Finite Group Databasebeta

← #26131913

Subgroups — #26131913

Order 12032.

690 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 127 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#20245077C212yesmaximal, complemented
2C2×#15688920C212yesmaximal, complemented
3#20243502C212yesmaximal, complemented
4C47×#3510C211yesmaximal, characteristic
5C2×#13093943C2214yes—
6#15688922—22no—
7#15688920—22no—
8#15688920—22no—
9#15688920—22no—
10#15688893—24no—
11C47×#559—22no—
12C47×#556—22no—
13C47×#551—24no—
14C47×#542—22no—
15C47×#536C2211yescharacteristic
16C47×#535C411yescharacteristic
17C47×#533C411yescharacteristic
18C47×#529—22no—
19C2×#124×C47C2211yescharacteristic
20C47×#519C2211yescharacteristic
21C47×#518C411yescharacteristic
22C4×#41×C47C411yescharacteristic
23C2×#10625329—24no—
24#13093943—24no—
25#13093943—24no—
26C2×#41×C47C2×C411yescharacteristic
27C22×Q8×C47C2311yescharacteristic
28C22×D4×C47C2×C411yescharacteristic
29C4⋊Q8×C47—21no—
30C4⋊Q8×C47D412yes—
31#127×C47—21no—
32#126×C47—24no—
33#126×C47—24no—
34#125×C47—21no—
35#124×C47—24no—
36#124×C47—21no—
37#124×C47—22no—
38#124×C47D412yes—
39#124×C47D412yes—
40#124×C47D412yes—
41#123×C47—24no—
42#123×C47—24no—
43#123×C47—22no—
44C22⋊Q8×C47—24no—
45C22⋊Q8×C47—22no—
46C22⋊Q8×C47—22no—
47#121×C47—22no—
48#120×C47—22no—
49C4×Q8×C47—21no—
50C4×Q8×C47—22no—
51C4×Q8×C47—22no—
52C4×Q8×C47—22no—
53C4×D4×C47—24no—
54C4×D4×C47—21no—
55C4×D4×C47—22no—
56C4×D4×C47—22no—
57C4×D4×C47—22no—
58#117×C47—24no—
59#117×C47—21no—
60#117×C47C2×C411yescharacteristic
61#117×C47C2×C411yescharacteristic
62#117×C47C2×C411yescharacteristic
63#117×C47D414yes—
64C2×C22⋊C4×C47—22no—
65C2×C22⋊C4×C47—22no—
66C2×C42×C47—21no—
67C2×C42×C47C2×C411yescharacteristic
68C24×C47—21no—
69#41×C47—42no—
70C2×Q8×C47—22no—
71C2×Q8×C47—42no—
72C2×Q8×C47C22⋊C411yescharacteristic
73C2×Q8×C47C2×D412yes—
74C2×Q8×C47C2×D412yes—
75C2×D4×C47—21no—
76C2×D4×C47—24no—
77C2×D4×C47—21no—
78C2×D4×C47—42no—
79C2×D4×C47C22⋊C412yes—
80C2×D4×C47C22⋊C411yescharacteristic
81C22×C4×C47—22no—
82C22×C4×C47—24no—
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47C22×C411yescharacteristic
86C22×C4×C47C2×D412yes—
87C4⋊C4×C47—22no—
88C4⋊C4×C47—22no—
89C4⋊C4×C47—22no—
90C4⋊C4×C47—22no—
91C4⋊C4×C47—21no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—24no—
95C4⋊C4×C47—24no—
96C22⋊C4×C47—24no—
97C22⋊C4×C47—22no—
98C22⋊C4×C47—24no—
99C22⋊C4×C47—21no—
100C22⋊C4×C47—24no—
101C22⋊C4×C47—24no—
102C22⋊C4×C47—24no—
103C22⋊C4×C47C22⋊C412yes—
104C22⋊C4×C47C22⋊C412yes—
105C42×C47—22no—
106C42×C47—21no—
107C42×C47—21no—
108C42×C47—21no—
109C42×C47—24no—
110C42×C47C22⋊C411yescharacteristic
111C42×C47C22⋊C411yescharacteristic
112C42×C47C22⋊C412yes—
113#10625329—44no—
114#10625329—44no—
115C2×C47⋊C8—44no—
116C23×C47—22no—
117C23×C47—21no—
118C23×C47—42no—
119C23×C47—82no—
120C23×C47C2×C22⋊C411yescharacteristic
121Q8×C47—42no—
122Q8×C47—42no—
123Q8×C47—41no—
124Q8×C47—82no—
125D4×C47—42no—
126D4×C47—41no—
127D4×C47—42no—
128D4×C47—42no—
129D4×C47—84no—
130C2×C4×C47—24no—
131C2×C4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47—21no—
134C2×C4×C47—22no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—22no—
138C2×C4×C47—42no—
139C2×C4×C47—82no—
140C2×C4×C47—84no—
141C2×C4×C47—81no—
142C2×C4×C47C2×C22⋊C411yescharacteristic
143C2×C4×C47C2×C22⋊C411yescharacteristic
144C2×C4×C47#12014yes—
145C47⋊C8—84no—
146#12729—471nomaximal, Sylow
147C22×C47—41no—
148C22×C47—42no—
149C22×C47—44no—
150C22×C47—41no—
151C22×C47—42no—
152C22×C47—162no—
153C22×C47#37911yescharacteristic
154C22×C47#41012yes—
155C4×C47—24no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—42no—
159C4×C47—41no—
