Finite Group Databasebeta

← C2×#20243104 #26213677

Subgroups — C2×#20243104 #26213677

Order 12032.

2398 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C47⋊C8C211yesmaximal, characteristic
2C22×#13093983C211yesmaximal, characteristic
3#20243104C218yesmaximal, direct product factor
4C2×#15688885C214yesmaximal, complemented
5C22×#117×C47C211yesmaximal, characteristic
6C23×C47⋊C8—24no—
7C23×C47⋊C8—24no—
8C23×C47⋊C8C418yescomplemented
9C23×C47⋊C8C2212yes—
10C23×C47⋊C8C2214yes—
11C2×#13093983—24no—
12C2×#13093983—22no—
13C2×#13093983C418yescomplemented
14C2×#13093983C2214yes—
15C22×C4⋊Dic47C2212yes—
16C22×C22⋊Dic47—21no—
17C22×C4×Dic47—21no—
18#15688885C22116yescomplemented
19C24×C4×C47C2211yescharacteristic
20C2×#117×C47—24no—
21C2×#117×C47—22no—
22C2×#117×C47C418yescomplemented
23C2×#117×C47C2214yes—
24C22×C4⋊C4×C47C2212yes—
25C22×C22⋊C4×C47—21no—
26C22×C42×C47—21no—
27C23×Dic47—22no—
28C22×C47⋊C8—232no—
29C22×C47⋊C8—24no—
30C22×C47⋊C8—24no—
31C22×C47⋊C8—28no—
32C22×C47⋊C8—28no—
33C22×C47⋊C8C2×C418yes—
34C22×C47⋊C8C2×C4116yescomplemented
35C22×C47⋊C8C2314yes—
36#13093983—216no—
37#13093983—28no—
38#13093983C2×C4116yescomplemented
39C2×C4⋊Dic47—22no—
40C2×C4⋊Dic47—24no—
41C2×C4⋊Dic47C2×C418yes—
42C2×C4⋊Dic47C2314yes—
43C2×C22⋊Dic47—28no—
44C2×C22⋊Dic47—24no—
45C2×C4×Dic47—28no—
46C2×C4×Dic47—24no—
47C25×C47D411yescharacteristic
48C23×C4×C47—22no—
49C23×C4×C47C2×C414yes—
50C23×C4×C47C2×C414yes—
51C23×C4×C47C2×C414yes—
52C23×C4×C47D412yes—
53C23×C4×C47D411yescharacteristic
54C23×C4×C47D414yes—
55C23×C4×C47D414yes—
56C23×C4×C47Q814yes—
57C23×C4×C47C2311yescharacteristic
58C23×C4×C47C2312yes—
59#117×C47—216no—
60#117×C47—28no—
61#117×C47C2×C4116yescomplemented
62C2×C4⋊C4×C47—22no—
63C2×C4⋊C4×C47—24no—
64C2×C4⋊C4×C47C2×C418yes—
65C2×C4⋊C4×C47C2314yes—
66C2×C22⋊C4×C47—28no—
67C2×C22⋊C4×C47—24no—
68C2×C42×C47—28no—
69C2×C42×C47—24no—
70C24×C47—28no—
71C24×C47C22⋊C414yes—
72C24×C47C22⋊C414yes—
73C24×C47C4⋊C414yes—
74C24×C47C2×D412yes—
75C24×C47C2×D411yescharacteristic
76C22×C4×C47—28no—
77C22×C4×C47—24no—
78C22×C4×C47—48no—
79C22×C4×C47C42116yes—
80C22×C4×C47C22⋊C414yes—
81C22×C4×C47C22⋊C418yes—
82C22×C4×C47C22⋊C414yes—
83C22×C4×C47C22⋊C418yes—
84C22×C4×C47C22⋊C4116yes—
85C22×C4×C47C4⋊C418yes—
86C22×C4×C47C4⋊C414yes—
87C22×C4×C47C4⋊C4116yes—
88C22×C4×C47C4⋊C4116yes—
89C22×C4×C47C22×C412yes—
90C22×C4×C47C22×C414yes—
91C22×C4×C47C22×C412yes—
92C22×C4×C47C22×C414yes—
93C22×C4×C47C22×C412yes—
94C22×C4×C47C22×C414yes—
95C22×C4×C47C2×D412yes—
