Finite Group Databasebeta

← #45667339

Subgroups — #45667339

Order 32896.

788 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551565C211yesmaximal, characteristic
2C2×#22551554C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D257C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊D257C211yesmaximal, characteristic, complemented
5C2×C4⋊C4×D257C211yesmaximal, characteristic, complemented
6#30656637C211yesmaximal, characteristic, complemented
7#30656632C211yesmaximal, characteristic, complemented
8#30656630C211yesmaximal, characteristic, complemented
9#30656627C211yesmaximal, characteristic, complemented
10#30656615C211yesmaximal, characteristic, complemented
11#30656613C211yesmaximal, characteristic, complemented
12#30656611C211yesmaximal, characteristic, complemented
13#30656588C211yesmaximal, characteristic, complemented
14#30656584C211yesmaximal, characteristic, complemented
15#385×C257C211yesmaximal, characteristic, complemented
16C22×D1028C2211yescharacteristic
17C22×C22⋊D257C2211yescharacteristic
18C22×C22⋊D257C2211yescharacteristic
19C22×C4×D257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21C22×C4×D257C2211yescharacteristic
22C22×C4×D257C2211yescharacteristic
23#22551565C418yescomplemented
24#22551554—24no—
25C4×C22⋊D257—24no—
26C4×C22⋊D257—24no—
27C2×C4⋊Dic257C2211yescharacteristic
28C2×#16861597C2211yescharacteristic
29C2×#16861597C2211yescharacteristic
30C2×#16861597C2211yescharacteristic
31C2×#16861597C2211yescharacteristic
32C4⋊C4×D257—24no—
33C2×C22⋊Dic257C2211yescharacteristic
34C2×C22⋊Dic257C2211yescharacteristic
35C2×#16861595C2211yescharacteristic
36C2×#16861595C2211yescharacteristic
37C2×#16861595C2211yescharacteristic
38C2×#16861595C2211yescharacteristic
39C2×#16861595C2211yescharacteristic
40C2×#16861595C2211yescharacteristic
41C2×C4×Dic257C2211yescharacteristic
42C2×C4×Dic257C2211yescharacteristic
43C2×C4×Dic257C2211yescharacteristic
44C2×C4×Dic257C2211yescharacteristic
45#22551499C2211yescharacteristic
46#22551499C2211yescharacteristic
47#22551499C2211yescharacteristic
48#22551499C2211yescharacteristic
49C23×C4×C257C2211yescharacteristic
50C2×C4⋊C4×C257C2211yescharacteristic
51C2×C4⋊C4×C257C2211yescharacteristic
52C2×C22⋊C4×C257C2211yescharacteristic
53C2×C22⋊C4×C257C2211yescharacteristic
54#95×C257C2211yescharacteristic
55#95×C257C2211yescharacteristic
56C23×D257C2311yescharacteristic
57C23×D257C2311yescharacteristic
58C2×D1028—22no—
59C2×D1028—22no—
60C2×D1028C2×C414yescomplemented
61C2×C22⋊D257—22no—
62C2×C22⋊D257—22no—
63C2×C22⋊D257—22no—
64C2×C22⋊D257—22no—
65C2×C22⋊D257C2×C414yescomplemented
66C2×C22⋊D257C2×C414yescomplemented
67C22×Dic257C2311yescharacteristic
68C22×Dic257C2311yescharacteristic
69C22×Dic257C2311yescharacteristic
70C22×Dic257C2311yescharacteristic
71C22×Dic257C2311yescharacteristic
72C22×Dic257C2311yescharacteristic
