Finite Group Databasebeta

← #26131764

Subgroups — #26131764

Order 12032.

1766 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1766 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×#15688916C214yesmaximal, complemented
2#20243303C212yesmaximal, complemented
3C22×#127×C47C211yesmaximal, characteristic
4C2×#13093939—22no—
5C2×#13093939C2214yescomplemented
6#15688979—22no—
7#15688916—28no—
8C23×D4×C47—22no—
9C23×D4×C47C411yescharacteristic, complemented
10C23×D4×C47C2211yescharacteristic
11C2×#127×C47—22no—
12C2×#127×C47—28no—
13C2×#127×C47C412yescomplemented
14C2×#127×C47C2212yes—
15C22×C42×C47C411yescharacteristic, complemented
16C2×C22⋊Dic47—24no—
17C2×C22⋊Dic47—44no—
18#13093939—28no—
19#13093939—44no—
20C25×C47—21no—
21C25×C47D412yes—
22C22×D4×C47—216no—
23C22×D4×C47—24no—
24C22×D4×C47—24no—
25C22×D4×C47—24no—
26C22×D4×C47—24no—
27C22×D4×C47—21no—
28C22×D4×C47—42no—
29C22×D4×C47—42no—
30C22×D4×C47—416no—
31C22×D4×C47—42no—
32C22×D4×C47—42no—
33C22×D4×C47C2×C411yescharacteristic
34C22×D4×C47C2×C412yes—
35C22×D4×C47D412yes—
36C22×D4×C47C2311yescharacteristic
37C23×C4×C47—21no—
38C23×C4×C47C2×C411yescharacteristic
39#127×C47—216no—
40#127×C47—24no—
41#127×C47—24no—
42#127×C47—24no—
43#127×C47D418yes—
44C2×C42×C47—21no—
45C2×C42×C47—24no—
46C2×C42×C47C2×C412yes—
47C24×C47—24no—
48C24×C47—22no—
49C24×C47—24no—
50C24×C47—24no—
51C24×C47—42no—
52C24×C47—48no—
53C24×C47—48no—
54C24×C47—42no—
55C24×C47—42no—
56C24×C47C22⋊C411yescharacteristic
57C24×C47C22⋊C412yes—
58C24×C47C2×D412yes—
59C2×D4×C47—24no—
60C2×D4×C47—216no—
61C2×D4×C47—24no—
62C2×D4×C47—216no—
63C2×D4×C47—24no—
64C2×D4×C47—22no—
65C2×D4×C47—22no—
66C2×D4×C47—41no—
67C2×D4×C47—41no—
68C2×D4×C47—42no—
69C2×D4×C47—41no—
70C2×D4×C47—41no—
71C2×D4×C47—416no—
72C2×D4×C47—416no—
73C2×D4×C47—416no—
74C2×D4×C47—416no—
75C2×D4×C47—464no—
76C2×D4×C47—42no—
77C2×D4×C47—42no—
78C2×D4×C47—42no—
79C2×D4×C47—42no—
80C2×D4×C47C22⋊C414yescomplemented
81C2×D4×C47C2×D414yes—
82C22×C4×C47—24no—
83C22×C4×C47—28no—
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47—22no—
87C22×C4×C47—22no—
88C22×C4×C47C22⋊C411yescharacteristic
89C22×C4×C47C22×C411yescharacteristic
90C22×Dic47—44no—
91C42×C47—24no—
92C42×C47—22no—
93C42×C47C22⋊C414yescomplemented
94C22⋊Dic47—44no—
95C22⋊Dic47—44no—
96C22⋊Dic47—88no—
97C23×C47—24no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—22no—
103C23×C47—24no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—48no—
107C23×C47—44no—
108C23×C47—44no—
109C23×C47—416no—
110C23×C47—416no—
111C23×C47—48no—
112C23×C47—48no—
113C23×C47—48no—
114C23×C47—42no—
115C23×C47—42no—
116C23×C47—84no—
117C23×C47—88no—
118C23×C47—88no—
119C23×C47—84no—
120C23×C47—84no—
121C23×C47#9912yes—
122C23×C47C2×C22⋊C411yescharacteristic
123C23×C47#12014yes—
124D4×C47—44no—
125D4×C47—48no—
126D4×C47—44no—
127D4×C47—432no—
128D4×C47—432no—
129D4×C47—48no—
130D4×C47—44no—
131D4×C47—44no—
132C2×C4×C47—216no—
133C2×C4×C47—28no—
134C2×C4×C47—28no—
135C2×C4×C47—21no—
136C2×C4×C47—24no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47#9912yes—
141C2×C4×C47C2×C22⋊C412yes—
142C2×Dic47—84no—
143C2×Dic47—88no—
