Finite Group Databasebeta

← #26132269

Subgroups — #26132269

Order 12032.

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

normal subgroups only

This group has 588 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×#15688916C212yesmaximal
2#20243376C212yesmaximal, complemented
3#20241513C212yesmaximal, complemented
4C47×#2328C211yesmaximal, characteristic
5C2×#13093939C2214yes—
6#15688983—22no—
7#15688916—24no—
8#15688890—22no—
9C2×#127×C47C2211yescharacteristic
10C2×C4×D4×C47C411yescharacteristic, complemented
11C2×C4×D4×C47C2211yescharacteristic
12C47×#390C411yescharacteristic, complemented
13C47×#390C2211yescharacteristic
14C47×#386—22no—
15C47×#386—22no—
16C47×#382C411yescharacteristic, complemented
17C47×#376C411yescharacteristic, complemented
18C2×C4⋊Dic47—42no—
19C2×C22⋊Dic47—24no—
20#13093939—24no—
21#13093939—24no—
22C47⋊#95—42no—
23C22×D4×C47—22no—
24C22×D4×C47C2×C411yescharacteristic
25C22×D4×C47C2311yescharacteristic
26C23×C4×C47—21no—
27C23×C4×C47D412yes—
28#127×C47—22no—
29#127×C47D412yes—
30#127×C47Q812yes—
31C4×D4×C47—42no—
32C4×D4×C47—42no—
33C2×C4⋊C4×C47C2×C411yescharacteristic
34C2×C4⋊C4×C47C2×C411yescharacteristic
35C2×C22⋊C4×C47—21no—
36C2×C22⋊C4×C47—22no—
37C2×C22⋊C4×C47—22no—
38C2×C22⋊C4×C47D412yes—
39C2×C42×C47C2×C411yescharacteristic
40C2×C42×C47C2×C411yescharacteristic
41C2×C42×C47C2×C411yescharacteristic
42#95×C47—21no—
43#95×C47—21no—
44#95×C47—22no—
45C24×C47—21no—
46C24×C47C2×D412yes—
47C2×D4×C47—24no—
48C2×D4×C47—24no—
49C2×D4×C47—21no—
50C2×D4×C47—41no—
51C2×D4×C47—41no—
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C2×D4×C47—82no—
55C2×D4×C47—82no—
56C2×D4×C47C4⋊C412yescomplemented
57C2×D4×C47C2×D411yescharacteristic
58C2×D4×C47C2×Q811yescharacteristic
59C2×D4×C47#4112yes—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—42no—
66C22×C4×C47—82no—
67C22×C4×C47—82no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C412yes—
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C22×Dic47—44no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C22⋊C4×C47—42no—
76C22⋊C4×C47—42no—
77C22⋊C4×C47—82no—
78C22⋊C4×C47—82no—
79C42×C47—22no—
80C42×C47—21no—
81C42×C47—41no—
82C42×C47C4⋊C412yescomplemented
83C22⋊Dic47—44no—
84C22⋊Dic47—44no—
85C4⋊Dic47—84no—
86C23×C47—22no—
87C23×C47—22no—
88C23×C47—22no—
89C23×C47—42no—
90C23×C47—42no—
91C23×C47—42no—
92C23×C47—42no—
93C23×C47—41no—
94C23×C47—41no—
95C23×C47C2×C22⋊C411yescharacteristic
96C23×C47#12112yes—
97C23×C47C22⋊Q812yes—
98D4×C47—41no—
99D4×C47—41no—
100D4×C47—42no—
101D4×C47—42no—
102D4×C47—42no—
103D4×C47—84no—
104D4×C47—84no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—22no—
108C2×C4×C47—21no—
109C2×C4×C47—22no—
110C2×C4×C47—41no—
111C2×C4×C47—44no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—42no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—82no—
122C2×C4×C47—84no—
123C2×C4×C47#9912yes—
124C2×C4×C47C2×C4⋊C411yescharacteristic
125C2×C4×C47#11711yescharacteristic
126C2×Dic47—84no—
127C2×Dic47—84no—
128C2×Dic47—84no—
129#12799—471nomaximal, Sylow
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—22no—
133C22×C47—22no—
134C22×C47—42no—
135C22×C47—42no—
136C22×C47—44no—
137C22×C47—42no—
138C22×C47—42no—
139C22×C47—42no—
140C22×C47—82no—
141C22×C47—82no—
142C22×C47—81no—
143C22×C47—81no—
144C22×C47#38011yescharacteristic
145C22×C47C2×#9911yescharacteristic
146C22×C47#41011yescharacteristic
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—22no—
150C4×C47—42no—
151C4×C47—42no—
152C4×C47—41no—
153C4×C47—81no—
154C4×C47—81no—
155Dic47—164no—
