Finite Group Databasebeta

← #26108671

Subgroups — #26108671

Order 12032.

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

normal subgroups only

This group has 692 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
1#20244209C211yesmaximal, characteristic, complemented
2#20242114C211yesmaximal, characteristic, complemented
3#20242008C212yesmaximal, complemented
4#20241926C212yesmaximal, complemented
5C47×#3544C211yesmaximal, characteristic
6#15689180C2212yes—
7C2×#13093948—22no—
8C2×#13093935C2211yescharacteristic
9#15688989C2211yescharacteristic
10#15688872—22no—
11#15688865—22no—
12#15688863—22no—
13#15688860—22no—
14#15688857—22no—
15C47×#563—22no—
16Q82×C47C411yescharacteristic, complemented
17C47×#555C411yescharacteristic, complemented
18C47×#554C412yescomplemented
19C47×#552—22no—
20C47×#547—22no—
21C47×#544—22no—
22C47×#542—22no—
23C47×#536—22no—
24C2×C4⋊Q8×C47C2211yescharacteristic
25C47×#529C2212yes—
26C22×C47⋊C8—21no—
27C2×C4⋊Dic47—21no—
28C4⋊(C47⋊C8)—22no—
29#13093948—44no—
30C22⋊(C47⋊C8)—22no—
31#13093935—22no—
32#13093935—22no—
33C47⋊#95—22no—
34C2×#41×C47—22no—
35C22×Q8×C47D412yes—
36C4⋊Q8×C47—22no—
37C4⋊Q8×C47—22no—
38C4⋊Q8×C47—22no—
39C4⋊Q8×C47—22no—
40C4⋊Q8×C47C2×C411yescharacteristic
41C4⋊Q8×C47C2×C411yescharacteristic
42C4⋊Q8×C47D412yes—
43#126×C47—24no—
44#126×C47—22no—
45#125×C47—24no—
46#125×C47—22no—
47#125×C47D412yes—
48#124×C47—22no—
49#124×C47—22no—
50#124×C47D412yes—
51#123×C47—24no—
52#123×C47—24no—
53#123×C47—22no—
54C22⋊Q8×C47—24no—
55C22⋊Q8×C47—24no—
56C22⋊Q8×C47—24no—
57C22⋊Q8×C47—22no—
58C22⋊Q8×C47D412yes—
59#121×C47—22no—
60#121×C47D412yes—
61C4×Q8×C47—22no—
62C4×Q8×C47C2×C412yes—
63C4×D4×C47—22no—
64C4×D4×C47—22no—
65C4×D4×C47C2×C412yes—
66#117×C47—22no—
67#117×C47—22no—
68#117×C47—22no—
69C2×C4⋊C4×C47—22no—
70C2×C4⋊C4×C47—22no—
71C2×C42×C47C2311yescharacteristic
72#41×C47—82no—
73#41×C47—82no—
74C2×Q8×C47—22no—
75C2×Q8×C47—22no—
76C2×Q8×C47—42no—
77C2×Q8×C47—42no—
78C2×Q8×C47C22⋊C412yes—
79C2×Q8×C47C22⋊C412yes—
80C2×Q8×C47C22⋊C412yes—
81C2×D4×C47—22no—
82C2×D4×C47—22no—
83C2×D4×C47C22⋊C412yes—
84C22×C4×C47—22no—
85C22×C4×C47—24no—
86C22×C4×C47—22no—
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D412yes—
89C22×Dic47—41no—
90C4⋊C4×C47—24no—
91C4⋊C4×C47—24no—
92C4⋊C4×C47—24no—
93C4⋊C4×C47—22no—
94C4⋊C4×C47—22no—
95C4⋊C4×C47—22no—
96C4⋊C4×C47—24no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—22no—
100C4⋊C4×C47C22⋊C412yes—
101C4⋊C4×C47C22⋊C412yes—
102C22⋊C4×C47—24no—
103C22⋊C4×C47—24no—
104C22⋊C4×C47—22no—
105C22⋊C4×C47—22no—
106C22⋊C4×C47—22no—
107C22⋊C4×C47—22no—
108C22⋊C4×C47—22no—
109C42×C47—22no—
110C42×C47—22no—
111C42×C47—22no—
112C42×C47C22×C411yescharacteristic
113C42×C47C2×D412yes—
114C42×C47C2×D411yescharacteristic
115C4⋊Dic47—42no—
116C2×C47⋊C8—22no—
117C2×C47⋊C8—41no—
118C2×C47⋊C8—41no—
119C2×C47⋊C8—42no—
120C23×C47—22no—
121C23×C47#12011yescharacteristic
122Q8×C47—42no—
123Q8×C47—42no—
124Q8×C47—42no—
125Q8×C47—82no—
126Q8×C47—82no—
127D4×C47—42no—
128D4×C47—42no—
129D4×C47—42no—
130D4×C47—82no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—42no—
139C2×C4×C47—44no—
140C2×C4×C47—42no—
141C2×C4×C47—42no—
142C2×C4×C47—84no—
143C2×C4×C47C2×C22⋊C412yes—
144C2×C4×C47C2×C22⋊C411yescharacteristic
145C2×C4×C47#12012yes—
146C2×C4×C47#12011yescharacteristic
147C2×Dic47—82no—
148C2×Dic47—81no—
149C47⋊C8—42no—
150C47⋊C8—82no—
151#8498—471nomaximal, Sylow
152C22×C47—22no—
153C22×C47—42no—
154C22×C47—42no—
155C22×C47—42no—
156C22×C47#37911yescharacteristic
157C22×C47#45311yescharacteristic
158C22×C47#45511yescharacteristic
159C4×C47—21no—
160C4×C47—21no—
161C4×C47—22no—
162C4×C47—22no—
163C4×C47—42no—
