Finite Group Databasebeta

← C2×#20251950 #27079898

Subgroups — C2×#20251950 #27079898

Order 12032.

2830 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D47C212yesmaximal
2C22×#13093981C211yesmaximal, characteristic
3#20251950C2116yesmaximal, direct product factor
4C2×#15689148C218yesmaximal, complemented
5C2×#15689147C211yesmaximal, characteristic, complemented
6C2×#15689146C212yesmaximal, complemented
7C2×C47×#423C211yesmaximal, characteristic, complemented
8C23×C4×D47C2211yescharacteristic
9C22×#10625329C2212yes—
10C22×#10625324C2214yes—
11C2×M16×D47—216no—
12C2×M16×D47C22116yes—
13C22×C8×D47C2214yes—
14C2×#13093981C414yes—
15C2×#13093981C418yes—
16C2×#13093981C2214yes—
17C2×#13093981C2218yes—
18C22×Dic47⋊C4C412yes—
19C22×(C2×D47)⋊C4C412yes—
20C22×C4×Dic47C2211yescharacteristic
21#15689148C22164yescomplemented
22#15689147C2218yescomplemented
23#15689146C22116yescomplemented
24C2×C4⋊(C47⋊C8)C2214yescomplemented
25C2×Dic47⋊C8C2218yescomplemented
26C22×M16×C47C2212yes—
27C22×C42×C47C2211yescharacteristic
28C47×#423C2218yescomplemented
29C2×C4⋊C8×C47C2214yescomplemented
30C24×D47C2×C411yescharacteristic
31C23×Dic47C2×C412yes—
32C23×Dic47C2×C411yescharacteristic
33C22×C4×D47C2×C412yes—
34C22×C4×D47C2×C414yes—
35C22×C4×D47D418yes—
36C22×C4×D47Q818yes—
37C22×C4×D47C2312yes—
38C22×C4×D47C2314yes—
39C2×#10625329—28no—
40C2×#10625329C2318yes—
41C2×#10625324—216no—
42C2×#10625324C23116yes—
43M16×D47—264no—
44C22×C47⋊C8C2314yes—
45C2×C8×D47—216no—
46C2×C8×D47C23116yes—
47#13093981C2×C4116yes—
48#13093981C2×C4132yes—
49#13093981C23116yes—
50C2×Dic47⋊C4C2×C418yes—
51C2×Dic47⋊C4C2×C4116yes—
52C2×(C2×D47)⋊C4C2×C418yes—
53C2×(C2×D47)⋊C4C2×C4116yes—
54C2×C4×Dic47C2×C412yes—
55C2×C4×Dic47C2×C414yes—
56C2×C4×Dic47C2312yes—
57C2×C4×Dic47C2314yes—
58C4⋊(C47⋊C8)C23116yescomplemented
59Dic47⋊C8C23132yescomplemented
60C23×C4×C47C2×C412yes—
61C23×C4×C47C2311yescharacteristic
62C2×M16×C47—28no—
63C2×M16×C47C2318yes—
64C22×C8×C47C2314yes—
65C2×C42×C47C2×C412yes—
66C2×C42×C47C2×C414yes—
67C2×C42×C47C2312yes—
68C2×C42×C47C2314yes—
69C4⋊C8×C47C23116yescomplemented
70C24×C47C22×C411yescharacteristic
71C23×D47—44no—
72C23×D47C4⋊C418yes—
73C23×D47C22×C412yes—
74C23×D47C22×C414yes—
75C22×C4×C47—44no—
76C22×C4×C47C22×C414yes—
77C22×C4×C47C22×C418yes—
78C22×C4×C47C22×C412yes—
79C22×C4×C47C22×C411yescharacteristic
80C22×C4×C47C2×D414yes—
81C22×C4×C47C2×Q814yes—
82C22×C4×C47C2412yes—
83C22×C4×C47C2411yescharacteristic
84C22×Dic47—44no—
85C22×Dic47C4⋊C418yes—
86C22×Dic47C22×C414yes—
87C22×Dic47C22×C418yes—
88C22×Dic47C22×C414yes—
89C22×Dic47C22×C412yes—
90C2×C4×D47C4⋊C4116yes—
91C2×C4×D47C4⋊C4132yes—
92C2×C4×D47C22×C414yes—
93C2×C4×D47C22×C418yes—
94C2×C4×D47C2×D418yes—
95C2×C4×D47C2×D4116yes—
96C2×C4×D47C2×Q818yes—
97C2×C4×D47C2×Q8116yes—
98C2×C4×D47C2414yes—
99M16×C47—216no—
100C2×C8×C47—28no—
101C2×C8×C47C2418yes—
102C42×C47C22×C414yes—
103C42×C47C22×C418yes—
104C42×C47C2414yes—
105(C2×D47)⋊C4C22×C4132yes—
106Dic47⋊C4C22×C4132yes—
107C4×Dic47C22×C414yes—
