Finite Group Databasebeta

← #26108012

Subgroups — #26108012

Order 12032.

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

normal subgroups only

This group has 738 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
1C2×#15689843C211yesmaximal, characteristic
2#20242061C212yesmaximal, complemented
3#20241816C212yesmaximal, complemented
4#20241709C211yesmaximal, characteristic
5C47×#1800C211yesmaximal, characteristic
6#15689843—24no—
7#15689843—24no—
8C2×#13094024C2211yescharacteristic
9C2×#13094017—22no—
10C2×#13094011—21no—
11C2×#13094003C412yes—
12C2×C4×Dic94C2211yescharacteristic
13C2×#13093989—21no—
14C2×#13093983C412yes—
15#15689176C2212yes—
16#15688870—24no—
17#15688853—24no—
18#15688847—22no—
19C2×C4×D4×C47C2211yescharacteristic
20C47×#423C2212yes—
21C2×#100×C47—21no—
22C47×#343—21no—
23C22×Dic94—21no—
24C22×Dic94C2×C411yescharacteristic
25#13094024—22no—
26#13094024D414yes—
27#13094017—216no—
28#13094011—28no—
29#13094003—24no—
30#13094003—24no—
31C4×Dic94—22no—
32C4×Dic94D414yes—
33#13093989—28no—
34#13093983—24no—
35#13093983—24no—
36C2×C4⋊Dic47—21no—
37C2×C4⋊Dic47C2×C411yescharacteristic
38C2×C4⋊Dic47C2×C412yes—
39C2×Dic47⋊C4—22no—
40C2×Dic47⋊C4—21no—
41C2×C22⋊Dic47—22no—
42C2×C22⋊Dic47—22no—
43C2×C4×Dic47—22no—
44C2×C4×Dic47—21no—
45C4⋊(C47⋊C8)—22no—
46C22⋊(C47⋊C8)—22no—
47C4×C47⋊C8—22no—
48C22×D4×C47D411yescharacteristic
49C23×C4×C47C2×C412yes—
50C2×M16×C47—22no—
51C4×D4×C47—22no—
52C4×D4×C47—22no—
53C2×C4⋊C4×C47D411yescharacteristic
54C2×C22⋊C4×C47D412yes—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—22no—
57C4⋊C8×C47—22no—
58#100×C47—42no—
59C24×C47C22⋊C412yes—
60C2×D4×C47—22no—
61C2×D4×C47—22no—
62C2×D4×C47—21no—
63C2×D4×C47—21no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—44no—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—22no—
74C2×Dic94—21no—
75C2×Dic94—24no—
76C2×Dic94—42no—
77C2×Dic94—42no—
78C2×Dic94—41no—
79C2×Dic94—41no—
80C2×Dic94C22⋊C412yes—
81M16×C47—42no—
82M16×C47—42no—
83C2×C8×C47—42no—
84C4⋊C4×C47—22no—
85C22⋊C4×C47—24no—
86C42×C47C2×D412yes—
87C42×C47C2×D412yes—
88C22⋊Dic47—28no—
89C22⋊Dic47—28no—
90C4⋊Dic47—21no—
91C4⋊Dic47—22no—
92C4⋊Dic47—24no—
93C4⋊Dic47C22⋊C412yes—
94C4⋊Dic47C22⋊C414yes—
95Dic47⋊C4—28no—
96Dic47⋊C4—24no—
97C4×Dic47—28no—
98C4×Dic47—24no—
99C2×C47⋊C8—42no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—22no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47C2×C22⋊C411yescharacteristic
106D4×C47—42no—
107D4×C47—42no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—41no—
111C2×C4×C47—44no—
112C2×C4×C47—44no—
113C2×C4×C47—42no—
114C2×C4×C47C2×C22⋊C411yescharacteristic
115C2×C4×C47C2×C22⋊C411yescharacteristic
116C2×C4×C47#12012yes—
117C2×C4×C47#12012yes—
118C2×C4×C47#13512yes—
119C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C47#13711yescharacteristic
121C2×Dic47—22no—
122C2×Dic47—22no—
123C2×Dic47—24no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128Dic94—24no—
129Dic94—42no—
130Dic94—44no—
131Dic94—44no—
132C8×C47—82no—
133C47⋊C8—82no—
134#8389—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—42no—
138C22×C47—42no—
139C22×C47—42no—
140C22×C47—82no—
141C22×C47—82no—
142C22×C47#37911yescharacteristic
143C22×C47#41611yescharacteristic
144C22×C47#41711yescharacteristic
145C4×C47—22no—
146C4×C47—21no—
147C4×C47—42no—
148C4×C47#41312yes—
149C4×C47#44912yes—
150C4×C47#45212yes—
151Dic47—42no—
152Dic47—42no—
153Dic47—44no—
