Finite Group Databasebeta

← #45667018

Subgroups — #45667018

Order 32896.

696 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×#22551564C211yesmaximal, characteristic
2C2×C4×C22⋊D257C211yesmaximal, characteristic, complemented
3#30656628C211yesmaximal, characteristic, complemented
4#30656625C211yesmaximal, characteristic, complemented
5#30656621C211yesmaximal, characteristic, complemented
6#30656618C211yesmaximal, characteristic, complemented
7#30656617C211yesmaximal, characteristic, complemented
8#30656616C211yesmaximal, characteristic, complemented
9#30656614C211yesmaximal, characteristic, complemented
10#30656613C211yesmaximal, characteristic, complemented
11#30656610C211yesmaximal, characteristic, complemented
12#30656608C211yesmaximal, characteristic, complemented
13#30656595C211yesmaximal, characteristic, complemented
14C2×#22551499C211yesmaximal, characteristic, complemented
15C2×C4×D4×C257C211yesmaximal, characteristic, complemented
16C22×C22⋊D257C2211yescharacteristic
17C22×C22⋊D257C2211yescharacteristic
18C23×Dic257C2211yescharacteristic
19C23×Dic257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21#22551564C418yescomplemented
22C4×C22⋊D257—24no—
23C2×#16861597C2211yescharacteristic
24C2×#16861597C2211yescharacteristic
25C2×#16861597C2211yescharacteristic
26C2×#16861597C2211yescharacteristic
27C2×C22⋊Dic257C2211yescharacteristic
28C2×C22⋊Dic257C2211yescharacteristic
29C2×C22⋊Dic257C2211yescharacteristic
30C2×C22⋊Dic257C2211yescharacteristic
31C2×#16861595C2211yescharacteristic
32C2×#16861595C2211yescharacteristic
33C2×#16861595C2211yescharacteristic
34C2×#16861595C2211yescharacteristic
35C2×C4×Dic257C2211yescharacteristic
36#22551499C2211yescharacteristic
37#22551499C2211yescharacteristic
38#22551499C2211yescharacteristic
39#22551499C2211yescharacteristic
40#22551499C2211yescharacteristic
41#22551499C2211yescharacteristic
42#22551499C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45#22551499C2211yescharacteristic
46C22×D4×C257C2211yescharacteristic
47C23×C4×C257C2211yescharacteristic
48C23×C4×C257C2211yescharacteristic
49C4×D4×C257—24no—
50C2×C4⋊C4×C257C2211yescharacteristic
51C2×C22⋊C4×C257C2211yescharacteristic
52C2×C22⋊C4×C257C2211yescharacteristic
53C2×C42×C257C2211yescharacteristic
54C23×D257C2311yescharacteristic
55C2×C22⋊D257—22no—
56C2×C22⋊D257—22no—
57C2×C22⋊D257—41no—
58C2×C22⋊D257—41no—
59C2×C22⋊D257C2×C414yescomplemented
60C2×C22⋊D257C2×C414yescomplemented
61C22×Dic257—22no—
62C22×Dic257—41no—
63C22×Dic257—41no—
64C22×Dic257—41no—
65C22×Dic257—41no—
66C22×Dic257C2×C414yescomplemented
67C22×Dic257C2311yescharacteristic
68C22×Dic257C2311yescharacteristic
69C22×Dic257C2311yescharacteristic
70C22×Dic257C2311yescharacteristic
71C22×Dic257C2311yescharacteristic
72C22×Dic257C2311yescharacteristic
73C22×Dic257C2311yescharacteristic
74C2×C4×D257—22no—
75C2×C4×D257—41no—
76C2×C4×D257—41no—
77#16861597—22no—
78#16861597—41no—
79#16861597—41no—
80#16861597C2×C414yescomplemented
81C22⋊Dic257—22no—
82C22⋊Dic257—41no—
83C22⋊Dic257—41no—
84C22⋊Dic257C2×C414yescomplemented
85#16861595—22no—
86#16861595—41no—
87#16861595—41no—
88#16861595C2×C414yescomplemented
89C4×Dic257—22no—
90C24×C257C2311yescharacteristic
91C24×C257C2311yescharacteristic
92C2×D4×C257—22no—
93C2×D4×C257—22no—
94C2×D4×C257C2×C414yescomplemented
95C22×C4×C257—22no—
96C22×C4×C257—41no—
97C22×C4×C257—41no—
98C22×C4×C257—41no—
99C22×C4×C257—41no—
100C22×C4×C257D414yescomplemented
101C22×C4×C257C2311yescharacteristic
102C22×C4×C257C2311yescharacteristic
