Finite Group Databasebeta

← #45667331

Subgroups — #45667331

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×#22551564C211yesmaximal, characteristic
2C2×D4×Dic257C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D257C211yesmaximal, characteristic, complemented
4C2×#22551552C211yesmaximal, characteristic, complemented
5C22×#16861597C211yesmaximal, characteristic, complemented
6#30656637C211yesmaximal, characteristic, complemented
7#30656634C211yesmaximal, characteristic, complemented
8#30656631C211yesmaximal, characteristic, complemented
9#30656626C211yesmaximal, characteristic, complemented
10#30656614C211yesmaximal, characteristic, complemented
11#30656611C211yesmaximal, characteristic, complemented
12#30656609C211yesmaximal, characteristic, complemented
13#30656588C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic257C211yesmaximal, characteristic, complemented
15#386×C257C211yesmaximal, characteristic, complemented
16C22×C22⋊D257C2211yescharacteristic
17C22×C22⋊D257C2211yescharacteristic
18C23×Dic257C2211yescharacteristic
19C23×Dic257C2211yescharacteristic
20C23×Dic257C2211yescharacteristic
21C22×C4×D257C2211yescharacteristic
22#22551564C418yescomplemented
23D4×Dic257—24no—
24C4×C22⋊D257—24no—
25#22551552—24no—
26C2×C4⋊Dic257C2211yescharacteristic
27C2×#16861597—24no—
28C2×#16861597C2211yescharacteristic
29C2×#16861597C2211yescharacteristic
30C2×#16861597C2211yescharacteristic
31C2×#16861597C2211yescharacteristic
32C2×#16861597C2211yescharacteristic
33C2×#16861597C2211yescharacteristic
34C2×C22⋊Dic257C2211yescharacteristic
35C2×C22⋊Dic257C2211yescharacteristic
36C2×C22⋊Dic257C2211yescharacteristic
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
49C22×D4×C257C2211yescharacteristic
50C23×C4×C257C2211yescharacteristic
51C2×C22⋊C4×C257C2211yescharacteristic
52C2×C22⋊C4×C257C2211yescharacteristic
53C2×C22⋊C4×C257C2211yescharacteristic
54#95×C257C2211yescharacteristic
55#95×C257C2211yescharacteristic
56C23×D257C2311yescharacteristic
57C2×C22⋊D257—22no—
58C2×C22⋊D257—22no—
59C2×C22⋊D257—22no—
60C2×C22⋊D257—22no—
61C2×C22⋊D257C2×C414yescomplemented
62C2×C22⋊D257C2×C414yescomplemented
63C22×Dic257—22no—
64C22×Dic257—22no—
65C22×Dic257—22no—
66C22×Dic257—22no—
67C22×Dic257—22no—
68C22×Dic257—22no—
69C22×Dic257—41no—
70C22×Dic257—41no—
71C22×Dic257C2×C414yescomplemented
72C22×Dic257C2311yescharacteristic
73C22×Dic257C2311yescharacteristic
74C22×Dic257C2311yescharacteristic
75C22×Dic257C2311yescharacteristic
76C22×Dic257C2311yescharacteristic
77C22×Dic257C2311yescharacteristic
78C22×Dic257C2311yescharacteristic
79C2×C4×D257—22no—
80C2×C4×D257—22no—
81C2×C4×D257—41no—
82C4⋊Dic257—22no—
83#16861597—22no—
84#16861597—22no—
85#16861597—22no—
86#16861597—22no—
87#16861597C2×C414yescomplemented
88#16861597D414yescomplemented
89C22⋊Dic257—22no—
90C22⋊Dic257C2×C414yescomplemented
91C22⋊Dic257D414yescomplemented
92#16861595—22no—
93#16861595—22no—
94#16861595—41no—
95#16861595C2×C414yescomplemented
96C4×Dic257—22no—
97C4×Dic257—22no—
98C4×Dic257—22no—
99C4×Dic257—22no—
100C24×C257C2311yescharacteristic
101C24×C257C2311yescharacteristic
102C2×D4×C257—22no—
103C2×D4×C257—22no—
104C2×D4×C257C2×C414yescomplemented
105C22×C4×C257—22no—
106C22×C4×C257—22no—
107C22×C4×C257—41no—
108C22×C4×C257C2311yescharacteristic
109C22×C4×C257C2311yescharacteristic
110C22×C4×C257C2311yescharacteristic
111C22×C4×C257C2311yescharacteristic
112C22×C4×C257C2311yescharacteristic