156#2328D4711yescharacteristic
157C2×#353—472no—
158#1799—472no—
159#1393—472no—
160C2×C47—21no—
161C2×C47—21no—
162C2×C47—42no—
163C2×C47—42no—
164C2×C47—82no—
165C2×C47#179911yescharacteristic
166C2×C47#212711yescharacteristic
167C2×C47#212811yescharacteristic
168C2×#127C2×D4711yescharacteristic
169C2×C4×D4Dic4711yescharacteristic, complemented
170C2×C4×D4C2×D4711yescharacteristic
171C2×#99—474no—
172#390Dic4711yescharacteristic, complemented
173#390C2×D4711yescharacteristic
174#386—22no—
175#386—22no—
176#382Dic4711yescharacteristic, complemented
177#380—942no—
178#376Dic4711yescharacteristic, complemented
179#353—944no—
180#342—942no—
181C47#1279911yescharacteristic, complemented, Sylow
182C22×D4—22no—
183C22×D4C2×Dic4711yescharacteristic
184C22×D4C22×D4711yescharacteristic
185C23×C4—21no—
186C23×C4C22⋊D4712yes—
187#127—22no—
188#127Dic9412yes—
189#127D18812yes—
190C4×D4—42no—
191C4×D4—42no—
192C2×C4⋊C4—1882no—
193C2×C4⋊C4C2×Dic4711yescharacteristic
194C2×C4⋊C4C2×Dic4711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—944no—
199C2×C22⋊C4C22⋊D4712yes—
200C2×C42C2×Dic4711yescharacteristic
201C2×C42C2×Dic4711yescharacteristic
202C2×C42C2×Dic4711yescharacteristic
203#99—944no—
204#99—944no—
205#95—21no—
206#95—22no—
207#95—21no—
208#95—1882no—
209C24—21no—
210C24C2×C22⋊D4712yes—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—82no—
219C2×D4—82no—
220C2×D4C4⋊Dic4712yescomplemented
221C2×D4C2×Dic9411yescharacteristic
222C2×D4C2×D18811yescharacteristic
223C2×D4#1062535012yes—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—1884no—
233C22×C4C22⋊Dic4711yescharacteristic
234C22×C4C22⋊Dic4712yes—
235C22×C4C22⋊Dic4711yescharacteristic
236C22×C4C22×Dic4711yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—3764no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—82no—
244C22⋊C4—1884no—
245C22⋊C4—1884no—
246C42—22no—
247C42—21no—
248C42—41no—
249C42C4⋊Dic4712yescomplemented
250C23—22no—
251C23—22no—
252C23—22no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23C2×C22⋊Dic4711yescharacteristic
260C23#1309399812yes—
261C23#1309400312yes—
262D4—41no—
263D4—41no—
264D4—42no—
265D4—42no—
266D4—42no—
267D4—84no—
268D4—84no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4—3764no—
288C2×C4—3764no—
289C2×C4—3764no—
290C2×C4#1309393912yes—
291C2×C4C2×C4⋊Dic4711yescharacteristic
292C2×C4#1309398311yescharacteristic
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22—82no—
305C22—81no—
306C22—81no—
307C22#1568898311yescharacteristic
308C22C2×#1309393911yescharacteristic
309C22#1568909711yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—41no—
316C4—81no—
317C4—81no—
318C4—7524no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2—42no—
323C2—82no—
324C2#2024337611yescharacteristic
325C2#2024505711yescharacteristic
326C2#2024506011yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC22×D4C22×D47
FittingC47×#2328C2
socleC22×C47C2×#99
prime radicalC2×#127×C47C22
cosocleC22×D4×C47C23
hypercenter#2328D47
hypocenterC47#12799
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #20243376C2semidirect
3. #20241513C2semidirect
10. C2×C4×D4×C47C4semidirect
12. C47×#390C4semidirect
16. C47×#382C4semidirect
17. C47×#376C4semidirect
56. C2×D4×C47C4⋊C4semidirect
82. C42×C47C4⋊C4semidirect
169. C2×C4×D4Dic47semidirect
172. #390Dic47semidirect
176. #382Dic47semidirect
178. #376Dic47semidirect
181. C47#12799semidirect
220. C2×D4C4⋊Dic47semidirect
249. C42C4⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156889162
#202433762
#202415132
C47×#23282
#1279947