5141no—
388C2×C4—5141no—
389C2×C4—5141no—
390C2×C4—5141no—
391C2×C4—5141no—
392C2×C4—5141no—
393C2×C4—10284no—
394C2×C4—10284no—
395C2×C4—10284no—
396C2×C4—10284no—
397C2×C4—10284no—
398C2×C4C2×C4×D25712yes—
399C2×C4C2×C4×D25712yes—
400C2×C4C2×C4×D25712yes—
401C2×C4D4×D25712yes—
402C2×C4#1686161912yes—
403C22—2574no—
404C22—5142no—
405C22—5142no—
406C22—5142no—
407C22—5142no—
408C22—5142no—
409C22—5142no—
410C22—5142no—
411C22—5142no—
412C22—5142no—
413C22—5142no—
414C22—5142no—
415C22—5142no—
416C22—10281no—
417C22—10281no—
418C22—10281no—
419C22—10281no—
420C22—10281no—
421C22—10281no—
422C22—10281no—
423C22—10281no—
424C22C22×C4×D25711yescharacteristic
425C22C2×D4×D25711yescharacteristic
426C22#41×D25711yescharacteristic
427C22#41×D25711yescharacteristic
428C22C2×#1686161911yescharacteristic
429C22#2255163811yescharacteristic
430C22#2255163811yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—5142no—
437C4—5142no—
438C4—5142no—
439C4—41no—
440C4—41no—
441C4—10281no—
442C4—10281no—
443C2—5144no—
444C2—10282no—
445C2—10282no—
446C2C4×D4×D25711yescharacteristic
447C2C4×#1686161911yescharacteristic
448C2#3065721111yescharacteristic
449C2#3065737811yescharacteristic
450C2#3065737911yescharacteristic
451C2#3065743811yescharacteristic
452C2#3065744011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#387×C257C2
socle, cosocleC23×C257C24
prime radicalC2×#22551567C2
hypercenter#387D257
hypocenterC257#2364
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×D1028C2semidirect
3. C2×#22551554C2semidirect
4. #30656629C2semidirect
5. #30656627C2semidirect
6. #30656623C2semidirect
7. #30656615C2semidirect
8. #30656615C2semidirect
9. #30656615C2semidirect
10. #30656615C2semidirect
11. C4×#16861595C2semidirect
12. #30656584C2semidirect
13. #30656584C2semidirect
14. #95×D257C2semidirect
15. #387×C257C2semidirect
24. #22551567C4semidirect
63. C2×D1028C2×C4semidirect
64. C2×D1028C2×C4semidirect
65. C2×D1028C2×C4semidirect
85. C2×C4×D257C2×C4semidirect
94. #16861595C2×C4semidirect
95. #16861595C2×C4semidirect
96. #16861595D4semidirect
108. C4⋊C4×C257C2×C4semidirect
226. C257#2364semidirect
234. #387D257semidirect
322. C4⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515672
C2×C4×D10282
C2×#225515542
#306566292
#306566272
#306566232
#306566152
C4×#168615952
#306565842
#95×D2572
#387×C2572
#2364257