164C4×C47—42no—
165C4×C47—42no—
166C4×C47—42no—
167C4×C47—42no—
168Dic47—161no—
169#3544D4711yescharacteristic
170#1960—471no—
171#1558—471no—
172#1532—472no—
173#1512—472no—
174C2×C47—21no—
175C2×C47—21no—
176C2×C47—82no—
177C2×C47#179211yescharacteristic
178C2×C47#179612yes—
179#563—22no—
180Q82Dic4711yescharacteristic, complemented
181#555Dic4711yescharacteristic, complemented
182#554Dic4712yescomplemented
183#552—22no—
184#547—22no—
185#544—22no—
186#542—22no—
187#536—22no—
188C2×C4⋊Q8C2×D4711yescharacteristic
189#529C2×D4712yes—
190#432—472no—
191C2×#103—942no—
192C2×#97—471no—
193#382—471no—
194#333—942no—
195#330—942no—
196#329—942no—
197#328—942no—
198#327—942no—
199C47#849811yescharacteristic, complemented, Sylow
200C2×#41—22no—
201C22×Q8C22⋊D4712yes—
202C22×C8—941no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8C2×Dic4711yescharacteristic
208C4⋊Q8C2×Dic4711yescharacteristic
209C4⋊Q8C22⋊D4712yes—
210#126—24no—
211#126—22no—
212#125—24no—
213#125—22no—
214#125C22⋊D4712yes—
215#124—22no—
216#124—22no—
217#124C22⋊D4712yes—
218#123—24no—
219#123—24no—
220#123—22no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—22no—
225C22⋊Q8C22⋊D4712yes—
226#121—22no—
227#121C22⋊D4712yes—
228C4×Q8—22no—
229C4×Q8C2×Dic4712yes—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4C2×Dic4712yes—
233#117—22no—
234#117—22no—
235#117—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—941no—
239C2×C42C22×D4711yescharacteristic
240C4⋊C8—942no—
241#103—1884no—
242C22⋊C8—942no—
243#97—942no—
244#97—942no—
245#95—942no—
246#41—82no—
247#41—82no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8C22⋊Dic4712yes—
253C2×Q8C22⋊Dic4712yes—
254C2×Q8C22⋊Dic4712yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C22⋊Dic4712yes—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—1881no—
262C22×C4C2×C22⋊D4711yescharacteristic
263C22×C4C2×C22⋊D4712yes—
264C2×C8—942no—
265C2×C8—1882no—
266C2×C8—1881no—
267C2×C8—1881no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—1882no—
279C4⋊C4C22⋊Dic4712yes—
280C4⋊C4C22⋊Dic4712yes—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42C22×Dic4711yescharacteristic
292C42C2×C22⋊D4712yes—
293C42C2×C22⋊D4711yescharacteristic
294C23—22no—
295C23#1309399311yescharacteristic
296Q8—42no—
297Q8—42no—
298Q8—42no—
299Q8—82no—
300Q8—82no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—82no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—84no—
317C2×C4—3762no—
318C2×C4—3761no—
319C2×C4C2×C22⋊Dic4712yes—
320C2×C4C2×C22⋊Dic4711yescharacteristic
321C2×C4#1309399312yes—
322C2×C4#1309399311yescharacteristic
323C8—3762no—
324C8—1882no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22#1568897911yescharacteristic
330C22#1568929111yescharacteristic
331C22#1568930111yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—7521no—
342C2—21no—
343C2—21no—
344C2—82no—
345C2#2024334211yescharacteristic
346C2#2024336312yes—
subgroupG/H
centerC22#15688979
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3544C2
socleC22×C47#379
cosocleC2×C42×C47C23
hypercenter#3544D47
hypocenterC47#8498
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3544

Complements

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

NH ≅ G/Ntype
1. #20244209C2semidirect
2. #20242114C2semidirect
3. #20242008C2semidirect
4. #20241926C2semidirect
16. Q82×C47C4semidirect
17. C47×#555C4semidirect
18. C47×#554C4semidirect
180. Q82Dic47semidirect
181. #555Dic47semidirect
182. #554Dic47semidirect
199. C47#8498semidirect

Maximal subgroups

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

subgroupindex
#202442092
#202421142
#202420082
#202419262
C47×#35442
#849847