Finite Group Databasebeta

← #26108014

Subgroups — #26108014

Order 12032.

706 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 131 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×#15689851C211yesmaximal, characteristic
2#20243384C211yesmaximal, characteristic
3#20242059C212yesmaximal
4#20241818C212yesmaximal
5C47×#1461C211yesmaximal, characteristic
6#15689851—24no—
7#15689851—24no—
8C2×#13094024C2211yescharacteristic
9C2×#13094022—21no—
10C2×#13094016C412yes—
11C2×#13094014—22no—
12C2×C4×Dic94C412yes—
13C2×C4×Dic94C2211yescharacteristic
14C2×#13093989—21no—
15#15689147C2212yes—
16C2×#13093943—21no—
17#15688892—21no—
18#15688868—24no—
19#15688855—24no—
20C2×C4×Q8×C47C2211yescharacteristic
21C47×#431C2212yes—
22C47×#324—22no—
23C22×Dic94—21no—
24C22×Dic94C2×C411yescharacteristic
25C2×#10625329—22no—
26#13094024—22no—
27#13094024D414yes—
28#13094022—28no—
29#13094016—24no—
30#13094016—24no—
31#13094014—216no—
32C4×Dic94—24no—
33C4×Dic94—22no—
34C4×Dic94—24no—
35C4×Dic94D414yes—
36#13093989—28no—
37C2×C4⋊Dic47—21no—
38C2×C4⋊Dic47C2×C412yes—
39C2×C4⋊Dic47C2×C411yescharacteristic
40C2×Dic47⋊C4—22no—
41C2×Dic47⋊C4—22no—
42C2×Dic47⋊C4—22no—
43C2×Dic47⋊C4—21no—
44C2×C4×Dic47—22no—
45C2×C4×Dic47—21no—
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—
59C2×Q8×C47—22no—
60C2×Q8×C47—22no—
61C2×Q8×C47—21no—
62C2×Q8×C47—21no—
63C22×C4×C47C22⋊C412yes—
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—22no—
71C2×Dic94—21no—
72C2×Dic94—24no—
73C2×Dic94—24no—
74C2×Dic94—42no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94C22⋊C412yes—
78C2×C8×C47—42no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—24no—
81C42×C47—22no—
82C42×C47—22no—
83C42×C47C2×D412yes—
84C42×C47C2×D412yes—
85C4⋊Dic47—21no—
86C4⋊Dic47—22no—
87C4⋊Dic47—24no—
88C4⋊Dic47C22⋊C414yes—
89C4⋊Dic47C22⋊C412yes—
90Dic47⋊C4—28no—
91Dic47⋊C4—28no—
92Dic47⋊C4—28no—
93Dic47⋊C4—24no—
94C4×Dic47—28no—
95C4×Dic47—24no—
96#10625329—42no—
97#10625329—42no—
98C2×C47⋊C8—42no—
99C23×C47C2×C22⋊C411yescharacteristic
100Q8×C47—42no—
101Q8×C47—42no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—41no—
108C2×C4×C47—42no—
109C2×C4×C47C2×C22⋊C411yescharacteristic
110C2×C4×C47C2×C22⋊C411yescharacteristic
111C2×C4×C47#12012yes—
112C2×C4×C47#12012yes—
113C2×C4×C47#13512yes—
114C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C47#13711yescharacteristic
116C2×Dic47—24no—
117C2×Dic47—22no—
118C2×Dic47—22no—
119C2×Dic47—24no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123Dic94—24no—
124Dic94—22no—
125Dic94—44no—
126Dic94—44no—
127Dic94#10114yescomplemented
128C8×C47—82no—
129C47⋊C8—82no—
130#8390—471nomaximal, Sylow
131C22×C47—21no—
132C22×C47—21no—
133C22×C47#37911yescharacteristic
134C22×C47#41611yescharacteristic
135C22×C47#41711yescharacteristic
136C4×C47—22no—
137C4×C47—21no—
138C4×C47—42no—
139C4×C47—42no—
140C4×C47C2×#10112yes—
141C4×C47#44912yes—
142C4×C47#45212yes—
143Dic47—24no—
144Dic47—42no—
145Dic47—44no—
146C2×#557—471no—
147#1801—471no—
148#1546—472no—
149#1487—472no—
150#1461D4711yescharacteristic
151C2×C47—21no—
152C2×C47—21no—
153C2×C47#178411yescharacteristic
154C2×C47#178811yescharacteristic