108C4×Dic47C22×C418yes—
109C4×Dic47C2414yes—
110#10625329—216no—
111C2×C47⋊C8—28no—
112C2×C47⋊C8C2418yes—
113#10625324—232no—
114C8×D47—232no—
115C23×C47—24no—
116C23×C47C2×C4⋊C414yes—
117C23×C47C23×C412yes—
118C23×C47C23×C411yescharacteristic
119C22×D47—28no—
120C22×D47—416no—
121C22×D47—48no—
122C22×D47C2×C4⋊C418yes—
123C22×D47C2×C4⋊C4116yes—
124C22×D47C23×C414yes—
125C2×C4×C47—28no—
126C2×C4×C47—48no—
127C2×C4×C47C2×C4⋊C418yes—
128C2×C4×C47C2×C4⋊C414yes—
129C2×C4×C47C23×C418yes—
130C2×C4×C47C23×C412yes—
131C2×C4×C47C23×C411yescharacteristic
132C2×C4×C47C22×D412yes—
133C2×C4×C47C22×D414yes—
134C2×C4×C47C22×Q814yes—
135C2×C4×C47C22×Q812yes—
136C2×C4×C47C2511yescharacteristic
137C2×Dic47—28no—
138C2×Dic47—48no—
139C2×Dic47C2×C4⋊C4116yes—
140C2×Dic47C2×C4⋊C418yes—
141C2×Dic47C23×C418yes—
142C2×Dic47C23×C414yes—
143C4×D47C2×C4⋊C4116yes—
144C4×D47C2×C4⋊C4132yes—
145C4×D47C22×D418yes—
146C4×D47C22×Q818yes—
147C8×C47—28no—
148C47⋊C8—28no—
149C2×#2905—471nomaximal, Sylow
150C22×C47—28no—
151C22×C47—24no—
152C22×C47C22×C4⋊C414yes—
153C22×C47C22×C4⋊C412yes—
154C22×C47#56814yes—
155C22×C47C24×C411yescharacteristic
156C2×D47—216no—
157C2×D47—28no—
158C2×D47—416no—
159C2×D47C22×C4⋊C418yes—
160C4×C47—28no—
161C4×C47C22×C4⋊C412yes—
162C4×C47C22×C4⋊C414yes—
163C4×C47C23×D411yescharacteristic
164C4×C47C23×Q811yescharacteristic
165Dic47—28no—
166Dic47C22×C4⋊C418yes—
167C23×M16—472no—
168C22×#117—471no—
169#2905—4716no—
170C2×#424—478no—
171C2×#423—471no—
172C2×#423—472no—
173C2×#423D4711yescharacteristic, complemented
174C2×C47—24no—
175C2×C47#290514yescomplemented
176C2×C47C23×C4⋊C411yescharacteristic
177C2×C47C2×#56812yes—
178D47—28no—
179C24×C4—471no—
180C22×M16—472no—
181C22×M16—4716no—
182C22×M16—474no—
183C22×M16—9416no—
184C22×M16C2×D4712yes—
185C23×C8—474no—
186C2×#117—474no—
187C2×#117—478no—
188C2×#117—474no—
189C2×#117—478no—
190C22×C4⋊C4—472no—
191C22×C22⋊C4—472no—
192C22×C42—471no—
193C22×C42C2×D4711yescharacteristic
194#424—4764no—
195#423—4716no—
196#423—478no—
197#423C2×D4718yescomplemented
198C2×C4⋊C8—478no—
199C2×C4⋊C8—474no—
200C2×C4⋊C8C2×D4714yescomplemented
201C47C2×#290511yescharacteristic, complemented, Sylow
202C25—471no—
203C23×C4—471no—
204C23×C4—472no—
205C23×C4—474no—
206C23×C4—478no—
207C23×C4—478no—
208C23×C4—474no—
209C23×C4—472no—
210C23×C4—472no—
211C23×C4C4×D4712yes—
212C23×C4C22×D4711yescharacteristic
213C2×M16—28no—
214C2×M16—478no—
215C2×M16—4716no—
216C2×M16—9416no—
217C2×M16—9464no—
218C2×M16—948no—
219C2×M16C22×D4718yes—
220C22×C8—474no—
221C22×C8—4716no—
222C22×C8—9416no—
223C22×C8C22×D4714yes—
224#117—4716no—
225#117—4732no—
226#117—4716no—
227C2×C4⋊C4—478no—
228C2×C4⋊C4—4716no—
229C2×C22⋊C4—478no—
230C2×C22⋊C4—4716no—
231C2×C42—472no—
232C2×C42—474no—
233C2×C42—474no—
234C2×C42—472no—
235C2×C42C4×D4712yes—
236C2×C42C4×D4714yes—
237C2×C42C22×D4714yes—
238C2×C42C22×D4712yes—
239C4⋊C8—4732no—
240C4⋊C8—4716no—
241C4⋊C8C22×D47116yescomplemented
242C24—472no—
243C24—478no—
244C24—474no—
245C24—1884no—
246C24C2×C4×D4711yescharacteristic