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D412yes—
98C22×C4×C47C2×D414yes—
99C22×C4×C47C2×D412yes—
100C22×C4×C47C2×D414yes—
101C22×C4×C47C2×Q814yes—
102C22×C4×C47C2×Q812yes—
103C22×C4×C47C2411yescharacteristic
104C22×Dic47—28no—
105C22×Dic47—24no—
106C22×Dic47—48no—
107C4⋊C4×C47—28no—
108C4⋊C4×C47—24no—
109C4⋊C4×C47C22×C418yes—
110C22⋊C4×C47—216no—
111C42×C47—216no—
112C22⋊Dic47—216no—
113C4⋊Dic47—28no—
114C4⋊Dic47—24no—
115C4⋊Dic47C22×C418yes—
116C4×Dic47—216no—
117C2×C47⋊C8—216no—
118C2×C47⋊C8—232no—
119C2×C47⋊C8—24no—
120C2×C47⋊C8—24no—
121C2×C47⋊C8C22×C418yes—
122C23×C47—28no—
123C23×C47—216no—
124C23×C47—216no—
125C23×C47—216no—
126C23×C47#95116yes—
127C23×C47C2×C22⋊C414yes—
128C23×C47C2×C22⋊C412yes—
129C23×C47C2×C22⋊C414yes—
130C23×C47C2×C22⋊C412yes—
131C23×C47C2×C4⋊C414yes—
132C23×C47C2×C4⋊C412yes—
133C23×C47#13518yes—
134C23×C47C22×D411yescharacteristic
135C2×C4×C47—28no—
136C2×C4×C47—48no—
137C2×C4×C47—416no—
138C2×C4×C47#95132yes—
139C2×C4×C47#95116yes—
140C2×C4×C47C2×C4218yes—
141C2×C4×C47C2×C4214yes—
142C2×C4×C47C2×C22⋊C418yes—
143C2×C4×C47C2×C22⋊C414yes—
144C2×C4×C47C2×C22⋊C414yes—
145C2×C4×C47C2×C22⋊C412yes—
146C2×C4×C47C2×C22⋊C414yes—
147C2×C4×C47C2×C22⋊C412yes—
148C2×C4×C47C2×C4⋊C418yes—
149C2×C4×C47C2×C4⋊C414yes—
150C2×C4×C47C2×C4⋊C412yes—
151C2×C4×C47C2×C4⋊C414yes—
152C2×C4×C47C2×C4⋊C418yes—
153C2×C4×C47C2×C4⋊C414yes—
154C2×C4×C47C23×C411yescharacteristic
155C2×C4×C47C23×C411yescharacteristic
156C2×C4×C47C23×C411yescharacteristic
157C2×C4×C47C22×D411yescharacteristic
158C2×C4×C47C22×D411yescharacteristic
159C2×C4×C47C22×Q811yescharacteristic
160C2×Dic47—28no—
161C2×Dic47—48no—
162C2×Dic47—416no—
163C47⋊C8—28no—
164C2×#1735—471nomaximal, Sylow
165C22×C47—28no—
166C22×C47—28no—
167C22×C47—28no—
168C22×C47—232no—
169C22×C47C2×#9518yes—
170C22×C47C2×#9514yes—
171C22×C47#41618yes—
172C22×C47#41618yes—
173C22×C47#42718yes—
174C22×C47C22×C22⋊C411yescharacteristic
175C22×C47C22×C22⋊C411yescharacteristic
176C22×C47C22×C4⋊C411yescharacteristic
177C22×C47C2×#13514yes—
178C4×C47—48no—
179C4×C47C2×#9518yes—
180C4×C47C2×#9514yes—
181C4×C47C22×C4211yescharacteristic
182C4×C47C22×C22⋊C411yescharacteristic
183C4×C47C22×C4⋊C411yescharacteristic
184C4×C47C22×C4⋊C411yescharacteristic
185Dic47—48no—
186C24×C8—471no—
187C22×#117—471no—
188C22×#117D4711yescharacteristic
189#1735—478no—
190C2×#340—474no—
191C2×C47—28no—
192C2×C47#173518yescomplemented
193C2×C47C22×#9511yescharacteristic
194C2×C47C2×#41612yes—
195C2×C47C2×#41612yes—
196C2×C47C2×#42712yes—
