Finite Group Databasebeta

← #26131786

Subgroups — #26131786

Order 12032.

1018 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×#13094022C211yesmaximal, characteristic
2C2×#15688921C214yesmaximal, complemented
3#20243312C211yesmaximal, characteristic
4C47×#1784C211yesmaximal, characteristic
5C23×Dic94C411yescharacteristic
6C2×#13094022—28no—
7C2×#13094022—21no—
8C2×#13094022—21no—
9C2×#13094022C412yes—
10C2×#13094022C2212yes—
11C22×Dic47⋊C4—21no—
12C22×Dic47⋊C4—21no—
13C22×C4×Dic47C411yescharacteristic
14C2×#13093943—21no—
15C2×#13093943C2212yes—
16#15689087—21no—
17#15688921—28no—
18C23×Q8×C47C2211yescharacteristic
19C2×#101×C47—21no—
20C2×#101×C47C2212yes—
21C47×#407—21no—
22C22×Dic94—21no—
23C22×Dic94—24no—
24C22×Dic94—42no—
25C22×Dic94—42no—
26C22×Dic94C2×C412yes—
27C23×Dic47—21no—
28C23×Dic47—21no—
29C2×#10625329—22no—
30#13094022—28no—
31#13094022—28no—
32#13094022—24no—
33#13094022—24no—
34#13094022—24no—
35#13094022D418yes—
36C2×Dic47⋊C4—28no—
37C2×Dic47⋊C4—28no—
38C2×Dic47⋊C4—22no—
39C2×Dic47⋊C4—22no—
40C2×Dic47⋊C4—22no—
41C2×Dic47⋊C4—22no—
42C2×C4×Dic47—21no—
43C2×C4×Dic47—24no—
44C2×C4×Dic47C2×C412yes—
45#13093943—24no—
46#13093943—42no—
47C22⋊(C47⋊C8)—42no—
48C22×Q8×C47—24no—
49C22×Q8×C47—42no—
50C22×Q8×C47—42no—
51C22×Q8×C47C2×C411yescharacteristic
52C22×Q8×C47D411yescharacteristic
53C22×Q8×C47D411yescharacteristic
54C22×Q8×C47C2311yescharacteristic
55C23×C4×C47C2×C411yescharacteristic
56C23×C4×C47D411yescharacteristic
57C23×C4×C47D411yescharacteristic
58C2×M16×C47—22no—
59#101×C47—24no—
60#101×C47—42no—
61C22⋊C8×C47—42no—
62C24×C47C22⋊C411yescharacteristic
63C2×Q8×C47—22no—
64C2×Q8×C47—22no—
65C2×Q8×C47—22no—
66C2×Q8×C47—22no—
67C2×Q8×C47—22no—
68C2×Q8×C47—41no—
69C2×Q8×C47—41no—
70C2×Q8×C47—41no—
71C2×Q8×C47—41no—
72C2×Q8×C47—41no—
73C2×Q8×C47—41no—
74C2×Q8×C47—416no—
75C2×Q8×C47C2×D414yes—
76C22×C4×C47—22no—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47—22no—
83C22×C4×C47—41no—
84C22×C4×C47—41no—
85C22×C4×C47C22⋊C411yescharacteristic
86C22×C4×C47C22⋊C411yescharacteristic
87C22×C4×C47C22⋊C411yescharacteristic
88C22×C4×C47C22×C411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×Dic47—24no—
92C22×Dic47—28no—
93C22×Dic47—22no—
94C22×Dic47—24no—
95C22×Dic47—22no—
96C22×Dic47—42no—
97C22×Dic47—42no—
98C2×Dic94—24no—
99C2×Dic94—22no—
100C2×Dic94—42no—
101C2×Dic94—416no—
102C2×Dic94—42no—
103C2×Dic94—42no—
104C2×Dic94—42no—
105C2×Dic94C22⋊C414yes—
106M16×C47—42no—
107M16×C47—42no—
108C2×C8×C47—42no—
109Dic47⋊C4—28no—
110Dic47⋊C4—28no—
111Dic47⋊C4—28no—
112Dic47⋊C4—28no—
113C4×Dic47—24no—
114C4×Dic47—22no—
115C4×Dic47C22⋊C414yes—
116#10625329—42no—
117#10625329—42no—
118C2×C47⋊C8—42no—
119C23×C47—24no—
120C23×C47—21no—
121C23×C47—21no—
122C23×C47#9912yes—
123C23×C47C2×C22⋊C411yescharacteristic
124Q8×C47—44no—
125Q8×C47—44no—
126Q8×C47—44no—
127Q8×C47—44no—
128C2×C4×C47—28no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—44no—
144C2×C4×C47—44no—
145C2×C4×C47#9912yes—
146C2×C4×C47C2×C22⋊C412yes—
147C2×C4×C47#12012yes—
148C2×C4×C47#12012yes—
149C2×Dic47—216no—
150C2×Dic47—28no—
151C2×Dic47—24no—
152C2×Dic47—24no—
153C2×Dic47—48no—
154C2×Dic47—42no—
155C2×Dic47—42no—
156Dic94—48no—
157Dic94—44no—
158Dic94—44no—
159C8×C47—82no—
160C47⋊C8—82no—
161#12700—471nomaximal, Sylow
162C22×C47—24no—
163C22×C47—24no—
164C22×C47—22no—
165C22×C47—22no—
166C22×C47—22no—
167C22×C47#35614yescomplemented
168C22×C47#37911yescharacteristic
169C22×C47C2×#9911yescharacteristic
170C22×C47C2×#9911yescharacteristic
171C4×C47—22no—
172C4×C47—22no—
173C4×C47—22no—
174C4×C47—22no—
175C4×C47—41no—
176C4×C47—41no—
