Finite Group Databasebeta

← #26108034

Subgroups — #26108034

Order 12032.

738 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689846C211yesmaximal, characteristic
2#20243384C211yesmaximal, characteristic, complemented
3#20242068C212yesmaximal, complemented
4#20241814C212yesmaximal
5C47×#1461C211yesmaximal, characteristic, complemented
6#15689846—24no—
7#15689846—24no—
8C2×#13094024C2211yescharacteristic
9C2×#13094019—22no—
10C2×#13094012C412yes—
11C2×#13094000—21no—
12C2×C4×Dic94C412yes—
13C2×C4×D188C2211yescharacteristic
14C2×#13093982—21no—
15#15689147C2212yes—
16C2×#13093943—21no—
17#15688892—21no—
18#15688871—24no—
19#15688852—24no—
20C2×C4×Q8×C47C2211yescharacteristic
21C47×#431C2212yes—
22C47×#324—22no—
23C22×Dic94C2×C412yes—
24C22×D188C2×C411yescharacteristic
25C22×C4×D47—21no—
26C2×#10625329—22no—
27#13094024—22no—
28#13094024D414yes—
29#13094019—216no—
30#13094012—24no—
31#13094012—24no—
32#13094000—28no—
33C4×Dic94—24no—
34C4×Dic94—24no—
35C4×D188—22no—
36C4×D188D414yes—
37#13093982—28no—
38C2×C4⋊Dic47—21no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47C2×C411yescharacteristic
41C2×Dic47⋊C4—22no—
42C2×(C2×D47)⋊C4—22no—
43C2×(C2×D47)⋊C4—22no—
44C2×(C2×D47)⋊C4—21no—
45C2×C4×Dic47—22no—
46C4⋊(C47⋊C8)—22no—
47C4⋊(C47⋊C8)—22no—
48#13093943—42no—
49C22×Q8×C47D411yescharacteristic
50C4×Q8×C47—22no—
51C4×Q8×C47—22no—
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C4⋊C4×C47D412yes—
54C2×C42×C47C2×C412yes—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—22no—
57C22⋊C8×C47—22no—
58C4×C8×C47—22no—
59C23×D47—21no—
60C2×Q8×C47—22no—
61C2×Q8×C47—22no—
62C2×Q8×C47—21no—
63C2×Q8×C47—21no—
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—21no—
70C22×Dic47—22no—
71C2×D188—21no—
72C2×D188—42no—
73C2×D188C22⋊C412yes—
74C2×C4×D47—24no—
75C2×C4×D47—42no—
76C2×C4×D47—41no—
77C2×C4×D47—41no—
78C2×Dic94—22no—
79C2×Dic94—44no—
80C2×Dic94C22⋊C414yes—
81C2×C8×C47—42no—
82C4⋊C4×C47—22no—
83C4⋊C4×C47—24no—
84C42×C47—22no—
85C42×C47—22no—
86C42×C47C2×D412yes—
87C42×C47C2×D412yes—
88(C2×D47)⋊C4—28no—
89(C2×D47)⋊C4—28no—
90(C2×D47)⋊C4—24no—
91C4⋊Dic47—21no—
92C4⋊Dic47—24no—
93C4⋊Dic47—24no—
94C4⋊Dic47C22⋊C412yes—
95Dic47⋊C4—28no—
96C4×Dic47—28no—
97#10625329—42no—
98#10625329—42no—
99C2×C47⋊C8—42no—
100C23×C47C2×C22⋊C411yescharacteristic
101C22×D47—22no—
102C22×D47—42no—
103C22×D47—42no—
104C22×D47—41no—
105C22×D47—41no—
106Q8×C47—42no—
107Q8×C47—42no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47C2×C22⋊C411yescharacteristic
116C2×C4×C47C2×C22⋊C411yescharacteristic
117C2×C4×C47#12012yes—
118C2×C4×C47#12012yes—
119C2×C4×C47#13512yes—
120C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C47#13711yescharacteristic
122C2×Dic47—22no—
123C2×Dic47—24no—
124C2×Dic47—44no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127D188—24no—
128D188—42no—
129C4×D47—44no—
130C4×D47—44no—
131Dic94—44no—
132Dic94—44no—
133C8×C47—82no—
134C47⋊C8—82no—
135#8394—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47#37911yescharacteristic
139C22×C47#41611yescharacteristic
140C22×C47#41711yescharacteristic
141C2×D47—42no—
142C2×D47—42no—
143C2×D47—42no—
144C2×D47—82no—
145C2×D47—82no—
146C4×C47—22no—
147C4×C47—21no—
148C4×C47—42no—
149C4×C47—42no—
150C4×C47#41312yes—
151C4×C47#44912yes—
152C4×C47#45212yes—
153Dic47—44no—
154Dic47—42no—
155C2×#556—471no—
156#1801—471no—
157#1548—472no—
158#1486—472no—
159#1461D4711yescharacteristic, complemented
160C2×C47—21no—
161C2×C47—21no—
162C2×C47#178511yescharacteristic
