Finite Group Databasebeta

← #26108019

Subgroups — #26108019

Order 12032.

966 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 147 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×D4×Dic94C211yesmaximal, characteristic
2#20243380C211yesmaximal, characteristic
3#20242068C212yesmaximal, complemented
4#20241818C212yesmaximal, complemented
5C47×#1460C211yesmaximal, characteristic
6C23×Dic94C412yes—
7D4×Dic94—24no—
8D4×Dic94—24no—
9C2×#13094024C2211yescharacteristic
10C2×#13094023—21no—
11C2×#13094003C412yes—
12C2×#13094001—22no—
13C2×C4×Dic94C2211yescharacteristic
14C2×D4×Dic47—21no—
15#15689147C2212yes—
16C2×#13093941—21no—
17#15688892—21no—
18#15688871—24no—
19#15688855—24no—
20C2×C4×D4×C47C2211yescharacteristic
21C47×#431C2212yes—
22C47×#323—22no—
23C22×Dic94—22no—
24C22×Dic94—24no—
25C22×Dic94—24no—
26C22×Dic94—48no—
27C22×Dic94C2×C411yescharacteristic
28C22×Dic94C2×C412yes—
29C23×Dic47—22no—
30C2×#10625329—22no—
31#13094024—22no—
32#13094024D414yes—
33#13094023—28no—
34#13094003—24no—
35#13094003—24no—
36#13094001—216no—
37C4×Dic94—22no—
38C4×Dic94D414yes—
39D4×Dic47—28no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47C2×C411yescharacteristic
43C2×Dic47⋊C4—22no—
44C2×Dic47⋊C4—21no—
45C2×C22⋊Dic47—22no—
46C2×C4×Dic47—21no—
47C4⋊(C47⋊C8)—22no—
48C4⋊(C47⋊C8)—22no—
49#13093941—42no—
50C22×D4×C47D411yescharacteristic
51C23×C4×C47C2×C412yes—
52C4×D4×C47—22no—
53C4×D4×C47—22no—
54C2×C4⋊C4×C47D411yescharacteristic
55C2×C22⋊C4×C47D412yes—
56C2×C42×C47C2311yescharacteristic
57C4⋊C8×C47—22no—
58C22⋊C8×C47—22no—
59C4×C8×C47—22no—
60C24×C47C22⋊C412yes—
61C2×D4×C47—22no—
62C2×D4×C47—22no—
63C2×D4×C47SD1612yes—
64C2×D4×C47Q1612yes—
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—44no—
68C22×C4×C47C22⋊C412yes—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—21no—
73C22×Dic47—28no—
74C22×Dic47—21no—
75C22×Dic47—22no—
76C22×Dic47—48no—
77C2×Dic94—21no—
78C2×Dic94—24no—
79C2×Dic94—28no—
80C2×Dic94—22no—
81C2×Dic94—416no—
82C2×Dic94—48no—
83C2×Dic94—48no—
84C2×Dic94—48no—
85C2×Dic94—44no—
86C2×Dic94C22⋊C412yes—
87C2×Dic94C22⋊C414yes—
88C2×C8×C47—42no—
89C4⋊C4×C47—22no—
90C22⋊C4×C47—24no—
91C42×C47C2×D412yes—
92C42×C47C2×D412yes—
93C22⋊Dic47—28no—
94C4⋊Dic47—21no—
95C4⋊Dic47—24no—
96C4⋊Dic47—24no—
97C4⋊Dic47C22⋊C412yes—
98Dic47⋊C4—28no—
99Dic47⋊C4—24no—
100C4×Dic47—24no—
101#10625329—42no—
102#10625329—42no—
103C2×C47⋊C8—42no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—42no—
107C23×C47—42no—
108C23×C47#10314yes—
109C23×C47C2×C22⋊C411yescharacteristic
110D4×C47—24no—
111D4×C47—24no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—41no—
115C2×C4×C47—44no—
116C2×C4×C47—44no—
117C2×C4×C47—42no—
118C2×C4×C47C2×C22⋊C411yescharacteristic
119C2×C4×C47C2×C22⋊C411yescharacteristic
120C2×C4×C47#12012yes—
121C2×C4×C47#12012yes—
122C2×C4×C47C2×SD1611yescharacteristic
123C2×C4×C47C2×Q1611yescharacteristic
124C2×C4×C47#13712yes—
125C2×Dic47—24no—
126C2×Dic47—22no—
127C2×Dic47—22no—
128C2×Dic47—24no—
129C2×Dic47—416no—
130C2×Dic47—44no—
131C2×Dic47—42no—
132Dic94—24no—
133Dic94—22no—
134Dic94—416no—
135Dic94—44no—
136Dic94—44no—
137Dic94#10014yescomplemented
138C8×C47—82no—
139C47⋊C8—82no—
140#8391—471nomaximal, Sylow
141C22×C47—21no—
142C22×C47—21no—
143C22×C47—24no—
144C22×C47—44no—
145C22×C47—44no—
146C22×C47—42no—
147C22×C47—42no—
148C22×C47#37911yescharacteristic
149C22×C47C2×#10311yescharacteristic
150C22×C47#41911yescharacteristic
151C4×C47—22no—
152C4×C47—21no—
153C4×C47—42no—
154C4×C47C2×#10012yes—
155C4×C47#44812yes—
156C4×C47#45112yes—
157Dic47—24no—
158Dic47—42no—
159Dic47—44no—
160C2×D4×Q8—471no—
161#1800—471no—
162#1548—472no—