155C2×C47#179611yescharacteristic
156#557—944no—
157#557—944no—
158C2×C4⋊Q8—471no—
159C2×C4⋊Q8—941no—
160C2×#125—472no—
161C2×#125—942no—
162C2×C4×Q8—472no—
163C2×C4×Q8—471no—
164C2×C4×Q8—941no—
165C2×C4×Q8C2×D4711yescharacteristic
166#431C2×D4712yes—
167#423—472no—
168C2×#101—941no—
169#343—941no—
170#332—944no—
171Q8⋊C8—944no—
172#324—22no—
173C47#839011yescharacteristic, complemented, Sylow
174C22×Q8—471no—
175C22×Q8—941no—
176C22×Q8D18811yescharacteristic
177C2×M16—942no—
178C4⋊Q8—474no—
179C4⋊Q8—948no—
180C4⋊Q8—942no—
181#125—944no—
182#125—944no—
183#125—9416no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—474no—
187C4×Q8—944no—
188C4×Q8—944no—
189C4×Q8—948no—
190C4×Q8—942no—
191C2×C4⋊C4—471no—
192C2×C4⋊C4—472no—
193C2×C4⋊C4—942no—
194C2×C4⋊C4—942no—
195C2×C4⋊C4—942no—
196C2×C4⋊C4—941no—
197C2×C4⋊C4—941no—
198C2×C4⋊C4D18811yescharacteristic
199C2×C4⋊C4C22⋊D4712yes—
200C2×C42—942no—
201C2×C42—941no—
202C2×C42C4×D4712yes—
203C2×C42C22×D4711yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—942no—
206C4⋊C8—942no—
207#101—1882no—
208C22⋊C8—22no—
209C4×C8—22no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—472no—
215C2×Q8—944no—
216C2×Q8—941no—
217C2×Q8—944no—
218C2×Q8—1882no—
219C2×Q8—1881no—
220C2×Q8—1881no—
221C22×C4—941no—
222C22×C4—941no—
223C22×C4—942no—
224C22×C4(C2×D47)⋊C412yes—
225C22×C4(C2×D47)⋊C412yes—
226C22×C4C2×C4×D4711yescharacteristic
227C22×C4C2×D18811yescharacteristic
228C22×C4C2×C22⋊D4711yescharacteristic
229M16—1882no—
230M16—1882no—
231C2×C8—42no—
232C2×C8—1882no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—472no—
236C4⋊C4—474no—
237C4⋊C4—948no—
238C4⋊C4—948no—
239C4⋊C4—948no—
240C4⋊C4—941no—
241C4⋊C4—942no—
242C4⋊C4—944no—
243C4⋊C4—944no—
244C42—22no—
245C42—22no—
246C42—948no—
247C42—944no—
248C42D4×D4712yes—
249C42D4×D4712yes—
250C23C2×(C2×D47)⋊C411yescharacteristic
251Q8—42no—
252Q8—42no—
253Q8—474no—
254Q8—944no—
255Q8—942no—
256Q8—1884no—
257Q8—1884no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—944no—
266C2×C4—942no—
267C2×C4—942no—
268C2×C4—944no—
269C2×C4—1882no—
270C2×C4—1882no—
271C2×C4—1882no—
272C2×C4C22⋊C4×D4711yescharacteristic
273C2×C4C22⋊C4×D4711yescharacteristic
274C2×C4#1309399212yes—
275C2×C4#1309399412yes—
276C2×C4#1309404612yes—
277C2×C4#1309405211yescharacteristic
278C2×C4#1309405711yescharacteristic
279C8—82no—
280C8—3762no—
281C22—21no—
282C22—21no—
283C22#1568897811yescharacteristic
284C22#1568912011yescharacteristic
285C22#1568912511yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—944no—
291C4—1884no—
292C4—1882no—
293C4#101×D4712yes—
294C4#1568927012yes—
295C4#1568928712yes—
296C2—21no—
297C2—21no—
298C2#2024331011yescharacteristic
299C2#2024332911yescharacteristic
300C2#2024336111yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1461C2
socleC22×C47#379
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1461D47
hypocenterC47#8390
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
127. Dic94#101semidirect
173. C47#8390semidirect

Maximal subgroups

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

subgroupindex
C2×#156898512
#202433842
#202420592
#202418182
C47×#14612
#839047