113C22⋊C4×C257—22no—
114C22⋊C4×C257—22no—
115C22⋊C4×C257—41no—
116C22×D257—21no—
117C22×D257—21no—
118C22×D257—21no—
119C22×D257—21no—
120C22×D257—21no—
121C22×D257—21no—
122C22×D257C22×C412yes—
123C22⋊D257—28no—
124C22⋊D257—28no—
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×Dic257—21no—
135C2×Dic257—28no—
136C2×Dic257—21no—
137C2×Dic257—21no—
138C2×Dic257—21no—
139C2×Dic257—44no—
140C2×Dic257—44no—
141C2×Dic257C22×C412yes—
142C2×Dic257C22×C412yes—
143C2×Dic257C22×C412yes—
144C2×Dic257C2×D412yes—
145C2×Dic257C2×D412yes—
146C2×Dic257C2×D412yes—
147C2×Dic257C2×D412yes—
148C2×Dic257#4112yes—
149C4×D257—44no—
150C23×C257—21no—
151C23×C257—21no—
152C23×C257—21no—
153C23×C257—21no—
154C23×C257—21no—
155C23×C257—21no—
156C23×C257—21no—
157C23×C257—21no—
158C23×C257—21no—
159C23×C257—21no—
160C23×C257C22×C412yes—
161C23×C257C22×C412yes—
162C23×C257C2×D412yes—
163C23×C257#4112yes—
164C23×C257C2411yescharacteristic
165D4×C257—28no—
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—21no—
178C2×C4×C257—21no—
179C2×C4×C257—44no—
180C2×C4×C257C22×C412yes—
181C2×C4×C257C2×D412yes—
182C22×C257—22no—
183C22×C257—22no—
184C22×C257—22no—
185C22×C257—22no—
186C22×C257—22no—
187C22×C257—22no—
188C22×C257—22no—
189C22×C257—22no—
190C22×C257—22no—
191C22×C257—41no—
192C22×C257—41no—
193C22×C257—41no—
194C22×C257—41no—
195C22×C257C4×D414yescomplemented
196C22×C257C23×C411yescharacteristic
197C22×C257C22×D411yescharacteristic
198C22×C257C22×D411yescharacteristic
199C22×C257C2×#4111yescharacteristic
200C22×C257C2×#4111yescharacteristic
201C22×C257H(5,𝔽2)11yescharacteristic
202C22×C2572−1+411yescharacteristic
203C2×D257—22no—
204C2×D257—22no—
205C2×D257—22no—
206C2×D257—41no—
207C2×D257—41no—
208C2×D257—41no—
209C2×D257—41no—
210C4×C257—22no—
211C4×C257—22no—
212C4×C257—41no—
213C4×C257—41no—
214C4×C257—41no—
215Dic257—22no—
216Dic257—22no—
217Dic257—22no—
218Dic257—22no—
219Dic257—22no—
220Dic257—41no—
221Dic257C4×D414yescomplemented
222C2×C257—24no—
223C2×C257—42no—
224C2×C257C2×C4×D411yescharacteristic
225C2×C257C2×C4×D411yescharacteristic
226C2×C257#52011yescharacteristic
227C2×C257D4211yescharacteristic
228C2×C257#54711yescharacteristic
229C2×C257#54811yescharacteristic
230C2×C257#55311yescharacteristic
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#386D25711yescharacteristic, complemented
245#385—2571no—
246#385—2571no—
247C4×C4⋊C4—2571no—
248C4×C22⋊C4—2571no—
249C22×D4—2571no—
250C22×D4—2571no—
251C22×D4C2×D25711yescharacteristic
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—2571no—
267C2×C4⋊C4—2571no—
268C2×C4⋊C4—5144no—
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⋊C4C2×D25711yescharacteristic
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—
290C24C22×D25711yescharacteristic
291C24C22×D25711yescharacteristic
292C2×D4—2574no—
293C2×D4—2574no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—5142no—
297C2×D4—5142no—
298C2×D4—5142no—
299C2×D4—5142no—
300C2×D4C4×D25714yescomplemented
301C22×C4—2571no—
302C22×C4—2574no—
303C22×C4—2571no—
304C22×C4—2571no—
305C22×C4—2571no—
306C22×C4—2571no—
307C22×C4—2571no—
308C22×C4—2571no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—5142no—
312C22×C4—5142no—
313C22×C4—5142no—
314C22×C4—5142no—
315C22×C4—5142no—
316C22×C4—5142no—
317C22×C4—5142no—
318C22×C4—5142no—
319C22×C4—41no—
320C22×C4—10281no—
321C22×C4—10281no—
322C22×C4—10281no—
323C22×C4C22×D25711yescharacteristic
324C22×C4C22×D25711yescharacteristic
