Finite Group Databasebeta

← C2×#20242996 #26203949

Subgroups — C2×#20242996 #26203949

Order 12032.

750 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
1C2×#15689526C211yesmaximal, characteristic
2C2×#15689526C211yesmaximal, characteristic
3C2×#15689475C211yesmaximal, characteristic
4C2×#15689474C211yesmaximal, characteristic
5C2×#15689473C211yesmaximal, characteristic
6#20242996C218yesmaximal, direct product factor
7C2×C8×C47⋊C8C211yesmaximal, characteristic
8C2×C47×#487C211yesmaximal, characteristic
9C2×#13094024C2211yescharacteristic
10C2×#13094024C2211yescharacteristic
11#15689526—22no—
12#15689526—22no—
13#15689526C2214yes—
14#15689526C2214yes—
15#15689475—22no—
16#15689475C2214yes—
17#15689474—22no—
18#15689474C2214yes—
19#15689473—22no—
20#15689473C2214yes—
21C2×#13093957—22no—
22C2×#13093957—22no—
23C2×#13093949—22no—
24C2×#13093948—22no—
25C2×#13093947—22no—
26C2×C4×C47⋊C8C2211yescharacteristic
27C2×C4×C47⋊C8C2211yescharacteristic
28C8×C47⋊C8C2214yes—
29C2×C4⋊Q8×C47C2211yescharacteristic
30C2×C4⋊Q8×C47C2211yescharacteristic
31C47×#487—22no—
32C47×#487C2214yes—
33C2×#103×C47—22no—
34C2×C4×C8×C47C2211yescharacteristic
35C22×Dic94—21no—
36C22×Dic94—21no—
37C22×C47⋊C8D412yes—
38C22×C47⋊C8D412yes—
39#13094024—21no—
40#13094024—21no—
41#13094024—21no—
42#13094024—21no—
43#13094024—21no—
44#13094024—21no—
45#13094024C2312yes—
46#13094024C2312yes—
47C2×C4⋊Dic47—21no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47—21no—
50C2×C4⋊Dic47—21no—
51#13093957—28no—
52#13093957—28no—
53#13093949—28no—
54#13093948—28no—
55#13093947—28no—
56C4×C47⋊C8D412yes—
57C4×C47⋊C8D412yes—
58C4×C47⋊C8C2312yes—
59C4×C47⋊C8C2312yes—
60C22×Q8×C47—21no—
61C22×Q8×C47—21no—
62C22×C8×C47D412yes—
63C4⋊Q8×C47—21no—
64C4⋊Q8×C47—21no—
65C4⋊Q8×C47—21no—
66C4⋊Q8×C47—21no—
67C4⋊Q8×C47—21no—
68C4⋊Q8×C47—21no—
69C4⋊Q8×C47C2312yes—
70C4⋊Q8×C47C2312yes—
71C2×C4⋊C4×C47—21no—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47—21no—
74C2×C4⋊C4×C47—21no—
75C2×C42×C47C2311yescharacteristic
76#103×C47—28no—
77C4×C8×C47D412yes—
78C4×C8×C47C2312yes—
79C2×Q8×C47—22no—
80C2×Q8×C47—22no—
81C2×Q8×C47—41no—
82C2×Q8×C47—41no—
83C2×Q8×C47—44no—
84C2×Q8×C47—44no—
85C22×C4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×Dic47—41no—
91C22×Dic47—41no—
92C2×Dic94—22no—
93C2×Dic94—22no—
94C2×Dic94—44no—
95C2×Dic94—44no—
96C2×Dic94—41no—
97C2×Dic94—41no—
98C2×C8×C47—24no—
99C2×C8×C47C2×D414yes—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—41no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C42×C47C2×D411yescharacteristic
109C42×C47C2×D411yescharacteristic
110C42×C47C2×D411yescharacteristic
111C42×C47C2411yescharacteristic
112C4⋊Dic47—22no—
113C4⋊Dic47—22no—
114C4⋊Dic47—22no—
115C4⋊Dic47—22no—
116C4⋊Dic47—41no—
117C4⋊Dic47—41no—
118C4⋊Dic47—41no—
119C4⋊Dic47—41no—
120C2×C47⋊C8—24no—
121C2×C47⋊C8Q1618yes—
122C2×C47⋊C8C2×D414yes—
123C2×C47⋊C8C2×D414yes—
124C23×C47#12711yescharacteristic
125Q8×C47—44no—
126Q8×C47—44no—
127Q8×C47—82no—
128Q8×C47—82no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—82no—
132C2×C4×C47—82no—
133C2×C4×C47#12711yescharacteristic
134C2×C4×C47#12711yescharacteristic
135C2×C4×C47#12711yescharacteristic
136C2×C4×C47C2×Q1614yes—
137C2×C4×C47#13514yes—
138C2×C4×C47#13514yes—
139C2×C4×C47C22×D411yescharacteristic
140C2×C4×C47C22×D411yescharacteristic
141C2×C4×C47C22×D411yescharacteristic
142C2×Dic47—42no—
143C2×Dic47—42no—
144C2×Dic47—82no—
145C2×Dic47—82no—
146Dic94—44no—
147Dic94—44no—
148Dic94—82no—
149Dic94—82no—
150C8×C47—24no—
151C47⋊C8—24no—
152C47⋊C8C2×Q1618yes—
153C2×#1710—471nomaximal, Sylow
154C22×C47#49412yes—
155C22×C47#49512yes—
156C22×C47#49512yes—
157C22×C47C2×#12711yescharacteristic
158C4×C47—82no—
159C4×C47—82no—
160C4×C47#49412yes—
161C4×C47#49512yes—
162C4×C47#49512yes—
163C4×C47C22×Q1612yes—
164C4×C47C2×#13512yes—
165C4×C47C2×#13512yes—