163#1487—472no—
164#1460D4711yescharacteristic
165C2×C47—21no—
166C2×C47—21no—
167C2×C47—44no—
168C2×C47#178311yescharacteristic
169C2×C47#178711yescharacteristic
170C2×C47#179611yescharacteristic
171C23×Q8—472no—
172D4×Q8—944no—
173D4×Q8—944no—
174C2×C4⋊Q8—471no—
175C2×C4⋊Q8—941no—
176C2×C22⋊Q8—472no—
177C2×C22⋊Q8—942no—
178C2×C4×Q8—471no—
179C2×C4×D4—941no—
180C2×C4×D4C2×D4711yescharacteristic
181#431C2×D4712yes—
182#423—472no—
183C2×#100—941no—
184#343—941no—
185#333—944no—
186Q8⋊C8—944no—
187#323—22no—
188C47#839111yescharacteristic, complemented, Sylow
189C22×Q8—471no—
190C22×Q8—472no—
191C22×Q8—944no—
192C22×Q8—944no—
193C22×Q8—942no—
194C22×Q8—1888no—
195C22×D4D18811yescharacteristic
196C23×C4—942no—
197C23×C4C4×D4712yes—
198C2×M16—942no—
199C4⋊Q8—474no—
200C4⋊Q8—948no—
201C4⋊Q8—942no—
202C22⋊Q8—9416no—
203C22⋊Q8—944no—
204C22⋊Q8—944no—
205C4×Q8—474no—
206C4×Q8—942no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—948no—
210C2×C4⋊C4—471no—
211C2×C4⋊C4—942no—
212C2×C4⋊C4—941no—
213C2×C4⋊C4—941no—
214C2×C4⋊C4—941no—
215C2×C4⋊C4D18811yescharacteristic
216C2×C22⋊C4—942no—
217C2×C22⋊C4C22⋊D4712yes—
218C2×C42—941no—
219C2×C42C22×D4711yescharacteristic
220C4⋊C8—22no—
221C4⋊C8—942no—
222C4⋊C8—942no—
223#100—1882no—
224C22⋊C8—22no—
225C4×C8—22no—
226C24(C2×D47)⋊C412yes—
227C2×Q8—472no—
228C2×Q8—474no—
229C2×Q8—944no—
230C2×Q8—948no—
231C2×Q8—941no—
232C2×Q8—942no—
233C2×Q8—1884no—
234C2×Q8—1888no—
235C2×Q8—1888no—
236C2×Q8—18816no—
237C2×Q8—1888no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4#1062532512yes—
241C2×D4Dic18812yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—44no—
245C22×C4—941no—
246C22×C4—948no—
247C22×C4—941no—
248C22×C4—942no—
249C22×C4—1888no—
250C22×C4(C2×D47)⋊C412yes—
251C22×C4C2×C4×D4711yescharacteristic
252C22×C4C2×D18811yescharacteristic
253C22×C4C2×C22⋊D4711yescharacteristic
254M16—1882no—
255M16—1882no—
256C2×C8—42no—
257C2×C8—1882no—
258C4⋊C4—22no—
259C4⋊C4—472no—
260C4⋊C4—948no—
261C4⋊C4—941no—
262C4⋊C4—944no—
263C4⋊C4—944no—
264C4⋊C4—944no—
265C22⋊C4—24no—
266C22⋊C4—948no—
267C42—944no—
268C42D4×D4712yes—
269C42D4×D4712yes—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23#1309394914yes—
275C23C2×(C2×D47)⋊C411yescharacteristic
276Q8—474no—
277Q8—944no—
278Q8—942no—
279Q8—1884no—
280Q8—18816no—
281Q8—1884no—
282D4—24no—
283D4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—944no—
291C2×C4—942no—
292C2×C4—942no—
293C2×C4—944no—
294C2×C4—18816no—
295C2×C4—1882no—
296C2×C4—1884no—
297C2×C4C22⋊C4×D4711yescharacteristic
298C2×C4C22⋊C4×D4711yescharacteristic
299C2×C4#1309399212yes—
300C2×C4#1309399412yes—
301C2×C4C2×#1062532511yescharacteristic
302C2×C4C2×Dic18811yescharacteristic
303C2×C4#1309405712yes—
304C8—82no—
305C8—3762no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—44no—
310C22—44no—
311C22—42no—
312C22—42no—
313C22#1568897811yescharacteristic
314C22C2×#1309394911yescharacteristic
315C22#1568913311yescharacteristic
316C4—21no—
317C4—22no—
318C4—42no—
319C4—944no—
320C4—1884no—
321C4—1882no—
322C4#100×D4712yes—
323C4#1568926412yes—
324C4#1568928112yes—
325C2—21no—
326C2—21no—
327C2—44no—
328C2#2024330711yescharacteristic
329C2#2024332411yescharacteristic
330C2#2024336111yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1460C2
socleC22×C47#379
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#1460D47
hypocenterC47#8391
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #20242068C2semidirect
4. #20241818C2semidirect
137. Dic94#100semidirect
188. C47#8391semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic942
#202433802
#202420682
#202418182
C47×#14602
#839147