Finite Group Databasebeta

← #26115802

Subgroups — #26115802

Order 12032.

510 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#15689364C211yesmaximal, characteristic
2#20244372C212yesmaximal
3#20243164C211yesmaximal, characteristic
4#20241959C212yesmaximal
5C47×#1807C211yesmaximal, characteristic
6C22×Dic188C411yescharacteristic, complemented
7C2×#13094003C2212yes—
8#15689364—21no—
9#15689364—21no—
10#15689364—21no—
11#15689364—21no—
12C2×#13093958C411yescharacteristic, complemented
13C2×#13093949C412yescomplemented
14C2×C22⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093934—21no—
16C4×C22⋊Dic47C2211yescharacteristic
17#15688889—24no—
18#15688877—21no—
19#15688859—24no—
20C23×C8×C47C2211yescharacteristic
21C2×#106×C47—21no—
22C47×#380C2212yes—
23C47×#340—21no—
24C22×Dic94C2×C412yes—
25C2×Dic188—21no—
26C2×Dic188—21no—
27C2×Dic188—42no—
28C22×C47⋊C8—21no—
29#13094003—22no—
30#13094003—22no—
31#13094003—22no—
32#13094003—22no—
33C2×C4⋊Dic47C2×C412yes—
34C2×Dic47⋊C4—22no—
35C2×C22⋊Dic47—22no—
36C2×C22⋊Dic47D412yes—
37C2×C4×Dic47—21no—
38#13093958—21no—
39#13093958—21no—
40#13093949—22no—
41#13093949—22no—
42C22⋊(C47⋊C8)—21no—
43C22⋊(C47⋊C8)D412yes—
44#13093934—42no—
45C47⋊#95—21no—
46C23×C4×C47C2311yescharacteristic
47C22×C8×C47—42no—
48C22×C8×C47C2×C411yescharacteristic
49C22×C8×C47C2×C411yescharacteristic
50C22×C8×C47D411yescharacteristic
51C22×C8×C47D411yescharacteristic
52C22×C8×C47D411yescharacteristic
53C22×C8×C47D411yescharacteristic
54C2×C4⋊C4×C47—22no—
55C2×C22⋊C4×C47—22no—
56#106×C47—42no—
57#95×C47—22no—
58C24×C47C2×D411yescharacteristic
59C22×C4×C47—41no—
60C22×C4×C47—41no—
61C22×C4×C47—42no—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47#4111yescharacteristic
67C22×C4×C47#4111yescharacteristic
68C22×Dic47—22no—
69C22×Dic47—22no—
70C22×Dic47—22no—
71C2×Dic94—22no—
72C2×Dic94—22no—
73C2×Dic94—42no—
74C2×Dic94—42no—
75C2×C8×C47—22no—
76C2×C8×C47—22no—
77C2×C8×C47—22no—
78C2×C8×C47—41no—
79C2×C8×C47—41no—
80C2×C8×C47—41no—
81C2×C8×C47C22⋊C411yescharacteristic
82C2×C8×C47C22⋊C411yescharacteristic
83C2×C8×C47C22⋊C411yescharacteristic
84C2×C8×C47C22⋊C411yescharacteristic
85C4⋊C4×C47—42no—
86C4⋊C4×C47—42no—
87C22⋊C4×C47—44no—
88C22⋊Dic47—24no—
89C22⋊Dic47—42no—
90C22⋊Dic47—42no—
91C4⋊Dic47—22no—
92C4⋊Dic47—22no—
93Dic47⋊C4—42no—
94Dic47⋊C4—42no—
95C4×Dic47—41no—
96C4×Dic47—41no—
97C2×C47⋊C8—22no—
98C2×C47⋊C8—41no—
99C2×C47⋊C8—41no—
100Dic188—42no—
101Dic188—42no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47C4×D411yescharacteristic
105C23×C47#12111yescharacteristic
106C23×C47#12711yescharacteristic
107C23×C47#13714yes—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—82no—
118C2×C4×C47—82no—
119C2×C4×C47—82no—
120C2×C4×C47C2×C22⋊C411yescharacteristic
121C2×C4×C47C4×D411yescharacteristic
122C2×C4×C47#12111yescharacteristic
123C2×C4×C47#12411yescharacteristic
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—82no—
130C2×Dic47—82no—
131Dic94—42no—
132Dic94—42no—
133Dic94—82no—
134C8×C47—21no—
135C8×C47—21no—
136C8×C47—21no—
137C8×C47#9912yes—
138C47⋊C8—42no—
139#9784—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—21no—
145C22×C47—21no—
146C22×C47—44no—
147C22×C47#39011yescharacteristic
148C22×C47#44112yes—
149C22×C47#47012yes—
150C22×C47#49712yes—
151C4×C47—21no—
152C4×C47—21no—
153C4×C47—21no—
154C4×C47—162no—
155C4×C47C2×#9911yescharacteristic
156C4×C47#41011yescharacteristic
157Dic47—82no—
158Dic47—82no—
159Dic47—82no—
160C2×#467—471no—
161#1996—472no—
162#1807D4711yescharacteristic
163#1752—471no—
164#1520—472no—
165C2×C47—22no—
166C2×C47—22no—
167C2×C47—22no—
168C2×C47#196411yescharacteristic
169C2×C47#197111yescharacteristic