166Dic47—82no—
167Dic47—82no—
168C2×#500—471no—
169C2×#500—471no—
170C2×#487—471no—
171C2×#487—471no—
172C2×#487—471no—
173C2×#487D4711yescharacteristic
174#1710—478no—
175C2×C82—471no—
176C2×C47#171014yescomplemented
177C2×C47C2×#49411yescharacteristic
178C2×C47C2×#49511yescharacteristic
179C2×C47C2×#49511yescharacteristic
180C2×C4⋊Q8—471no—
181C2×C4⋊Q8—471no—
182C2×C4⋊Q8C2×D4711yescharacteristic
183C2×C4⋊Q8C2×D4711yescharacteristic
184#500—474no—
185#500—474no—
186#500—942no—
187#500—942no—
188#487—22no—
189#487—474no—
190#487—474no—
191#487—474no—
192#487—942no—
193#487—942no—
194#487—942no—
195#487C2×D4714yes—
196C2×#107—942no—
197C2×#107—942no—
198C2×#103—22no—
199C2×#103—942no—
200C2×#103—942no—
201C2×#103—942no—
202C2×C4×C8—471no—
203C2×C4×C8—471no—
204C2×C4×C8C2×D4711yescharacteristic
205C82—474no—
206C47C2×#171011yescharacteristic, complemented, Sylow
207C22×Q8—21no—
208C22×Q8—21no—
209C22×Q8—941no—
210C22×Q8—941no—
211C22×C8—472no—
212C22×C8—472no—
213C22×C8C22⋊D4712yes—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—472no—
221C4⋊Q8—472no—
222C4⋊Q8—941no—
223C4⋊Q8—941no—
224C4⋊Q8—941no—
225C4⋊Q8—941no—
226C4⋊Q8—941no—
227C4⋊Q8—941no—
228C4⋊Q8C22×D4712yes—
229C4⋊Q8C22×D4712yes—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—941no—
235C2×C4⋊C4—941no—
236C2×C4⋊C4—941no—
237C2×C4⋊C4—941no—
238C2×C42C22×D4711yescharacteristic
239#107—948no—
240#107—948no—
241#103—28no—
242#103—948no—
243#103—948no—
244#103—948no—
245C4×C8—472no—
246C4×C8—472no—
247C4×C8—472no—
248C4×C8—472no—
249C4×C8C22⋊D4712yes—
250C4×C8C22×D4712yes—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—942no—
258C2×Q8—942no—
259C2×Q8—1884no—
260C2×Q8—1884no—
261C2×Q8—1881no—
262C2×Q8—1881no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—1881no—
266C22×C4—1881no—
267C22×C4D4×D4711yescharacteristic
268C22×C4D4×D4711yescharacteristic
269C22×C4C2×C22⋊D4711yescharacteristic
270C2×C8—24no—
271C2×C8—478no—
272C2×C8—474no—
273C2×C8—474no—
274C2×C8—944no—
275C2×C8C2×C22⋊D4714yes—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—942no—
285C4⋊C4—942no—
286C4⋊C4—942no—
287C4⋊C4—942no—
288C4⋊C4—1881no—
289C4⋊C4—1881no—
290C4⋊C4—1881no—
291C4⋊C4—1881no—
292C42D4×D4711yescharacteristic
293C42D4×D4711yescharacteristic
294C42C2×C22⋊D4711yescharacteristic
295C42C23×D4711yescharacteristic
296C23#1309402011yescharacteristic
297Q8—44no—
298Q8—44no—
299Q8—82no—
300Q8—82no—
301Q8—3762no—
302Q8—3762no—
303Q8—1884no—
304Q8—1884no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—3762no—
310C2×C4—3762no—
311C2×C4—1882no—
312C2×C4—1882no—
313C2×C4#1309402011yescharacteristic
314C2×C4#1309402011yescharacteristic
315C2×C4#1309402011yescharacteristic
316C2×C4Q16×D4714yes—
317C2×C4#1309404414yes—
318C2×C4#1309404514yes—
319C2×C4C2×D4×D4711yescharacteristic
320C2×C4C2×D4×D4711yescharacteristic
321C2×C4C22×C22⋊D4711yescharacteristic
322C8—24no—
323C8—478no—
324C8—944no—
325C22#1568950112yes—
326C22#1568950412yes—
327C22#1568950512yes—
328C22C2×#1309402011yescharacteristic
329C4—82no—
330C4—82no—
331C4—3762no—
332C4—3762no—
333C4#1568950112yes—
334C4#1568950412yes—
335C4#1568950512yes—
336C4C2×Q16×D4712yes—
337C4C2×#1309404412yes—
338C4C2×#1309404512yes—
339C2#2024299614yesdirect product factor
340C2C2×#1568950111yescharacteristic
341C2C2×#1568950411yescharacteristic
342C2C2×#1568950511yescharacteristic
subgroupG/H
centerC23#13094020
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#487C2
socle, prime radicalC23×C47#127
hypercenterC2×#487D47
hypocenterC47C2×#1710
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
6. #20242996C2direct (N × H)
176. C2×C47#1710semidirect
206. C47C2×#1710semidirect
339. C2#20242996direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156895262
C2×#156894752
C2×#156894742
C2×#156894732
#202429962
C2×C8×C47⋊C82
C2×C47×#4872
C2×#171047