Finite Group Databasebeta

← #26108047

Subgroups — #26108047

Order 12032.

738 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 147 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×Q8×Dic94C211yesmaximal, characteristic
2#20242074C212yesmaximal
3#20241816C212yesmaximal
4#20241712C211yesmaximal, characteristic
5C47×#1801C211yesmaximal, characteristic
6Q8×Dic94—24no—
7Q8×Dic94—24no—
8C2×#13094024C412yes—
9C2×#13094024C2211yescharacteristic
10C2×#13094023—22no—
11C2×#13094023—21no—
12C2×Q8×Dic47—21no—
13C2×C4×Dic94C412yes—
14C2×C4×Dic94C2211yescharacteristic
15#15689176C2212yes—
16#15688868—24no—
17#15688853—24no—
18#15688849—22no—
19C2×C4×Q8×C47C2211yescharacteristic
20C47×#423C2212yes—
21C2×#101×C47—21no—
22C47×#343—21no—
23C22×Dic94—22no—
24C22×Dic94—21no—
25C22×Dic94C2×C411yescharacteristic
26#13094024—22no—
27#13094024—24no—
28#13094024—24no—
29#13094024D414yes—
30#13094023—216no—
31#13094023—28no—
32Q8×Dic47—28no—
33C4×Dic94—24no—
34C4×Dic94—22no—
35C4×Dic94—24no—
36C4×Dic94D414yes—
37C2×C4⋊Dic47—22no—
38C2×C4⋊Dic47—21no—
39C2×C4⋊Dic47C2×C412yes—
40C2×C4⋊Dic47C2×C411yescharacteristic
41C2×Dic47⋊C4—22no—
42C2×Dic47⋊C4—21no—
43C2×C4×Dic47—22no—
44C2×C4×Dic47—21no—
45C4⋊(C47⋊C8)—22no—
46C22⋊(C47⋊C8)—22no—
47C4×C47⋊C8—22no—
48C22×Q8×C47D411yescharacteristic
49C2×M16×C47—22no—
50C4×Q8×C47—22no—
51C4×Q8×C47—22no—
52C2×C4⋊C4×C47D412yes—
53C2×C4⋊C4×C47D411yescharacteristic
54C2×C42×C47C2×C412yes—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—22no—
57C4⋊C8×C47—22no—
58#101×C47—42no—
59C2×Q8×C47—22no—
60C2×Q8×C47—22no—
61C2×Q8×C47SD1612yes—
62C2×Q8×C47Q1612yes—
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—22no—
70C22×Dic47—21no—
71C2×Dic94—21no—
72C2×Dic94—24no—
73C2×Dic94—28no—
74C2×Dic94—24no—
75C2×Dic94—82no—
76C2×Dic94—84no—
77C2×Dic94C22⋊C412yes—
78M16×C47—42no—
79M16×C47—42no—
80C2×C8×C47—42no—
81C4⋊C4×C47—24no—
82C4⋊C4×C47—22no—
83C42×C47—22no—
84C42×C47—22no—
85C42×C47C2×D412yes—
86C42×C47C2×D412yes—
87C4⋊Dic47—28no—
88C4⋊Dic47—21no—
89C4⋊Dic47—22no—
90C4⋊Dic47—24no—
91C4⋊Dic47C22⋊C414yes—
92C4⋊Dic47C22⋊C412yes—
93Dic47⋊C4—28no—
94Dic47⋊C4—24no—
95C4×Dic47—28no—
96C4×Dic47—24no—
97C2×C47⋊C8—42no—
98C23×C47C2×C22⋊C411yescharacteristic
99Q8×C47—24no—
100Q8×C47—24no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—41no—
106C2×C4×C47—42no—
107C2×C4×C47#10314yes—
108C2×C4×C47C2×C22⋊C411yescharacteristic
109C2×C4×C47C2×C22⋊C411yescharacteristic
110C2×C4×C47#12012yes—
111C2×C4×C47#12012yes—
112C2×C4×C47C2×SD1611yescharacteristic
113C2×C4×C47C2×Q1611yescharacteristic
114C2×C4×C47#13712yes—
115C2×Dic47—24no—
116C2×Dic47—22no—
117C2×Dic47—24no—
118C2×Dic47—22no—
119C2×Dic47—44no—
120C2×Dic47—42no—
121Dic94—24no—
122Dic94—22no—
123Dic94—84no—
124Dic94—88no—
125Dic94#10114yescomplemented
126C8×C47—82no—
127C47⋊C8—82no—
128#8396—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47#37911yescharacteristic
132C22×C47C2×#10311yescharacteristic
133C22×C47#41911yescharacteristic
134C4×C47—22no—
135C4×C47—21no—
136C4×C47—24no—
137C4×C47—42no—
138C4×C47C2×#10112yes—
139C4×C47#44812yes—
140C4×C47#45112yes—
141Dic47—24no—
142Dic47—44no—
143Dic47—42no—
144C2×Q82—471no—
145#1801D4711yescharacteristic
146#1549—472no—
147#1487—472no—
148#1461—471no—