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—22no—
80C2×C4×D257—22no—
81C2×C4×D257—41no—
82C2×C4×D257—41no—
83C2×C4×D257—41no—
84C2×C4×D257C2×C414yescomplemented
85C4⋊Dic257—22no—
86#16861597—22no—
87#16861597—22no—
88#16861597C2×C414yescomplemented
89#16861597D414yescomplemented
90C22⋊Dic257—22no—
91C22⋊Dic257—22no—
92#16861595—22no—
93#16861595—22no—
94#16861595—22no—
95#16861595—41no—
96#16861595C2×C414yescomplemented
97#16861595D414yescomplemented
98C4×Dic257—22no—
99C4×Dic257—22no—
100C4×Dic257—22no—
101C4×Dic257—22no—
102C24×C257C2311yescharacteristic
103C22×C4×C257—22no—
104C22×C4×C257—22no—
105C22×C4×C257—41no—
106C22×C4×C257C2311yescharacteristic
107C22×C4×C257C2311yescharacteristic
108C22×C4×C257C2311yescharacteristic
109C22×C4×C257C2311yescharacteristic
110C22×C4×C257C2311yescharacteristic
111C22×C4×C257C2311yescharacteristic
112C4⋊C4×C257—22no—
113C4⋊C4×C257—22no—
114C22⋊C4×C257—41no—
115C22⋊C4×C257C2×C414yescomplemented
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×D257C2×D412yes—
129C22×D257#4112yes—
130D1028—28no—
131C22⋊D257—28no—
132C22⋊D257—28no—
133C2×Dic257—21no—
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×Dic257C22×C412yes—
146C2×Dic257C22×C412yes—
147C2×Dic257C2×D412yes—
148C2×Dic257C2×D412yes—
149C2×Dic257C2×D412yes—
150C2×Dic257#4112yes—
151C4×D257—28no—
152C4×D257—44no—
153C4×D257—44no—
154C4×D257—44no—
155C23×C257—21no—
156C23×C257—21no—
157C23×C257—21no—
158C23×C257—21no—
159C23×C257—21no—
160C23×C257—21no—
161C23×C257C22×C412yes—
162C23×C257C2411yescharacteristic
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×C257—21no—
172C2×C4×C257—21no—
173C2×C4×C257—21no—
174C2×C4×C257—21no—
175C2×C4×C257—21no—
176C2×C4×C257—21no—
177C2×C4×C257—44no—
178C2×C4×C257C22×C412yes—
179C2×C4×C257C22×C412yes—
180C2×C4×C257C2×D412yes—
181C2×C4×C257C2×D412yes—
182C22×C257—22no—
183C22×C257—22no—
184C22×C257—22no—
185C22×C257—41no—
186C22×C257—41no—
187C22×C257—41no—
188C22×C257—41no—
189C22×C257C23×C411yescharacteristic
190C22×C257C22×D411yescharacteristic
191C22×C257C22×D411yescharacteristic
192C22×C257C2×#4111yescharacteristic
193C22×C257C2×#4111yescharacteristic
194C22×C257H(5,𝔽2)11yescharacteristic
195C22×C2572−1+411yescharacteristic
196C2×D257—22no—
197C2×D257—22no—
198C2×D257—22no—
199C2×D257—22no—
200C2×D257—22no—
201C2×D257—22no—
202C2×D257—22no—
203C2×D257—22no—
204C2×D257—22no—
205C2×D257—41no—
206C2×D257—41no—
207C2×D257—41no—
208C2×D257—41no—
209C2×D257C4×D414yescomplemented
210C4×C257—22no—
211C4×C257—22no—
212C4×C257—22no—
213C4×C257—22no—
214C4×C257—41no—
215C4×C257—41no—
216Dic257—22no—
217Dic257—22no—
218Dic257—22no—
219Dic257—41no—
220Dic257—41no—
221Dic257C4×D414yescomplemented
222C2×C257—42no—
223C2×C257C2×C4×D411yescharacteristic
224C2×C257C2×C4×D411yescharacteristic
225C2×C257#52011yescharacteristic
