Finite Group Databasebeta

← #26131821

Subgroups — #26131821

Order 12032.

952 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#20245077C214yesmaximal, complemented
2#20243312C212yesmaximal, complemented
3C47×#3486C211yesmaximal, characteristic
4C2×#13093943—22no—
5C2×#13093943C2214yes—
6#15689087—22no—
7#15688922—24no—
8#15688920—24no—
9C2×2−1+4×C47C411yescharacteristic
10C23×Q8×C47C2211yescharacteristic
11D4×Q8×C47—24no—
12C47×#548—24no—
13C47×#536—22no—
14C47×#536C2212yes—
15C47×#525—22no—
16C47×#525C412yes—
17C47×#519C411yescharacteristic
18C2×#10625329—24no—
19#13093943—28no—
20#13093943—44no—
21C22⋊(C47⋊C8)—44no—
222−1+4×C47—44no—
23C2×#41×C47—22no—
24C2×#41×C47—22no—
25C2×#41×C47C2×C412yes—
26C22×Q8×C47—22no—
27C22×Q8×C47—24no—
28C22×Q8×C47—44no—
29C22×Q8×C47C2×C411yescharacteristic
30C22×Q8×C47D412yes—
31C22×Q8×C47C2311yescharacteristic
32C23×C4×C47C2×C411yescharacteristic
33C23×C4×C47D412yes—
34C4⋊Q8×C47—24no—
35C4⋊Q8×C47D414yes—
36#124×C47—24no—
37#124×C47D414yes—
38#123×C47—22no—
39#123×C47—24no—
40C22⋊Q8×C47—24no—
41C22⋊Q8×C47—28no—
42C22⋊Q8×C47—22no—
43C22⋊Q8×C47—24no—
44#121×C47—24no—
45#121×C47—22no—
46#120×C47—22no—
47C4×Q8×C47—24no—
48C4×D4×C47—28no—
49C4×D4×C47—24no—
50#117×C47—22no—
51#117×C47C2×C412yes—
52C24×C47C22⋊C411yescharacteristic
53#41×C47—44no—
54#41×C47—44no—
55#41×C47—48no—
56#41×C47—44no—
57C2×Q8×C47—22no—
58C2×Q8×C47—24no—
59C2×Q8×C47—24no—
60C2×Q8×C47—22no—
61C2×Q8×C47—24no—
62C2×Q8×C47—42no—
63C2×Q8×C47—42no—
64C2×Q8×C47—416no—
65C2×Q8×C47—48no—
66C2×Q8×C47C22⋊C412yes—
67C2×Q8×C47C2×D414yes—
68C2×D4×C47—24no—
69C2×D4×C47—22no—
70C2×D4×C47—22no—
71C2×D4×C47C22⋊C412yes—
72C22×C4×C47—24no—
73C22×C4×C47—24no—
74C22×C4×C47—22no—
75C22×C4×C47—22no—
76C22×C4×C47—24no—
77C22×C4×C47—42no—
78C22×C4×C47—42no—
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22⋊C412yes—
81C22×C4×C47C22×C411yescharacteristic
82C22×C4×C47C2×D412yes—
83C4⋊C4×C47—24no—
84C4⋊C4×C47—24no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—28no—
87C22⋊C4×C47—24no—
88C22⋊C4×C47—24no—
89C22⋊C4×C47—22no—
90C22⋊C4×C47—28no—
91C42×C47—24no—
92C42×C47C22⋊C414yes—
93#10625329—44no—
94#10625329—44no—
95C2×C47⋊C8—44no—
96C23×C47—21no—
97C23×C47—22no—
98C23×C47—22no—
99C23×C47—42no—
100C23×C47C2×C22⋊C411yescharacteristic
101Q8×C47—28no—
102Q8×C47—44no—
103Q8×C47—48no—
104Q8×C47—42no—
105Q8×C47—44no—
106D4×C47—42no—
107D4×C47—44no—
108D4×C47—48no—
109D4×C47—44no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—24no—
116C2×C4×C47—21no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—48no—
120C2×C4×C47—44no—
121C2×C4×C47—48no—
122C2×C4×C47—44no—
123C2×C4×C47#9914yes—
124C2×C4×C47C2×C22⋊C412yes—
125C2×C4×C47#12014yes—
126C47⋊C8—84no—
127#12710—471nomaximal, Sylow
128C22×C47—22no—
129C22×C47—24no—
130C22×C47—41no—
131C22×C47—41no—
132C22×C47—42no—
133C22×C47—44no—
134C22×C47—41no—
135C22×C47—42no—
136C22×C47#37911yescharacteristic
137C22×C47C2×#9912yes—
138C4×C47—24no—
139C4×C47—24no—
140C4×C47—22no—
141C4×C47—44no—
142C4×C47—41no—
143C4×C47—42no—
144#3486D4711yescharacteristic
145#2133—474no—
146#1784—472no—
147C2×C47—21no—
148C2×C47—22no—
