Finite Group Databasebeta

← #26107692

Subgroups — #26107692

Order 12032.

710 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 151 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#20244325C211yesmaximal, characteristic, complemented
2#20241775C212yesmaximal, complemented
3#20241770C212yesmaximal, complemented
4#20241712C211yesmaximal, characteristic, complemented
5C47×#3461C211yesmaximal, characteristic
6#15689176C2212yes—
7C2×C4⋊(C47⋊C8)C2211yescharacteristic
8C2×C4×C47⋊C8C2211yescharacteristic
9#15688878—22no—
10Q8⋊(C47⋊C8)—22no—
11Q8⋊(C47⋊C8)—22no—
12D4⋊(C47⋊C8)—22no—
13D4⋊(C47⋊C8)—22no—
14#15688849—22no—
15C47×#544C411yescharacteristic
16C47×#542—22no—
17C47×#541—22no—
18C47×#539—22no—
19C47×#537C411yescharacteristic
20C47×#536C412yes—
21C47×#533—22no—
22C47×#529—22no—
23C47×#529—22no—
24C4×#41×C47C2212yes—
25C2×C4×Q8×C47C2211yescharacteristic
26C22×C47⋊C8—21no—
27C22×C47⋊C8—21no—
28C4⋊(C47⋊C8)—22no—
29C4⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31C4×C47⋊C8—22no—
32C4×C47⋊C8—22no—
33C2×#41×C47C2×C412yes—
34C22×Q8×C47C2×C411yescharacteristic
35C4⋊Q8×C47—22no—
36C4⋊Q8×C47—22no—
37#126×C47—24no—
38#126×C47—24no—
39#125×C47—22no—
40#125×C47—24no—
41#125×C47—22no—
42#124×C47—22no—
43#124×C47—22no—
44#123×C47—24no—
45#123×C47—24no—
46C22⋊Q8×C47—24no—
47C22⋊Q8×C47—22no—
48C22⋊Q8×C47—22no—
49C22⋊Q8×C47—22no—
50C22⋊Q8×C47—22no—
51#121×C47—22no—
52#121×C47—22no—
53C4×Q8×C47—22no—
54C4×Q8×C47—22no—
55C4×Q8×C47—22no—
56C4×Q8×C47D412yes—
57C4×Q8×C47D412yes—
58C4×D4×C47—22no—
59C4×D4×C47—22no—
60C4×D4×C47—22no—
61C4×D4×C47D412yes—
62C4×D4×C47D412yes—
63#117×C47—24no—
64#117×C47—22no—
65#117×C47C2×C412yes—
66C2×C4⋊C4×C47C2×C411yescharacteristic
67C2×C4⋊C4×C47D412yes—
68C2×C42×C47—22no—
69C2×C42×C47D412yes—
70C2×C42×C47C2311yescharacteristic
71#41×C47—28no—
72C2×Q8×C47—22no—
73C2×Q8×C47—22no—
74C2×Q8×C47—24no—
75C2×Q8×C47C22⋊C412yes—
76C2×D4×C47—22no—
77C2×D4×C47—22no—
78C2×D4×C47C22⋊C412yes—
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47C22⋊C412yes—
83C22×C4×C47C22⋊C412yes—
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C4⋊C4×C47—24no—
88C4⋊C4×C47—24no—
89C4⋊C4×C47—22no—
90C4⋊C4×C47—22no—
91C4⋊C4×C47—24no—
92C4⋊C4×C47—22no—
93C4⋊C4×C47—24no—
94C4⋊C4×C47—22no—
95C4⋊C4×C47—22no—
96C4⋊C4×C47C22⋊C412yes—
97C4⋊C4×C47C22⋊C412yes—
98C22⋊C4×C47—24no—
99C22⋊C4×C47—24no—
100C22⋊C4×C47—24no—
101C22⋊C4×C47—22no—
102C22⋊C4×C47—22no—
103C22⋊C4×C47—22no—
104C42×C47—22no—
105C42×C47—22no—
106C42×C47—22no—
107C42×C47—22no—
108C42×C47—22no—
109C42×C47C2×D412yes—
110C42×C47C2×D412yes—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—41no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8—42no—
115C2×C47⋊C8—42no—
116C2×C47⋊C8—41no—
117C23×C47—22no—
118C23×C47C2×C22⋊C411yescharacteristic
119Q8×C47—24no—
120Q8×C47—44no—
121Q8×C47—82no—
122D4×C47—24no—
123D4×C47—44no—
124D4×C47—82no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47—24no—
133C2×C4×C47—41no—
134C2×C4×C47—44no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—82no—
139C2×C4×C47C2×C22⋊C412yes—
140C2×C4×C47#12011yescharacteristic
141C2×C4×C47#12012yes—
142C2×C4×C47#12011yescharacteristic
143C2×C4×C47#13114yes—
144C47⋊C8—42no—
145C47⋊C8—82no—
146C47⋊C8—81no—
147#8321—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—42no—
151C22×C47—42no—
152C22×C47—42no—
153C22×C47#37911yescharacteristic
154C22×C47#40812yes—
155C4×C47—21no—
156C4×C47—22no—
157C4×C47—42no—
158C4×C47—42no—
159C4×C47—42no—
160C4×C47—42no—