197C24×C4C2×D4711yescharacteristic
198C23×C8—478no—
199C23×C8—472no—
200C23×C8—474no—
201C23×C8—944no—
202C23×C8—944no—
203C2×#117—24no—
204C2×#117—22no—
205C2×#117—478no—
206C2×#117—474no—
207C2×#117—944no—
208C2×#117—942no—
209C2×#117Dic4718yescomplemented
210C2×#117C2×D4714yes—
211C22×C4⋊C4—472no—
212C22×C4⋊C4C2×D4712yes—
213C22×C22⋊C4—21no—
214C22×C22⋊C4—941no—
215C22×C42—21no—
216C22×C42—941no—
217#340—4716no—
218C47C2×#173511yescharacteristic, complemented, Sylow
219C25C22⋊D4711yescharacteristic
220C23×C4—22no—
221C23×C4—942no—
222C23×C4Dic9414yes—
223C23×C4C4×D4714yes—
224C23×C4C4×D4714yes—
225C23×C4D18814yes—
226C23×C4C2×Dic4714yes—
227C23×C4C22⋊D4712yes—
228C23×C4C22⋊D4714yes—
229C23×C4C22⋊D4711yescharacteristic
230C23×C4C22×D4711yescharacteristic
231C23×C4C22×D4712yes—
232C22×C8—478no—
233C22×C8—4716no—
234C22×C8—474no—
235C22×C8—9432no—
236C22×C8—944no—
237C22×C8—948no—
238C22×C8—944no—
239C22×C8—948no—
240#117—216no—
241#117—28no—
242#117—4716no—
243#117—9416no—
244#117—948no—
245#117C2×Dic47116yescomplemented
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—478no—
249C2×C4⋊C4—474no—
250C2×C4⋊C4—944no—
251C2×C4⋊C4—942no—
252C2×C4⋊C4C2×Dic4718yes—
253C2×C4⋊C4C22×D4714yes—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—948no—
257C2×C22⋊C4—944no—
258C2×C42—28no—
259C2×C42—24no—
260C2×C42—948no—
261C2×C42—944no—
262C24—28no—
263C24Dic47⋊C414yes—
264C24(C2×D47)⋊C414yes—
265C24C22⋊Dic4714yes—
266C24C2×C22⋊D4712yes—
267C24C2×C22⋊D4711yescharacteristic
268C22×C4—28no—
269C22×C4—24no—
270C22×C4—48no—
271C22×C4—948no—
272C22×C4—944no—
273C22×C4—1888no—
274C22×C4C4×Dic47116yes—
275C22×C4Dic47⋊C418yes—
276C22×C4Dic47⋊C414yes—
277C22×C4Dic47⋊C4116yes—
278C22×C4C4⋊Dic47116yes—
279C22×C4(C2×D47)⋊C414yes—
280C22×C4(C2×D47)⋊C418yes—
281C22×C4(C2×D47)⋊C4116yes—
282C22×C4C22⋊Dic4714yes—
283C22×C4C22⋊Dic4718yes—
284C22×C4C2×Dic9414yes—
285C22×C4C2×Dic9412yes—
286C22×C4C2×C4×D4714yes—
287C22×C4C2×C4×D4712yes—
288C22×C4C2×C4×D4712yes—
289C22×C4C2×C4×D4714yes—
290C22×C4C2×D18812yes—
291C22×C4C2×D18814yes—
292C22×C4C22×Dic4712yes—
293C22×C4C22×Dic4714yes—
294C22×C4C2×C22⋊D4712yes—
295C22×C4C2×C22⋊D4711yescharacteristic
296C22×C4C2×C22⋊D4712yes—
297C22×C4C2×C22⋊D4714yes—
298C22×C4C23×D4711yescharacteristic
299C2×C8—478no—
300C2×C8—9416no—
301C2×C8—9432no—
302C2×C8—944no—
303C2×C8—944no—
304C4⋊C4—28no—
305C4⋊C4—24no—
306C4⋊C4—478no—
307C4⋊C4—948no—
308C4⋊C4—944no—
309C4⋊C4C22×Dic4718yes—
310C22⋊C4—216no—
311C22⋊C4—9416no—
312C42—216no—
313C42—9416no—
314C23—216no—