325C22×C4C22×D25711yescharacteristic
326C22×C4C22×D25711yescharacteristic
327C22×C4C22×D25711yescharacteristic
328C4⋊C4—2574no—
329C4⋊C4—2574no—
330C4⋊C4—5142no—
331C4⋊C4—5142no—
332C4⋊C4—5142no—
333C4⋊C4—5142no—
334C4⋊C4—5142no—
335C22⋊C4—2574no—
336C22⋊C4—2574no—
337C22⋊C4—2574no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—5142no—
341C22⋊C4—5142no—
342C22⋊C4—5142no—
343C22⋊C4—41no—
344C22⋊C4—10281no—
345C42—5142no—
346C42—5142no—
347C42—5142no—
348C42—5142no—
349C23—2572no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—5141no—
361C23—5141no—
362C23—5141no—
363C23—5141no—
364C23—5141no—
365C23—5141no—
366C23C2×C4×D25712yes—
367C23C2×C4×D25712yes—
368C23D4×D25712yes—
369C23#1686161712yes—
370C23C23×D25711yescharacteristic
371D4—28no—
372D4—5148no—
373D4—5148no—
374C2×C4—2572no—
375C2×C4—2572no—
376C2×C4—2572no—
377C2×C4—2572no—
378C2×C4—2572no—
379C2×C4—2572no—
380C2×C4—2572no—
381C2×C4—2572no—
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—21no—
395C2×C4—5148no—
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—5141no—
408C2×C4—5141no—
409C2×C4—44no—
410C2×C4—10284no—
411C2×C4—10284no—
412C2×C4—10284no—
413C2×C4C2×C4×D25712yes—
414C2×C4D4×D25712yes—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—5142no—
425C22—5142no—
426C22—5142no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—10281no—
432C22—10281no—
433C22—10281no—
434C22—10281no—
435C22#2255155214yescomplemented
436C22C22×C4×D25711yescharacteristic
437C22C2×D4×D25711yescharacteristic
438C22C2×D4×D25711yescharacteristic
439C22#41×D25711yescharacteristic
440C22C2×#1686161711yescharacteristic
441C22#2255163711yescharacteristic
442C22#2255163911yescharacteristic
443C4—2574no—
444C4—22no—
445C4—22no—
446C4—5142no—
447C4—5142no—
448C4—5142no—
449C4—5142no—
450C4—5142no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—10281no—
455C2—24no—
456C2—42no—
457C2—10282no—
458C2C4×D4×D25711yescharacteristic
459C2C2×#2255155211yescharacteristic
460C2#3065721611yescharacteristic
461C2#3065736911yescharacteristic
462C2#3065738511yescharacteristic
463C2#3065739311yescharacteristic
464C2#3065743011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#386×C257C2
socle, cosocleC23×C257C24
prime radicalC2×#22551564C2
hypercenter#386D257
hypocenterC257#2361
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic257C2semidirect
3. C2×C4×C22⋊D257C2semidirect
4. C2×#22551552C2semidirect
5. C22×#16861597C2semidirect
6. #30656637C2semidirect
7. #30656634C2semidirect
8. #30656631C2semidirect
9. #30656626C2semidirect
10. #30656614C2semidirect
11. #30656611C2semidirect
12. #30656609C2semidirect
13. #30656588C2semidirect
14. C22⋊C4×Dic257C2semidirect
15. #386×C257C2semidirect
22. #22551564C4semidirect
61. C2×C22⋊D257C2×C4semidirect
62. C2×C22⋊D257C2×C4semidirect
71. C22×Dic257C2×C4semidirect
87. #16861597C2×C4semidirect
88. #16861597D4semidirect
90. C22⋊Dic257C2×C4semidirect
91. C22⋊Dic257D4semidirect
95. #16861595C2×C4semidirect
104. C2×D4×C257C2×C4semidirect
195. C22×C257C4×D4semidirect
221. Dic257C4×D4semidirect
232. C257#2361semidirect
244. #386D257semidirect
300. C2×D4C4×D257semidirect
435. C22#22551552semidirect

Maximal subgroups

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

subgroupindex
C2×#225515642
C2×D4×Dic2572
C2×C4×C22⋊D2572
C2×#225515522
C22×#168615972
#306566372
#306566342
#306566312
#306566262
#306566142
#306566112
#306566092
#306565882
C22⋊C4×Dic2572
#386×C2572
#2361257