Finite Group Databasebeta

← #26109751

Subgroups — #26109751

Order 12032.

656 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 99 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#20245879C211yesmaximal, characteristic
2#20243316C211yesmaximal, characteristic
3#20241240C214yesmaximal, complemented
4C47×#1782C211yesmaximal, characteristic
5C2×#13093983C411yescharacteristic, complemented
6C22×C4×Dic47C411yescharacteristic, complemented
7C2×#13093943—21no—
8C2×#13093941—21no—
9#15689087C2212yes—
10#15689030—21no—
11#15689026—21no—
12#15689003—21no—
13#15689003—21no—
14C4×C22⋊Dic47C412yescomplemented
15C4×C22⋊Dic47C2212yes—
16C22×#41×C47C2211yescharacteristic
17C2×#99×C47—21no—
18C2×#99×C47—21no—
19C47×#381C2212yes—
20C23×Dic47—21no—
21C2×#10625329—22no—
22#13093983—42no—
23C2×C4⋊Dic47—21no—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—21no—
26C2×C22⋊Dic47—22no—
27C2×C4×Dic47—21no—
28C2×C4×Dic47—42no—
29C2×C4×Dic47C2×C412yes—
30C2×C4×Dic47C2×C412yes—
31#13093943—42no—
32#13093941—42no—
33C22⋊(C47⋊C8)—24no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C2×#41×C47—24no—
37C2×#41×C47—42no—
38C2×#41×C47—42no—
39C22×Q8×C47D411yescharacteristic
40C22×D4×C47D411yescharacteristic
41C22×D4×C47D411yescharacteristic
42C22×D4×C47D411yescharacteristic
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47C2311yescharacteristic
46C2×C22⋊C4×C47—22no—
47#99×C47—42no—
48#99×C47—42no—
49#95×C47—24no—
50C24×C47C22⋊C411yescharacteristic
51C24×C47C22⋊C411yescharacteristic
52C24×C47C2×D411yescharacteristic
53#41×C47—416no—
54C2×Q8×C47—22no—
55C2×Q8×C47—22no—
56C2×Q8×C47—41no—
57C2×D4×C47—22no—
58C2×D4×C47—22no—
59C2×D4×C47—22no—
60C2×D4×C47—22no—
61C2×D4×C47—22no—
62C2×D4×C47—22no—
63C2×D4×C47—41no—
64C2×D4×C47—41no—
65C2×D4×C47—41no—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47—42no—
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47#4114yes—
80C22×Dic47—22no—
81C22×Dic47—22no—
82C22×Dic47—44no—
83C22×Dic47—42no—
84C22⋊C4×C47—42no—
85C22⋊C4×C47—42no—
86C22⋊Dic47—44no—
87C22⋊Dic47—44no—
88C22⋊Dic47—42no—
89C4⋊Dic47—42no—
90C4×Dic47—42no—
91C4×Dic47—42no—
92C4×Dic47—42no—
93#10625329—42no—
94#10625329—42no—
95C2×C47⋊C8—42no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47—42no—
106C23×C47#9912yes—
107C23×C47C2×C22⋊C411yescharacteristic
108C23×C47#12312yes—
109Q8×C47—44no—
110D4×C47—44no—
111D4×C47—44no—
112D4×C47—44no—
113C2×C4×C47—28no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—44no—
124C2×C4×C47—44no—
125C2×C4×C47—82no—
126C2×C4×C47—84no—
127C2×C4×C47#9912yes—
128C2×C4×C47#11712yes—
129C2×C4×C47#12312yes—
130C2×Dic47—24no—
131C2×Dic47—42no—
132C2×Dic47—48no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C47⋊C8—82no—
138#8739—471nomaximal, Sylow
139C22×C47—22no—
140C22×C47—22no—
141C22×C47—22no—
142C22×C47—24no—
143C22×C47—42no—
144C22×C47—42no—
145C22×C47—42no—
146C22×C47—41no—
147C22×C47—42no—
148C22×C47—81no—
149C22×C47—81no—
150C22×C47—81no—
151C22×C47—81no—
152C22×C47#38111yescharacteristic
153C22×C47C2×#9911yescharacteristic
154C22×C47C2×#9911yescharacteristic
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—22no—
158C4×C47—41no—
159C4×C47—162no—
160Dic47—44no—
161Dic47—82no—
162#2297—471no—
163#1785—471no—
164#1782D4711yescharacteristic