170C2×C47#197611yescharacteristic
171C22×Q16—471no—
172C23×C8C2×D4711yescharacteristic
173C2×C22⋊Q8—472no—
174#467—941no—
175#467—941no—
176#467—941no—
177#467—941no—
178C2×#107—471no—
179C2×#106—21no—
180C2×#103—472no—
181C2×C22⋊C8—471no—
182C2×#97—941no—
183#380C2×D4712yes—
184C4×C22⋊C4—471no—
185#342—944no—
186#340—21no—
187#336—941no—
188#328—944no—
189C47#978411yescharacteristic, complemented, Sylow
190C22×Q8—472no—
191C23×C4C22×D4711yescharacteristic
192C2×Q16—941no—
193C2×Q16—941no—
194C2×Q16—1882no—
195C22×C8—42no—
196C22×C8—941no—
197C22×C8C4×D4711yescharacteristic
198C22×C8C4×D4711yescharacteristic
199C22×C8D18811yescharacteristic
200C22×C8D18811yescharacteristic
201C22×C8C22⋊D4711yescharacteristic
202C22×C8C22⋊D4711yescharacteristic
203C22⋊Q8—942no—
204C22⋊Q8—942no—
205C22⋊Q8—942no—
206C22⋊Q8—942no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—472no—
209C2×C4⋊C4—942no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—472no—
212C2×C22⋊C4—942no—
213C2×C42—941no—
214#107—941no—
215#107—941no—
216#106—42no—
217#103—942no—
218#103—942no—
219C22⋊C8—472no—
220C22⋊C8—941no—
221#97—1882no—
222#95—22no—
223#95—941no—
224C24D4×D4711yescharacteristic
225C2×Q8—942no—
226C2×Q8—942no—
227C2×Q8—1882no—
228C2×Q8—1882no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—942no—
233C22×C4—942no—
234C22×C4—942no—
235C22×C4C2×C4×D4711yescharacteristic
236C22×C4C2×D18811yescharacteristic
237C22×C4D4×D4711yescharacteristic
238C22×C4#1062535411yescharacteristic
239C22×C4#1062535411yescharacteristic
240C22×C4C2×C22⋊D4711yescharacteristic
241Q16—1882no—
242Q16—1882no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—942no—
250C2×C8—1881no—
251C2×C8—1881no—
252C2×C8(C2×D47)⋊C411yescharacteristic
253C2×C8(C2×D47)⋊C411yescharacteristic
254C2×C8(C2×D47)⋊C411yescharacteristic
255C2×C8(C2×D47)⋊C411yescharacteristic
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—942no—
259C4⋊C4—942no—
260C4⋊C4—1882no—
261C4⋊C4—1882no—
262C22⋊C4—44no—
263C22⋊C4—944no—
264C22⋊C4—1882no—
265C22⋊C4—1882no—
266C42—1881no—
267C42—1881no—
268C23—41no—
269C23—41no—
270C23#1309398611yescharacteristic
271C23#1309399911yescharacteristic
272C23#1309402011yescharacteristic
273C23#1309405314yes—
274Q8—3762no—
275Q8—1882no—
276Q8—1882no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—3762no—
290C2×C4—3762no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4—1882no—
294C2×C4—1882no—
295C2×C4—1882no—
296C2×C4C2×(C2×D47)⋊C411yescharacteristic
297C2×C4#1309398611yescharacteristic
298C2×C4#1309399911yescharacteristic
299C2×C4#1309401311yescharacteristic
300C8—21no—
301C8—21no—
302C8—21no—
303C8—1882no—
304C8#1309393812yes—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—44no—
312C22#1568902411yescharacteristic
313C22#1568922912yes—
314C22#1568938212yes—
315C22#1568951412yes—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—162no—
320C4—3762no—
321C4—3762no—
322C4—3762no—
323C4C2×#1309393811yescharacteristic
324C4#1568909611yescharacteristic
325C2—22no—
326C2—22no—
327C2—22no—
328C2#2024422311yescharacteristic
329C2#2024425711yescharacteristic
330C2#2024428411yescharacteristic
subgroupG/H
centerC22#15689024
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1807C2
socleC22×C47#390
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1807D47
hypocenterC47#9784
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
6. C22×Dic188C4semidirect
12. C2×#13093958C4semidirect
13. C2×#13093949C4semidirect
189. C47#9784semidirect

Maximal subgroups

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

subgroupindex
C2×#156893642
#202443722
#202431642
#202419592
C47×#18072
#978447