163C2×C47#178811yescharacteristic
164C2×C47#179611yescharacteristic
165D47—82no—
166#556—944no—
167#556—944no—
168C2×C4⋊Q8—471no—
169C2×#126—942no—
170C2×#124—472no—
171C2×C22⋊Q8—941no—
172C2×C4×Q8—472no—
173C2×C4×Q8C2×D4711yescharacteristic
174C2×C4×D4—471no—
175C2×#117—941no—
176#431C2×D4712yes—
177#423—472no—
178C2×#101—941no—
179#343—941no—
180#333—944no—
181D4⋊C8—944no—
182#324—22no—
183C47#839411yescharacteristic, complemented, Sylow
184C22×Q8—472no—
185C22×Q8D18811yescharacteristic
186C22×D4—471no—
187C23×C4—941no—
188C2×M16—942no—
189C4⋊Q8—474no—
190C4⋊Q8—942no—
191#126—9416no—
192#124—944no—
193#124—944no—
194C22⋊Q8—948no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—944no—
198C4×Q8—944no—
199C4×D4—474no—
200C4×D4—942no—
201#117—948no—
202C2×C4⋊C4—471no—
203C2×C4⋊C4—942no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4—941no—
206C2×C4⋊C4D18811yescharacteristic
207C2×C4⋊C4C22⋊D4712yes—
208C2×C22⋊C4—942no—
209C2×C22⋊C4—942no—
210C2×C22⋊C4—941no—
211C2×C42—942no—
212C2×C42C4×D4712yes—
213C2×C42C22×D4711yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—942no—
216C4⋊C8—942no—
217#101—1882no—
218C22⋊C8—22no—
219C4×C8—22no—
220C24—941no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—474no—
226C2×Q8—942no—
227C2×Q8—1884no—
228C2×D4—472no—
229C2×D4—941no—
230C2×D4—1882no—
231C22×C4—941no—
232C22×C4—942no—
233C22×C4—944no—
234C22×C4—1882no—
235C22×C4—1881no—
236C22×C4—1881no—
237C22×C4(C2×D47)⋊C412yes—
238C22×C4(C2×D47)⋊C412yes—
239C22×C4C2×C4×D4711yescharacteristic
240C22×C4C2×D18811yescharacteristic
241C22×C4C2×C22⋊D4711yescharacteristic
242M16—1882no—
243M16—1882no—
244C2×C8—42no—
245C2×C8—1882no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—472no—
249C4⋊C4—948no—
250C4⋊C4—941no—
251C4⋊C4—944no—
252C4⋊C4—944no—
253C22⋊C4—948no—
254C22⋊C4—948no—
255C22⋊C4—944no—
256C42—22no—
257C42—22no—
258C42—948no—
259C42D4×D4712yes—
260C42D4×D4712yes—
261C23—942no—
262C23—1882no—
263C23—1882no—
264C23—1881no—
265C23—1881no—
266C23C2×(C2×D47)⋊C411yescharacteristic
267Q8—42no—
268Q8—42no—
269Q8—1884no—
270Q8—1884no—
271D4—944no—
272D4—1882no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—942no—
281C2×C4—944no—
282C2×C4—1884no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1884no—
286C2×C4—1884no—
287C2×C4C22⋊C4×D4711yescharacteristic
288C2×C4C22⋊C4×D4711yescharacteristic
289C2×C4#1309399212yes—
290C2×C4#1309399412yes—
291C2×C4#1309404612yes—
292C2×C4#1309405211yescharacteristic
293C2×C4#1309405711yescharacteristic
294C8—82no—
295C8—3762no—
296C22—21no—
297C22—21no—
298C22—3762no—
299C22—3762no—
300C22—1882no—
301C22—1882no—
302C22—1882no—
303C22#1568897811yescharacteristic
304C22#1568912011yescharacteristic
305C22#1568912511yescharacteristic
306C4—21no—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—1884no—
311C4—1882no—
312C4#1568910612yes—
313C4#1568927012yes—
314C4#1568928712yes—
315C2—21no—
316C2—21no—
317C2—3762no—
318C2#2024331511yescharacteristic
319C2#2024332911yescharacteristic
320C2#2024336111yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1461C2
socleC22×C47#379
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1461D47
hypocenterC47#8394
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #20243384C2semidirect
3. #20242068C2semidirect
5. C47×#1461C2semidirect
159. #1461D47semidirect
183. C47#8394semidirect

Maximal subgroups

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

subgroupindex
C2×#156898462
#202433842
#202420682
#202418142
C47×#14612
#839447