165#1284—474no—
166C2×C47—21no—
167C2×C47—21no—
168C2×C47—42no—
169C2×C47—42no—
170C2×C47—81no—
171C2×C47#181911yescharacteristic
172C2×C47#182712yes—
173C22×#41C2×D4711yescharacteristic
174C2×#117—471no—
175C22×C42—471no—
176C2×#101—941no—
177C2×#100—941no—
178C2×#99—21no—
179C2×#99—21no—
180#407—472no—
181#391—941no—
182#390—941no—
183#386—941no—
184#386—941no—
185#381C2×D4712yes—
186C4×C22⋊C4—472no—
187C4×C22⋊C4—472no—
188C47#873911yescharacteristic, complemented, Sylow
189C2×#41—24no—
190C2×#41—42no—
191C2×#41—42no—
192C22×Q8D18811yescharacteristic
193C22×D4D18811yescharacteristic
194C22×D4C22⋊D4711yescharacteristic
195C22×D4C22⋊D4711yescharacteristic
196C23×C4—941no—
197C23×C4C4×D4711yescharacteristic
198C23×C4C4×D4711yescharacteristic
199C23×C4C22×D4711yescharacteristic
200C2×M16—942no—
201#117—1882no—
202C2×C4⋊C4—941no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—941no—
205C2×C22⋊C4—942no—
206C2×C22⋊C4—942no—
207C2×C42—472no—
208C2×C42—472no—
209C2×C42—941no—
210C2×C42—1882no—
211#101—1882no—
212#100—1882no—
213#99—42no—
214#99—42no—
215C22⋊C8—944no—
216#95—24no—
217#95—942no—
218#95—942no—
219C24(C2×D47)⋊C411yescharacteristic
220C24(C2×D47)⋊C411yescharacteristic
221C24C2×C22⋊D4711yescharacteristic
222#41—416no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—942no—
245C22×C4—942no—
246C22×C4—1884no—
247C22×C4—1882no—
248C22×C4(C2×D47)⋊C411yescharacteristic
249C22×C4(C2×D47)⋊C411yescharacteristic
250C22×C4C2×C4×D4711yescharacteristic
251C22×C4C2×D18811yescharacteristic
252C22×C4#1062535214yes—
253M16—1882no—
254M16—1882no—
255C2×C8—1882no—
256C4⋊C4—1882no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—1884no—
260C22⋊C4—1884no—
261C22⋊C4—1882no—
262C42—1882no—
263C42—1882no—
264C42—1882no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23#1309393812yes—
276C23C2×(C2×D47)⋊C411yescharacteristic
277C23#1309400512yes—
278Q8—44no—
279D4—44no—
280D4—44no—
281D4—44no—
282C2×C4—28no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—82no—
295C2×C4—84no—
296C2×C4—944no—
297C2×C4—1882no—
298C2×C4—1882no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4—1888no—
302C2×C4—1882no—
303C2×C4#1309393812yes—
304C2×C4#1309398012yes—
305C2×C4#1309400812yes—
306C8—3762no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—24no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—41no—
315C22—42no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22#1568898411yescharacteristic
321C22C2×#1309393811yescharacteristic
322C22C2×#1309393811yescharacteristic
323C4—22no—
324C4—22no—
325C4—22no—
326C4—41no—
327C4—162no—
328C4—3762no—
329C4—1884no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2—81no—
335C2#2024346011yescharacteristic
336C2#2024348912yes—
subgroupG/H
centerC22#15688984
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1782C2
socleC22×C47#381
prime radicalC22×#41×C47C22
cosocleC23×C4×C47C23
hypercenter#1782D47
hypocenterC47#8739
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
3. #20241240C2semidirect
5. C2×#13093983C4semidirect
6. C22×C4×Dic47C4semidirect
14. C4×C22⋊Dic47C4semidirect
188. C47#8739semidirect

Maximal subgroups

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

subgroupindex
#202458792
#202433162
#202412402
C47×#17822
#873947