315C23—216no—
316C23—28no—
317C23—216no—
318C23C47⋊#95116yes—
319C23C2×(C2×D47)⋊C414yes—
320C23C2×(C2×D47)⋊C412yes—
321C23C2×C22⋊Dic4714yes—
322C23C2×C22⋊Dic4712yes—
323C23C2×Dic47⋊C414yes—
324C23C2×Dic47⋊C412yes—
325C23#1309404418yes—
326C23C22×C22⋊D4711yescharacteristic
327C2×C4—28no—
328C2×C4—416no—
329C2×C4—48no—
330C2×C4—948no—
331C2×C4—18816no—
332C2×C4—1888no—
333C2×C4C47⋊#95132yes—
334C2×C4C47⋊#95116yes—
335C2×C4C2×C4×Dic4718yes—
336C2×C4C2×C4×Dic4714yes—
337C2×C4C2×(C2×D47)⋊C414yes—
338C2×C4C2×(C2×D47)⋊C412yes—
339C2×C4C2×(C2×D47)⋊C418yes—
340C2×C4C2×(C2×D47)⋊C414yes—
341C2×C4C2×C22⋊Dic4714yes—
342C2×C4C2×C22⋊Dic4712yes—
343C2×C4C2×Dic47⋊C412yes—
344C2×C4C2×Dic47⋊C414yes—
345C2×C4C2×Dic47⋊C418yes—
346C2×C4C2×Dic47⋊C414yes—
347C2×C4C2×C4⋊Dic4718yes—
348C2×C4C2×C4⋊Dic4714yes—
349C2×C4C22×C4×D4711yescharacteristic
350C2×C4C22×C4×D4711yescharacteristic
351C2×C4C23×Dic4711yescharacteristic
352C2×C4C22×C22⋊D4711yescharacteristic
353C2×C4C22×D18811yescharacteristic
354C2×C4C22×Dic9411yescharacteristic
355C8—948no—
356C22—28no—
357C22—232no—
358C22—28no—
359C22—28no—
360C22C2×C47⋊#9518yes—
361C22C2×C47⋊#9514yes—
362C22#1568911718yes—
363C22#1568911918yes—
364C22#1568915918yes—
365C22C22×(C2×D47)⋊C411yescharacteristic
366C22C22×C22⋊Dic4711yescharacteristic
367C22C22×Dic47⋊C411yescharacteristic
368C22C2×#1309404414yes—
369C4—48no—
370C4—1888no—
371C4C2×C47⋊#9518yes—
372C4C2×C47⋊#9514yes—
373C4C22×C4×Dic4711yescharacteristic
374C4C22×(C2×D47)⋊C411yescharacteristic
375C4C22×Dic47⋊C411yescharacteristic
376C4C22×C4⋊Dic4711yescharacteristic
377C2—28no—
378C2#2024310418yesdirect product factor
379C2C22×C47⋊#9511yescharacteristic
380C2C2×#1568911712yes—
381C2C2×#1568911912yes—
382C2C2×#1568915912yes—
subgroupG/H
centerC23×C4C22⋊D47
commutatorC4×C47C22×C42
FrattiniC22×C4C23×D47
FittingC22×#117×C47C2
socleC24×C47C2×D4
prime radicalC25×C47D4
cosocleC22×C4×C47C24
hypercenterC22×#117D47
hypocenterC47C2×#1735
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. #20243104C2direct (N × H)
4. C2×#15688885C2semidirect
8. C23×C47⋊C8C4semidirect
13. C2×#13093983C4semidirect
18. #15688885C22semidirect
22. C2×#117×C47C4semidirect
34. C22×C47⋊C8C2×C4semidirect
38. #13093983C2×C4semidirect
61. #117×C47C2×C4semidirect
192. C2×C47#1735semidirect
209. C2×#117Dic47semidirect
218. C47C2×#1735semidirect
245. #117C2×Dic47semidirect
378. C2#20243104direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C47⋊C82
C22×#130939832
#202431042
C2×#156888852
C22×#117×C472
C2×#173547