161C4×C47—42no—
162C4×C47#40814yes—
163C4×C47#45411yescharacteristic
164C4×C47#45611yescharacteristic
165#3461D4711yescharacteristic
166#1984—471no—
167#1477—472no—
168#1476—472no—
169#1461—471no—
170C2×C47—22no—
171C2×C47—82no—
172C2×C47#178011yescharacteristic
173C2×C47#179311yescharacteristic
174C2×C47#179411yescharacteristic
175#544Dic4711yescharacteristic
176#542—22no—
177#541—22no—
178#539—22no—
179#537Dic4711yescharacteristic
180#536Dic4712yes—
181#533—22no—
182#529—22no—
183#529—22no—
184C4×#41C2×D4712yes—
185C2×C4×Q8C2×D4711yescharacteristic
186#431—472no—
187C2×C4⋊C8—471no—
188C2×C4×C8—471no—
189#336—942no—
190Q8⋊C8—942no—
191Q8⋊C8—942no—
192D4⋊C8—942no—
193D4⋊C8—942no—
194#324—942no—
195C47#832111yescharacteristic, complemented, Sylow
196C2×#41C2×Dic4712yes—
197C22×Q8C2×Dic4711yescharacteristic
198C22×C8—941no—
199C22×C8—941no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202#126—24no—
203#126—24no—
204#125—22no—
205#125—24no—
206#125—22no—
207#124—22no—
208#124—22no—
209#123—24no—
210#123—24no—
211C22⋊Q8—24no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216#121—22no—
217#121—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8C22⋊D4712yes—
222C4×Q8C22⋊D4712yes—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4C22⋊D4712yes—
227C4×D4C22⋊D4712yes—
228#117—24no—
229#117—22no—
230#117C2×Dic4712yes—
231C2×C4⋊C4C2×Dic4711yescharacteristic
232C2×C4⋊C4C22⋊D4712yes—
233C2×C42—22no—
234C2×C42C22⋊D4712yes—
235C2×C42C22×D4711yescharacteristic
236C4⋊C8—942no—
237C4⋊C8—942no—
238C22⋊C8—942no—
239C4×C8—942no—
240C4×C8—942no—
241#41—28no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8C22⋊Dic4712yes—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4C22⋊Dic4712yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4C22⋊Dic4712yes—
253C22×C4C22⋊Dic4712yes—
254C22×C4C22×Dic4711yescharacteristic
255C22×C4C2×C22⋊D4711yescharacteristic
256C22×C4C2×C22⋊D4711yescharacteristic
257C2×C8—942no—
258C2×C8—1882no—
259C2×C8—1882no—
260C2×C8—1881no—
261C2×C8—1881no—
262C2×C8—1881no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4C22⋊Dic4712yes—
273C4⋊C4C22⋊Dic4712yes—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42C2×C22⋊D4712yes—
286C42C2×C22⋊D4712yes—
287C23—22no—
288C23C2×C22⋊Dic4711yescharacteristic
289Q8—24no—
290Q8—44no—
291Q8—82no—
292D4—24no—
293D4—44no—
294D4—82no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4C2×C22⋊Dic4712yes—
310C2×C4#1309399311yescharacteristic
311C2×C4#1309399312yes—
312C2×C4#1309399311yescharacteristic
313C2×C4#1309403114yes—
314C8—3762no—
315C8—3761no—
316C8—1882no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22#1568897911yescharacteristic
323C22#1568909012yes—
324C4—21no—
325C4—22no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—42no—
330C4—42no—
331C4#1568909014yes—
332C4#1568929611yescharacteristic
333C4#1568930611yescharacteristic
334C2—22no—
335C2—82no—
336C2#2024330011yescharacteristic
337C2#2024334711yescharacteristic
338C2#2024335211yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3461C2
socleC22×C47#379
prime radicalC47×#537C4
cosocleC2×C42×C47C23
hypercenter#3461D47
hypocenterC47#8321
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
1. #20244325C2semidirect
2. #20241775C2semidirect
3. #20241770C2semidirect
4. #20241712C2semidirect
195. C47#8321semidirect

Maximal subgroups

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

subgroupindex
#202443252
#202417752
#202417702
#202417122
C47×#34612
#832147