149C2×C47—42no—
150C2×C47—82no—
151C2×C47#178111yescharacteristic
152C2×2−1+4Dic4711yescharacteristic
153C23×Q8C2×D4711yescharacteristic
154D4×Q8—24no—
155#548—24no—
156#536—22no—
157#536C2×D4712yes—
158#525—22no—
159#525Dic4712yes—
160#519Dic4711yescharacteristic
161C2×#101—474no—
162C2×#101—942no—
163#407—942no—
164#356—944no—
165#355—944no—
166C47#1271011yescharacteristic, complemented, Sylow
1672−1+4—44no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41C2×Dic4712yes—
171C22×Q8—22no—
172C22×Q8—24no—
173C22×Q8—44no—
174C22×Q8C2×Dic4711yescharacteristic
175C22×Q8C22⋊D4712yes—
176C22×Q8C22×D4711yescharacteristic
177C23×C4C2×Dic4711yescharacteristic
178C23×C4C22⋊D4712yes—
179C2×M16—944no—
180C4⋊Q8—24no—
181C4⋊Q8C22⋊D4714yes—
182#124—24no—
183#124C22⋊D4714yes—
184#123—22no—
185#123—24no—
186C22⋊Q8—24no—
187C22⋊Q8—28no—
188C22⋊Q8—22no—
189C22⋊Q8—24no—
190#121—24no—
191#121—22no—
192#120—22no—
193C4×Q8—24no—
194C4×D4—28no—
195C4×D4—24no—
196#117—22no—
197#117C2×Dic4712yes—
198#101—948no—
199#101—1884no—
200C22⋊C8—1884no—
201C24C22⋊Dic4711yescharacteristic
202#41—44no—
203#41—44no—
204#41—48no—
205#41—44no—
206C2×Q8—22no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—22no—
210C2×Q8—24no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—416no—
214C2×Q8—48no—
215C2×Q8C22⋊Dic4712yes—
216C2×Q8C2×C22⋊D4714yes—
217C2×D4—24no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4C22⋊Dic4712yes—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C22⋊Dic4711yescharacteristic
229C22×C4C22⋊Dic4712yes—
230C22×C4C22×Dic4711yescharacteristic
231C22×C4C2×C22⋊D4712yes—
232M16—1884no—
233M16—1884no—
234C2×C8—1884no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—28no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—22no—
242C22⋊C4—28no—
243C42—24no—
244C42C22⋊Dic4714yes—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—42no—
249C23C2×C22⋊Dic4711yescharacteristic
250Q8—28no—
251Q8—44no—
252Q8—48no—
253Q8—42no—
254Q8—44no—
255D4—42no—
256D4—44no—
257D4—48no—
258D4—44no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—48no—
269C2×C4—44no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4#1309393914yes—
273C2×C4C2×C22⋊Dic4712yes—
274C2×C4#1309399314yes—
275C8—3764no—
276C22—22no—
277C22—24no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—44no—
282C22—41no—
283C22—42no—
284C22#1568897911yescharacteristic
285C22C2×#1309393912yes—
286C4—24no—
287C4—24no—
288C4—22no—
289C4—44no—
290C4—41no—
291C4—42no—
292C2—21no—
293C2—22no—
294C2—42no—
295C2—82no—
296C2#2024330311yescharacteristic
subgroupG/H
centerC2#20243303
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
Fitting, prime radicalC47×#3486C2
socleC2×C47#1781
cosocleC22×Q8×C47C23
hypercenter#3486D47
hypocenterC47#12710
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C23 ◁ C23×Q8 ◁ #3486

Complements

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

NH ≅ G/Ntype
1. #20245077C2semidirect
2. #20243312C2semidirect
166. C47#12710semidirect

Maximal subgroups

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

subgroupindex
#202450772
#202433122
C47×#34862
#1271047