103C22×C4×C257C2311yescharacteristic
104C22×C4×C257C2311yescharacteristic
105C22×C4×C257C2311yescharacteristic
106C4⋊C4×C257—22no—
107C22⋊C4×C257—22no—
108C22⋊C4×C257—22no—
109C42×C257—22no—
110C22×D257—21no—
111C22×D257—21no—
112C22×D257—21no—
113C22×D257—21no—
114C22×D257—21no—
115C22×D257—21no—
116C22×D257C22×C412yes—
117C22⋊D257—28no—
118C22⋊D257—44no—
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×Dic257—21no—
135C2×Dic257—21no—
136C2×Dic257—21no—
137C2×Dic257—44no—
138C2×Dic257—44no—
139C2×Dic257C22×C412yes—
140C2×Dic257C22×C412yes—
141C2×Dic257C22×C412yes—
142C4×D257—44no—
143C23×C257—21no—
144C23×C257—21no—
145C23×C257—21no—
146C23×C257—21no—
147C23×C257—21no—
148C23×C257—21no—
149C23×C257—21no—
150C23×C257—21no—
151C23×C257—21no—
152C23×C257—21no—
153C23×C257C22×C412yes—
154C23×C257C22×C412yes—
155C23×C257C2×D412yes—
156C23×C257#4112yes—
157C23×C257C2411yescharacteristic
158D4×C257—28no—
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×C257—44no—
172C2×C4×C257—44no—
173C2×C4×C257C22×C412yes—
174C2×C4×C257C2×D412yes—
175C2×C4×C257C2×D412yes—
176C22×C257—22no—
177C22×C257—22no—
178C22×C257—22no—
179C22×C257—22no—
180C22×C257—22no—
181C22×C257—22no—
182C22×C257—22no—
183C22×C257—22no—
184C22×C257—22no—
185C22×C257—41no—
186C22×C257—41no—
187C22×C257—41no—
188C22×C257—41no—
189C22×C257C4×D414yes—
190C22×C257C23×C411yescharacteristic
191C22×C257C22×D411yescharacteristic
192C22×C257C2×#4111yescharacteristic
193C22×C257H(5,𝔽2)11yescharacteristic
194C22×C257H(5,𝔽2)11yescharacteristic
195C22×C257H(5,𝔽2)11yescharacteristic
196C22×C2572−1+411yescharacteristic
197C2×D257—22no—
198C2×D257—22no—
199C2×D257—22no—
200C2×D257—41no—
201C2×D257—41no—
202C2×D257—41no—
203C2×D257—41no—
204C4×C257—22no—
205C4×C257—22no—
206C4×C257—22no—
207C4×C257—41no—
208C4×C257—41no—
209Dic257—22no—
210Dic257—22no—
211Dic257—22no—
212Dic257—41no—
213Dic257—41no—
214Dic257—41no—
215Dic257—41no—
216C2×C257—24no—
217C2×C257—42no—
218C2×C257C2×C4×D411yescharacteristic
219C2×C257#51811yescharacteristic
220C2×C257#52011yescharacteristic
221C2×C257#53411yescharacteristic
222C2×C257#53711yescharacteristic
223C2×C257#53811yescharacteristic
224C2×C257#53911yescharacteristic
225D257—42no—
226C257#232511yescharacteristic, complemented, Sylow
227#2325—2571nomaximal, Sylow
228C2×#121—2571no—
229C2×C4×D4—2571no—
230C2×C4×D4D25711yescharacteristic, complemented
231#388—2571no—
232#388—2571no—
233#387—2571no—
234#387—2571no—
235#386—2571no—
236#386—2571no—
237#386—2571no—
238#386—2571no—
239#385—2571no—
240#385—2571no—
241#380—2571no—
242C2×#95—2571no—
243C22×D4—2571no—
244C22×D4—2571no—
245C22×D4C2×D25711yescharacteristic
246C23×C4—2571no—
247C23×C4—2571no—
248C23×C4—2571no—
249C23×C4C2×D25711yescharacteristic
250C23×C4C2×D25711yescharacteristic
251#121—2578no—
252C4×D4—24no—
253C4×D4—5144no—
254C2×C4⋊C4—2571no—
255C2×C4⋊C4—2571no—
256C2×C4⋊C4—2571no—
257C2×C4⋊C4—2571no—
258C2×C4⋊C4C2×D25711yescharacteristic
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×C22⋊C4C2×D25711yescharacteristic
268C2×C22⋊C4C2×D25711yescharacteristic
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#95—2571no—
278#95—2571no—
279#95—2571no—
280#95—2571no—
281C24—2571no—
282C24C22×D25711yescharacteristic
283C24C22×D25711yescharacteristic
284C2×D4—2574no—
285C2×D4—2574no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—5142no—
289C2×D4—5142no—
290C2×D4—10281no—
291C2×D4—10281no—
292C2×D4C4×D25714yescomplemented
293C22×C4—2571no—
294C22×C4—2571no—
295C22×C4—2571no—
296C22×C4—2574no—
297C22×C4—2571no—