226C2×C257D4211yescharacteristic
227C2×C257#54711yescharacteristic
228C2×C257#54811yescharacteristic
229C2×C257#55311yescharacteristic
230D257—24no—
231D257—42no—
232C257#236111yescharacteristic, complemented, Sylow
233#2361—2571nomaximal, Sylow
234C2×#121—2571no—
235C2×C4×D4—2571no—
236C2×C4×D4—2571no—
237C2×C4×D4—2571no—
238C22×C4⋊C4—2571no—
239#390—2571no—
240#389—2571no—
241#388—2571no—
242#388—2571no—
243#386—2571no—
244#386—2571no—
245#385—2571no—
246#385D25711yescharacteristic, complemented
247C4×C4⋊C4—2571no—
248C4×C22⋊C4—2571no—
249C22×D4—2571no—
250C22×D4—2571no—
251C22×D4—2571no—
252C23×C4—2571no—
253C23×C4—2571no—
254C23×C4—2571no—
255C23×C4—2571no—
256C23×C4C2×D25711yescharacteristic
257#121—2578no—
258C4×D4—5144no—
259C4×D4—5144no—
260C4×D4—5144no—
261C2×C4⋊C4—2571no—
262C2×C4⋊C4—2571no—
263C2×C4⋊C4—2571no—
264C2×C4⋊C4—2571no—
265C2×C4⋊C4—2571no—
266C2×C4⋊C4—5144no—
267C2×C4⋊C4C2×D25711yescharacteristic
268C2×C4⋊C4C2×D25711yescharacteristic
269C2×C22⋊C4—2571no—
270C2×C22⋊C4—2571no—
271C2×C22⋊C4—2571no—
272C2×C22⋊C4—2571no—
273C2×C22⋊C4—2571no—
274C2×C22⋊C4—2571no—
275C2×C22⋊C4—2571no—
276C2×C22⋊C4—2571no—
277C2×C22⋊C4C2×D25711yescharacteristic
278C2×C22⋊C4C2×D25711yescharacteristic
279C2×C42—2571no—
280C2×C42—2571no—
281C2×C42—2571no—
282C2×C42—2571no—
283#95—2571no—
284#95—2571no—
285#95—2571no—
286#95—2571no—
287#95C2×D25711yescharacteristic
288#95C2×D25711yescharacteristic
289C24—2571no—
290C24—2571no—
291C24C22×D25711yescharacteristic
292C2×D4—2574no—
293C2×D4—2574no—
294C2×D4—2574no—
295C2×D4—5142no—
296C2×D4—5142no—
297C2×D4—5142no—
298C2×D4—5142no—
299C2×D4—5142no—
300C2×D4—5142no—
301C22×C4—2571no—
302C22×C4—2571no—
303C22×C4—2571no—
304C22×C4—2571no—
305C22×C4—2574no—
306C22×C4—2571no—
307C22×C4—2571no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—5142no—
311C22×C4—5142no—
312C22×C4—5142no—
313C22×C4—5142no—
314C22×C4—5142no—
315C22×C4—5142no—
316C22×C4—5142no—
317C22×C4—5142no—
318C22×C4—41no—
319C22×C4—10281no—
320C22×C4—10281no—
321C22×C4—10281no—
322C22×C4C22×D25711yescharacteristic
323C22×C4C22×D25711yescharacteristic
324C22×C4C22×D25711yescharacteristic
325C22×C4C22×D25711yescharacteristic
326C22×C4C22×D25711yescharacteristic
327C22×C4C22×D25711yescharacteristic
328C4⋊C4—2574no—
329C4⋊C4—2574no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—5142no—
333C4⋊C4—5142no—
334C4⋊C4—5142no—
335C22⋊C4—2574no—
336C22⋊C4—2574no—
337C22⋊C4—5142no—
338C22⋊C4—5142no—
339C22⋊C4—5142no—
340C22⋊C4—5142no—
341C22⋊C4—5142no—
342C22⋊C4—41no—
343C22⋊C4—10281no—
344C22⋊C4C4×D25714yescomplemented
345C42—5142no—
346C42—5142no—
347C42—5142no—
348C42—5142no—
349C23—2572no—
350C23—2572no—
351C23—2572no—
352C23—2572no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—5141no—
360C23—5141no—
361C23—5141no—
362C23—5141no—
363C23—5141no—
364C23—5141no—
365C23—5141no—
366C23—5141no—