247C22×C4—44no—
248C22×C4—478no—
249C22×C4—474no—
250C22×C4—472no—
251C22×C4—4716no—
252C22×C4—4732no—
253C22×C4—478no—
254C22×C4—474no—
255C22×C4—478no—
256C22×C4—4716no—
257C22×C4—4716no—
258C22×C4—478no—
259C22×C4—474no—
260C22×C4—474no—
261C22×C4—478no—
262C22×C4—1884no—
263C22×C4C2×C4×D4714yes—
264C22×C4C2×C4×D4718yes—
265C22×C4C2×C4×D4711yescharacteristic
266C22×C4C2×C4×D4712yes—
267C22×C4D4×D4714yes—
268C22×C4Q8×D4714yes—
269C22×C4C23×D4712yes—
270C22×C4C23×D4711yescharacteristic
271M16—216no—
272M16—9432no—
273M16—9416no—
274C2×C8—28no—
275C2×C8—478no—
276C2×C8—9432no—
277C2×C8—948no—
278C2×C8C23×D4718yes—
279C4⋊C4—4732no—
280C22⋊C4—4732no—
281C42—478no—
282C42—474no—
283C42—474no—
284C42C2×C4×D4718yes—
285C42C2×C4×D4714yes—
286C42C23×D4714yes—
287C23—24no—
288C23—478no—
289C23—474no—
290C23—4716no—
291C23—948no—
292C23—18816no—
293C23—1888no—
294C23C4⋊C4×D4714yes—
295C23C22×C4×D4712yes—
296C23C22×C4×D4711yescharacteristic
297C2×C4—28no—
298C2×C4—48no—
299C2×C4—4716no—
300C2×C4—478no—
301C2×C4—474no—
302C2×C4—4716no—
303C2×C4—4732no—
304C2×C4—478no—
305C2×C4—478no—
306C2×C4—478no—
307C2×C4—948no—
308C2×C4—1888no—
309C2×C4C4⋊C4×D4718yes—
310C2×C4C4⋊C4×D4714yes—
311C2×C4C22×C4×D4718yes—
312C2×C4C22×C4×D4712yes—
313C2×C4C22×C4×D4711yescharacteristic
314C2×C4C2×D4×D4712yes—
315C2×C4C2×D4×D4714yes—
316C2×C4C2×Q8×D4714yes—
317C2×C4C2×Q8×D4712yes—
318C2×C4C24×D4711yescharacteristic
319C8—28no—
320C8—948no—
321C22—28no—
322C22—24no—
323C22—478no—
324C22—9416no—
325C22—948no—
326C22—18816no—
327C22C2×C4⋊C4×D4714yes—
328C22C2×C4⋊C4×D4712yes—
329C22#1568988914yes—
330C22C23×C4×D4711yescharacteristic
331C4—28no—
332C4—478no—
333C4—948no—
334C4C2×C4⋊C4×D4714yes—
335C4C2×C4⋊C4×D4712yes—
336C4C22×D4×D4711yescharacteristic
337C4C22×Q8×D4711yescharacteristic
338C2—24no—
339C2—948no—
340C2#2025195014yesdirect product factor
341C2C22×C4⋊C4×D4711yescharacteristic
342C2C2×#1568988912yes—
subgroupG/H
centerC22×C4C23×D47
commutatorC22×C47C24×C4
FrattiniC2×C4C24×D47
FittingC2×C47×#423C2
socleC23×C47C23×C4
prime radicalC24×D47C2×C4
cosocleC2×C4×C47C25
hypercenterC2×#423D47
hypocenterC47C2×#2905
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
3. #20251950C2direct (N × H)
4. C2×#15689148C2semidirect
5. C2×#15689147C2semidirect
6. C2×#15689146C2semidirect
7. C2×C47×#423C2semidirect
21. #15689148C22semidirect
22. #15689147C22semidirect
23. #15689146C22semidirect
24. C2×C4⋊(C47⋊C8)C22semidirect
25. C2×Dic47⋊C8C22semidirect
28. C47×#423C22semidirect
29. C2×C4⋊C8×C47C22semidirect
58. C4⋊(C47⋊C8)C23semidirect
59. Dic47⋊C8C23semidirect
69. C4⋊C8×C47C23semidirect
173. C2×#423D47semidirect
175. C2×C47#2905semidirect
197. #423C2×D47semidirect
200. C2×C4⋊C8C2×D47semidirect
201. C47C2×#2905semidirect
241. C4⋊C8C22×D47semidirect
340. C2#20251950direct (N × H)

Maximal subgroups

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

subgroupindex
C22×M16×D472
C22×#130939812
#202519502
C2×#156891482
C2×#156891472
C2×#156891462
C2×C47×#4232
C2×#290547