298C22×C4—2571no—
299C22×C4—2571no—
300C22×C4—2571no—
301C22×C4—22no—
302C22×C4—5142no—
303C22×C4—5142no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—10281no—
309C22×C4—10281no—
310C22×C4—10281no—
311C22×C4—10281no—
312C22×C4—10281no—
313C22×C4—10281no—
314C22×C4C22⋊D25714yescomplemented
315C22×C4C22×D25711yescharacteristic
316C22×C4C22×D25711yescharacteristic
317C22×C4C22×D25711yescharacteristic
318C22×C4C22×D25711yescharacteristic
319C22×C4C22×D25711yescharacteristic
320C4⋊C4—2574no—
321C4⋊C4—22no—
322C4⋊C4—5142no—
323C4⋊C4—10281no—
324C4⋊C4—10281no—
325C22⋊C4—2574no—
326C22⋊C4—2574no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—5142no—
330C22⋊C4—5142no—
331C22⋊C4—10281no—
332C22⋊C4—10281no—
333C22⋊C4—10281no—
334C22⋊C4—10281no—
335C42—22no—
336C42—5142no—
337C23—2572no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—5141no—
349C23—5141no—
350C23—5141no—
351C23—5141no—
352C23—5141no—
353C23—5141no—
354C23C2×C4×D25712yes—
355C23C2×C4×D25712yes—
356C23C2×C22⋊D25712yes—
357C23#1686161812yes—
358C23C23×D25711yescharacteristic
359D4—28no—
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—5141no—
378C2×C4—5141no—
379C2×C4—5141no—
380C2×C4—5141no—
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—5141no—
394C2×C4—5141no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—10284no—
398C2×C4—10284no—
399C2×C4—10284no—
400C2×C4C2×C4×D25712yes—
401C2×C4C2×C22⋊D25712yes—
402C2×C4C2×C22⋊D25712yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—5142no—
413C22—5142no—
414C22—5142no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—10281no—
420C22—10281no—
421C22—10281no—
422C22—10281no—
423C22C4×C22⋊D25714yes—
424C22C22×C4×D25711yescharacteristic
425C22C22×C22⋊D25711yescharacteristic
426C22C2×#1686161811yescharacteristic
427C22#2255163711yescharacteristic
428C22#2255163711yescharacteristic
429C22#2255163811yescharacteristic
430C22#2255163911yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—5142no—
435C4—5142no—
436C4—5142no—
437C4—41no—
438C4—41no—
439C4—10281no—
440C4—10281no—
441C4—10281no—
442C4—10281no—
443C2—24no—
444C2—42no—
445C2—10282no—
446C2C2×C4×C22⋊D25711yescharacteristic
447C2#3065720911yescharacteristic
448C2#3065721611yescharacteristic
449C2#3065729611yescharacteristic
450C2#3065731511yescharacteristic
451C2#3065732011yescharacteristic
452C2#3065732911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
FittingC2×C4×D4×C257C2
socle, cosocleC23×C257C24
prime radicalC2×#22551564C2
hypercenterC2×C4×D4D257
hypocenterC257#2325
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D257C2semidirect
3. #30656628C2semidirect
4. #30656625C2semidirect
5. #30656621C2semidirect
6. #30656618C2semidirect
7. #30656617C2semidirect
8. #30656616C2semidirect
9. #30656614C2semidirect
10. #30656613C2semidirect
11. #30656610C2semidirect
12. #30656608C2semidirect
13. #30656595C2semidirect
14. C2×#22551499C2semidirect
15. C2×C4×D4×C257C2semidirect
21. #22551564C4semidirect
59. C2×C22⋊D257C2×C4semidirect
60. C2×C22⋊D257C2×C4semidirect
66. C22×Dic257C2×C4semidirect
80. #16861597C2×C4semidirect
84. C22⋊Dic257C2×C4semidirect
88. #16861595C2×C4semidirect
94. C2×D4×C257C2×C4semidirect
100. C22×C4×C257D4semidirect
226. C257#2325semidirect
230. C2×C4×D4D257semidirect
292. C2×D4C4×D257semidirect
314. C22×C4C22⋊D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515642
C2×C4×C22⋊D2572
#306566282
#306566252
#306566212
#306566182
#306566172
#306566162
#306566142
#306566132
#306566102
#306566082
#306565952
C2×#225514992
C2×C4×D4×C2572
#2325257