149C2×C47—21no—
150C2×C47—21no—
151C2×C47#178411yescharacteristic
152C2×C47#178711yescharacteristic
153C2×C47#179511yescharacteristic
154Q82—944no—
155Q82—944no—
156C2×C4⋊Q8—472no—
157C2×C4⋊Q8—471no—
158C2×C4⋊Q8—942no—
159C2×C4⋊Q8—941no—
160C2×C4×Q8—472no—
161C2×C4×Q8—471no—
162C2×C4×Q8—941no—
163C2×C4×Q8C2×D4711yescharacteristic
164#431—472no—
165#423C2×D4712yes—
166C2×#101—21no—
167#343—21no—
168#332—944no—
169Q8⋊C8—944no—
170#324—942no—
171C47#839611yescharacteristic, complemented, Sylow
172C22×Q8—471no—
173C22×Q8—942no—
174C22×Q8—941no—
175C22×Q8C22⋊D4711yescharacteristic
176C2×M16—22no—
177C4⋊Q8—474no—
178C4⋊Q8—944no—
179C4⋊Q8—944no—
180C4⋊Q8—9416no—
181C4⋊Q8—948no—
182C4⋊Q8—942no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—474no—
186C4×Q8—944no—
187C4×Q8—944no—
188C4×Q8—948no—
189C4×Q8—942no—
190C2×C4⋊C4—471no—
191C2×C4⋊C4—472no—
192C2×C4⋊C4—942no—
193C2×C4⋊C4—942no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4D18812yes—
197C2×C4⋊C4C22⋊D4711yescharacteristic
198C2×C42—942no—
199C2×C42—941no—
200C2×C42C4×D4712yes—
201C2×C42C22×D4711yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—942no—
205#101—42no—
206C22⋊C8—942no—
207C4×C8—942no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—472no—
211C2×Q8—944no—
212C2×Q8—948no—
213C2×Q8—941no—
214C2×Q8—944no—
215C2×Q8—3762no—
216C2×Q8—3764no—
217C2×Q8#1062533512yes—
218C2×Q8#1062533712yes—
219C22×C4—941no—
220C22×C4—942no—
221C22×C4—941no—
222C22×C4(C2×D47)⋊C412yes—
223C22×C4(C2×D47)⋊C412yes—
224C22×C4C2×C4×D4711yescharacteristic
225C22×C4C2×D18811yescharacteristic
226C22×C4C2×C22⋊D4711yescharacteristic
227M16—42no—
228M16—42no—
229C2×C8—42no—
230C2×C8—1882no—
231C4⋊C4—22no—
232C4⋊C4—24no—
233C4⋊C4—472no—
234C4⋊C4—474no—
235C4⋊C4—948no—
236C4⋊C4—948no—
237C4⋊C4—941no—
238C4⋊C4—942no—
239C4⋊C4—944no—
240C4⋊C4—944no—
241C42—22no—
242C42—22no—
243C42—948no—
244C42—944no—
245C42D4×D4712yes—
246C42D4×D4712yes—
247C23C2×(C2×D47)⋊C411yescharacteristic
248Q8—24no—
249Q8—24no—
250Q8—474no—
251Q8—944no—
252Q8—942no—
253Q8—3764no—
254Q8—3768no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—944no—
262C2×C4—942no—
263C2×C4—944no—
264C2×C4—942no—
265C2×C4—1884no—
266C2×C4—1882no—
267C2×C4#1309394714yes—
268C2×C4C22⋊C4×D4711yescharacteristic
269C2×C4C22⋊C4×D4711yescharacteristic
270C2×C4#1309399212yes—
271C2×C4#1309399412yes—
272C2×C4C2×#1062533511yescharacteristic
273C2×C4C2×#1062533711yescharacteristic
274C2×C4#1309405412yes—
275C8—82no—
276C8—3762no—
277C22—21no—
278C22—21no—
279C22#1568897811yescharacteristic
280C22C2×#1309394711yescharacteristic
281C22#1568913111yescharacteristic
282C4—21no—
283C4—24no—
284C4—22no—
285C4—42no—
286C4—944no—
287C4—1884no—
288C4—1882no—
289C4#101×D4712yes—
290C4#1568926112yes—
291C4#1568927812yes—
292C2—21no—
293C2—21no—
294C2#2024331011yescharacteristic
295C2#2024332211yescharacteristic
296C2#2024335911yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1801C2
socleC22×C47#379
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1801D47
hypocenterC47#8396
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
125. Dic94#101semidirect
171. C47#8396semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic942
#202420742
#202418162
#202417122
C47×#18012
#839647