177Dic47—28no—
178Dic47—44no—
179C22×C4⋊Q8—471no—
180C2×#356—474no—
181#1784—471no—
182#1784D4711yescharacteristic
183C2×C47—21no—
184C2×C47—24no—
185C2×C47—21no—
186C2×C47#178111yescharacteristic
187C2×C47C2×#35612yes—
188C23×Q8—471no—
189C23×Q8C2×D4711yescharacteristic
190C2×C4⋊Q8—472no—
191C2×C4⋊Q8—472no—
192C2×C4⋊Q8—948no—
193C2×C4⋊Q8—941no—
194C2×C4⋊Q8—941no—
195C22×C4⋊C4—941no—
196C22×C4⋊C4—941no—
197C22×C42—471no—
198C2×#101—21no—
199C2×#101—472no—
200C2×#101—941no—
201C2×#101C2×D4712yes—
202#407—21no—
203#407—941no—
204#356—948no—
205C47#1270011yescharacteristic, complemented, Sylow
206C22×Q8—24no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×Q8—472no—
210C22×Q8—944no—
211C22×Q8—941no—
212C22×Q8—1882no—
213C22×Q8—1882no—
214C22×Q8C4×D4711yescharacteristic
215C22×Q8D18811yescharacteristic
216C22×Q8C22⋊D4711yescharacteristic
217C22×Q8C22×D4711yescharacteristic
218C23×C4—941no—
219C23×C4—941no—
220C23×C4C4×D4711yescharacteristic
221C23×C4D18811yescharacteristic
222C23×C4C22⋊D4711yescharacteristic
223C2×M16—22no—
224C2×M16—942no—
225C4⋊Q8—478no—
226C4⋊Q8—944no—
227C4⋊Q8—944no—
228C4⋊Q8—948no—
229C4⋊Q8—948no—
230C4⋊Q8—944no—
231C2×C4⋊C4—948no—
232C2×C4⋊C4—948no—
233C2×C4⋊C4—942no—
234C2×C4⋊C4—942no—
235C2×C4⋊C4—942no—
236C2×C4⋊C4—942no—
237C2×C42—472no—
238C2×C42—944no—
239C2×C42—941no—
240#101—24no—
241#101—42no—
242#101—944no—
243#101—1882no—
244C22⋊C8—42no—
245C22⋊C8—1882no—
246C24(C2×D47)⋊C411yescharacteristic
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—416no—
259C2×Q8—474no—
260C2×Q8—944no—
261C2×Q8—942no—
262C2×Q8—18816no—
263C2×Q8—1882no—
264C2×Q8—1882no—
265C2×Q8—1882no—
266C2×Q8—1882no—
267C2×Q8D4×D4714yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—944no—
278C22×C4—948no—
279C22×C4—942no—
280C22×C4—944no—
281C22×C4—942no—
282C22×C4—1882no—
283C22×C4—1882no—
284C22×C4(C2×D47)⋊C411yescharacteristic
285C22×C4(C2×D47)⋊C411yescharacteristic
286C22×C4(C2×D47)⋊C411yescharacteristic
287C22×C4C2×C4×D4711yescharacteristic
288C22×C4C2×D18811yescharacteristic
289C22×C4C2×C22⋊D4711yescharacteristic
290M16—42no—
291M16—42no—
292M16—1882no—
293M16—1882no—
294C2×C8—42no—
295C2×C8—1882no—
296C4⋊C4—948no—
297C4⋊C4—948no—
298C4⋊C4—948no—
299C4⋊C4—948no—
300C42—474no—
301C42—944no—
302C42—942no—
303C23—21no—
304C23—21no—
305C23—24no—
306C23#1309393812yes—
307C23C2×(C2×D47)⋊C411yescharacteristic
308Q8—44no—
309Q8—44no—
310Q8—44no—
311Q8—44no—
312Q8—1884no—
313Q8—1888no—
314Q8—1884no—
315C2×C4—28no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—9416no—
333C2×C4—948no—
334C2×C4—944no—
335C2×C4—944no—
336C2×C4—1888no—
337C2×C4—1882no—
338C2×C4—1882no—
339C2×C4#1309393812yes—
340C2×C4C22⋊C4×D4712yes—
341C2×C4#1309399212yes—
342C2×C4#1309399412yes—
343C8—82no—
344C8—3762no—
345C22—24no—
346C22—24no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22#1568892114yescomplemented
351C22#1568897811yescharacteristic
352C22C2×#1309393811yescharacteristic
353C22C2×#1309393811yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—948no—
361C4—1884no—
362C2—21no—
363C2—21no—
364C2—24no—
365C2#2024330211yescharacteristic
366C2C2×#1568892112yes—
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC47×#1784C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC22×Q8×C47C23
hypercenter#1784D47
hypocenterC47#12700
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. C2×#15688921C2semidirect
167. C22×C47#356semidirect
205. C47#12700semidirect
350. C22#15688921semidirect

Maximal subgroups

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

subgroupindex
C22×#130940222
C2×#156889212
#202433122
C47×#17842
#1270047