367C23—5141no—
368C23—5141no—
369C23C2×C4×D25712yes—
370C23C23×D25711yescharacteristic
371D4—5148no—
372D4—5148no—
373D4—5148no—
374C2×C4—2572no—
375C2×C4—2572no—
376C2×C4—2572no—
377C2×C4—2572no—
378C2×C4—2572no—
379C2×C4—2572no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—5148no—
395C2×C4—5141no—
396C2×C4—5141no—
397C2×C4—5141no—
398C2×C4—5141no—
399C2×C4—5141no—
400C2×C4—5141no—
401C2×C4—5141no—
402C2×C4—5141no—
403C2×C4—5141no—
404C2×C4—5141no—
405C2×C4—5141no—
406C2×C4—5141no—
407C2×C4—44no—
408C2×C4—10284no—
409C2×C4—10284no—
410C2×C4—10284no—
411C2×C4C2×C4×D25712yes—
412C2×C4C2×C4×D25712yes—
413C2×C4D4×D25712yes—
414C2×C4D4×D25712yes—
415C22—2574no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—5142no—
420C22—5142no—
421C22—5142no—
422C22—5142no—
423C22—5142no—
424C22—5142no—
425C22—5142no—
426C22—5142no—
427C22—5142no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—10281no—
433C22—10281no—
434C22—10281no—
435C22—10281no—
436C22C22×C4×D25711yescharacteristic
437C22C2×D4×D25711yescharacteristic
438C22C2×D4×D25711yescharacteristic
439C22#41×D25711yescharacteristic
440C22#41×D25711yescharacteristic
441C22#2255163711yescharacteristic
442C22#2255164011yescharacteristic
443C4—2574no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—5142no—
449C4—5142no—
450C4—5142no—
451C4—41no—
452C4—41no—
453C4—10281no—
454C4—10281no—
455C2—5144no—
456C2—42no—
457C2—10282no—
458C2C4×D4×D25711yescharacteristic
459C2C4×D4×D25711yescharacteristic
460C2#3065721811yescharacteristic
461C2#3065736911yescharacteristic
462C2#3065738911yescharacteristic
463C2#3065739711yescharacteristic
464C2#3065743211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#385×C257C2
socle, cosocleC23×C257C24
prime radicalC2×#22551565C2
hypercenter#385D257
hypocenterC257#2361
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#22551554C2semidirect
3. C2×C4×C22⋊D257C2semidirect
4. C2×C4×C22⋊D257C2semidirect
5. C2×C4⋊C4×D257C2semidirect
6. #30656637C2semidirect
7. #30656632C2semidirect
8. #30656630C2semidirect
9. #30656627C2semidirect
10. #30656615C2semidirect
11. #30656613C2semidirect
12. #30656611C2semidirect
13. #30656588C2semidirect
14. #30656584C2semidirect
15. #385×C257C2semidirect
23. #22551565C4semidirect
60. C2×D1028C2×C4semidirect
65. C2×C22⋊D257C2×C4semidirect
66. C2×C22⋊D257C2×C4semidirect
84. C2×C4×D257C2×C4semidirect
88. #16861597C2×C4semidirect
89. #16861597D4semidirect
96. #16861595C2×C4semidirect
97. #16861595D4semidirect
115. C22⋊C4×C257C2×C4semidirect
209. C2×D257C4×D4semidirect
221. Dic257C4×D4semidirect
232. C257#2361semidirect
246. #385D257semidirect
344. C22⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515652
C2×#225515542
C2×C4×C22⋊D2572
C2×C4⋊C4×D2572
#306566372
#306566322
#306566302
#306566272
#306566152
#306566132
#306566112
#306565882
#306565842
#385×C2572
#2361257