154C2×#556—471no—
155#1800D4711yescharacteristic
156#1546—472no—
157#1487—472no—
158#1460—471no—
159C2×C47—21no—
160C2×C47—21no—
161C2×C47—82no—
162C2×C47#178511yescharacteristic
163C2×C47#178811yescharacteristic
164C2×C47#179611yescharacteristic
165#556—944no—
166#556—944no—
167C2×C4⋊Q8—471no—
168C2×#126—942no—
169C2×#124—941no—
170C2×C22⋊Q8—472no—
171C2×C4×Q8—471no—
172C2×C4×Q8—941no—
173C2×C4×D4C2×D4711yescharacteristic
174C2×#117—472no—
175#431—472no—
176#423C2×D4712yes—
177C2×#100—21no—
178#343—21no—
179#332—944no—
180Q8⋊C8—944no—
181#323—942no—
182C47#838911yescharacteristic, complemented, Sylow
183C22×Q8—471no—
184C22×Q8—941no—
185C22×D4C22⋊D4711yescharacteristic
186C23×C4C4×D4712yes—
187C2×M16—22no—
188C4⋊Q8—474no—
189C4⋊Q8—942no—
190#126—9416no—
191#124—948no—
192C22⋊Q8—944no—
193C22⋊Q8—944no—
194C4×Q8—474no—
195C4×Q8—948no—
196C4×Q8—942no—
197C4×D4—22no—
198C4×D4—22no—
199#117—944no—
200#117—944no—
201C2×C4⋊C4—471no—
202C2×C4⋊C4—472no—
203C2×C4⋊C4—942no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4—941no—
206C2×C4⋊C4C22⋊D4711yescharacteristic
207C2×C22⋊C4—942no—
208C2×C22⋊C4—942no—
209C2×C22⋊C4D18812yes—
210C2×C42—942no—
211C2×C42—941no—
212C2×C42C22×D4711yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—942no—
216#100—42no—
217C22⋊C8—942no—
218C4×C8—942no—
219C24(C2×D47)⋊C412yes—
220C2×Q8—472no—
221C2×Q8—941no—
222C2×Q8—944no—
223C2×Q8—1882no—
224C2×Q8—1882no—
225C2×Q8—1881no—
226C2×Q8—1881no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4—941no—
235C22×C4—942no—
236C22×C4—941no—
237C22×C4(C2×D47)⋊C412yes—
238C22×C4C2×C4×D4711yescharacteristic
239C22×C4C2×D18811yescharacteristic
240C22×C4C2×C22⋊D4711yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—42no—
244C2×C8—1882no—
245C4⋊C4—22no—
246C4⋊C4—472no—
247C4⋊C4—474no—
248C4⋊C4—948no—
249C4⋊C4—941no—
250C4⋊C4—942no—
251C4⋊C4—944no—
252C4⋊C4—944no—
253C22⋊C4—24no—
254C22⋊C4—948no—
255C22⋊C4—948no—
256C42—948no—
257C42—944no—
258C42D4×D4712yes—
259C42D4×D4712yes—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C2×(C2×D47)⋊C411yescharacteristic
266Q8—944no—
267Q8—1882no—
268Q8—1884no—
269Q8—1884no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—942no—
279C2×C4—944no—
280C2×C4—942no—
281C2×C4—1882no—
282C2×C4—1882no—
283C2×C4—1882no—
284C2×C4—1882no—
285C2×C4C22⋊C4×D4711yescharacteristic
286C2×C4C22⋊C4×D4711yescharacteristic
287C2×C4#1309399212yes—
288C2×C4#1309399412yes—
289C2×C4#1309404412yes—
290C2×C4#1309405011yescharacteristic
291C2×C4#1309405511yescharacteristic
292C8—82no—
293C8—3762no—
294C22—21no—
295C22—21no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22#1568897811yescharacteristic
302C22#1568911711yescharacteristic
303C22#1568912211yescharacteristic
304C4—21no—
305C4—22no—
306C4—42no—
307C4—1882no—
308C4—1884no—
309C4—1882no—
310C4#1568910412yes—
311C4#1568926612yes—
312C4#1568928312yes—
313C2—21no—
314C2—21no—
315C2—82no—
316C2#2024331311yescharacteristic
317C2#2024332611yescharacteristic
318C2#2024336411yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1800C2
socleC22×C47#379
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#1800D47
hypocenterC47#8389
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #20242061C2semidirect
3. #20241816C2semidirect
182. C47#8389semidirect

Maximal subgroups

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

subgroupindex
C2×#156898432
#202420612
#202418162
#202417092
C47×#18002
#838947