160C4×C47—42no—
161C4×C47—41no—
162C4×C47—42no—
163C4×C47—41no—
164#3510D4711yescharacteristic
165#2133—472no—
166C2×#355—472no—
167#1830—472no—
168C2×C47—21no—
169C2×C47—22no—
170C2×C47—81no—
171C2×C47—82no—
172C2×C47#178211yescharacteristic
173#559—22no—
174#556—22no—
175#551—24no—
176#542—22no—
177#536C2×D4711yescharacteristic
178#535Dic4711yescharacteristic
179#533Dic4711yescharacteristic
180#529—22no—
181C2×#124C2×D4711yescharacteristic
182#519C2×D4711yescharacteristic
183#518Dic4711yescharacteristic
184C4×#41Dic4711yescharacteristic
185C2×#101—474no—
186#356—942no—
187#355—942no—
188#355—942no—
189#355—942no—
190#344—944no—
191C47#1272911yescharacteristic, complemented, Sylow
192C2×#41C2×Dic4711yescharacteristic
193C22×Q8C22×D4711yescharacteristic
194C22×D4C2×Dic4711yescharacteristic
195C2×M16—944no—
196C4⋊Q8—21no—
197C4⋊Q8C22⋊D4712yes—
198#127—21no—
199#126—24no—
200#126—24no—
201#125—21no—
202#124—24no—
203#124—21no—
204#124—22no—
205#124C22⋊D4712yes—
206#124C22⋊D4712yes—
207#124C22⋊D4712yes—
208#123—24no—
209#123—24no—
210#123—22no—
211C22⋊Q8—24no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214#121—22no—
215#120—22no—
216C4×Q8—21no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×D4—24no—
221C4×D4—21no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225#117—24no—
226#117—21no—
227#117C2×Dic4711yescharacteristic
228#117C2×Dic4711yescharacteristic
229#117C2×Dic4711yescharacteristic
230#117C22⋊D4714yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C42—21no—
234C2×C42C2×Dic4711yescharacteristic
235#101—944no—
236#101—944no—
237C24—21no—
238#41—42no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8C22⋊Dic4711yescharacteristic
242C2×Q8C2×C22⋊D4712yes—
243C2×Q8C2×C22⋊D4712yes—
244C2×D4—21no—
245C2×D4—24no—
246C2×D4—21no—
247C2×D4—42no—
248C2×D4C22⋊Dic4712yes—
249C2×D4C22⋊Dic4711yescharacteristic
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C22×Dic4711yescharacteristic
255C22×C4C2×C22⋊D4712yes—
256M16—1884no—
257M16—1884no—
258C2×C8—1884no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—22no—
270C22⋊C4—24no—
271C22⋊C4—21no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4C22⋊Dic4712yes—
276C22⋊C4C22⋊Dic4712yes—
277C42—22no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—24no—
282C42C22⋊Dic4711yescharacteristic
283C42C22⋊Dic4711yescharacteristic
284C42C22⋊Dic4712yes—
285C23—22no—
286C23—21no—
287C23—42no—
288C23—82no—
289C23C2×C22⋊Dic4711yescharacteristic
290Q8—42no—
291Q8—42no—
292Q8—41no—
293Q8—82no—
294D4—42no—
295D4—41no—
296D4—42no—
297D4—42no—
298D4—84no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4—81no—
311C2×C4C2×C22⋊Dic4711yescharacteristic
312C2×C4C2×C22⋊Dic4711yescharacteristic
313C2×C4#1309399314yes—
314C8—3764no—
315C22—41no—
316C22—42no—
317C22—44no—
318C22—41no—
319C22—42no—
320C22—162no—
321C22#1568897911yescharacteristic
322C22#1568909712yes—
323C4—24no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—41no—
332C2—21no—
333C2—22no—
334C2—82no—
335C2—81no—
336C2#2024330611yescharacteristic
subgroupG/H
centerC2#20243306
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC47×#3510C2
socleC2×C47#1782
prime radicalC47×#535C4
cosocleC22×Q8×C47C23
hypercenter#3510D47
hypocenterC47#12729
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C23 ◁ #519 ◁ #3510

Complements

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

NH ≅ G/Ntype
1. #20245077C2semidirect
2. C2×#15688920C2semidirect
3. #20243502C2semidirect
191. C47#12729semidirect

Maximal subgroups

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

subgroupindex
#202450772
C2×#156889202
#202435022
C47×#35102
#1272947