144#12693—471nomaximal, Sylow
145C22×C47—24no—
146C22×C47—24no—
147C22×C47—22no—
148C22×C47—22no—
149C22×C47—22no—
150C22×C47—24no—
151C22×C47—44no—
152C22×C47—44no—
153C22×C47—48no—
154C22×C47—44no—
155C22×C47—44no—
156C22×C47—42no—
157C22×C47—44no—
158C22×C47—82no—
159C22×C47—816no—
160C22×C47—82no—
161C22×C47—816no—
162C22×C47—84no—
163C22×C47—84no—
164C22×C47#35314yescomplemented
165C22×C47#37911yescharacteristic
166C22×C47C2×#9911yescharacteristic
167C22×C47C2×#9911yescharacteristic
168C4×C47—28no—
169C4×C47—22no—
170C4×C47—22no—
171Dic47—164no—
172C22×#127D4711yescharacteristic
173C2×#353—474no—
174#1781—472no—
175C2×C47—21no—
176C2×C47—24no—
177C2×C47—21no—
178C2×C47—44no—
179C2×C47—82no—
180C2×C47—84no—
181C2×C47#178111yescharacteristic
182C2×C47C2×#35312yes—
183C23×D4—22no—
184C23×D4Dic4711yescharacteristic, complemented
185C23×D4C2×D4711yescharacteristic
186C2×#127—22no—
187C2×#127—28no—
188C2×#127Dic4712yescomplemented
189C2×#127C2×D4712yes—
190C22×C42Dic4711yescharacteristic, complemented
191C2×#99—474no—
192C2×#99—942no—
193#379—942no—
194#353—948no—
195C47#1269311yescharacteristic, complemented, Sylow
196C25—21no—
197C25C22⋊D4712yes—
198C22×D4—216no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—21no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—416no—
208C22×D4—42no—
209C22×D4C2×Dic4711yescharacteristic
210C22×D4C2×Dic4712yes—
211C22×D4C22⋊D4712yes—
212C22×D4C22×D4711yescharacteristic
213C23×C4—21no—
214C23×C4C2×Dic4711yescharacteristic
215#127—216no—
216#127—24no—
217#127—24no—
218#127—24no—
219#127C22⋊D4718yes—
220C2×C22⋊C4—944no—
221C2×C22⋊C4—1884no—
222C2×C42—21no—
223C2×C42—24no—
224C2×C42C2×Dic4712yes—
225#99—948no—
226#99—1884no—
227C24—24no—
228C24—22no—
229C24—24no—
230C24—24no—
231C24—42no—
232C24—48no—
233C24—42no—
234C24—48no—
235C24—42no—
236C24C22⋊Dic4712yes—
237C24C22⋊Dic4711yescharacteristic
238C24C2×C22⋊D4712yes—
239C2×D4—24no—
240C2×D4—216no—
241C2×D4—24no—
242C2×D4—216no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—416no—
254C2×D4—416no—
255C2×D4—416no—
256C2×D4—416no—
257C2×D4—464no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4C22⋊Dic4714yescomplemented
261C2×D4C2×C22⋊D4714yes—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—1884no—
269C22×C4C22⋊Dic4711yescharacteristic
270C22×C4C22×Dic4711yescharacteristic
271C22⋊C4—3768no—
272C22⋊C4—1884no—
273C22⋊C4—1884no—
274C42—24no—
275C42—22no—
276C42C22⋊Dic4714yescomplemented
277C23—21no—
278C23—21no—
279C23—24no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—24no—
284C23—22no—
285C23—22no—
286C23—48no—
287C23—44no—
288C23—44no—
289C23—416no—
290C23—416no—
291C23—48no—
292C23—48no—
293C23—42no—
294C23—42no—
295C23—48no—
296C23—84no—
297C23—88no—
298C23—88no—
299C23—84no—
300C23—84no—
301C23#1309393912yes—
302C23C2×C22⋊Dic4711yescharacteristic
303C23#1309399314yes—
304D4—44no—
305D4—48no—
306D4—44no—
307D4—432no—
308D4—432no—
309D4—48no—
310D4—44no—
311D4—44no—
312C2×C4—216no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—21no—
316C2×C4—24no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—3764no—
321C2×C4—3768no—
322C2×C4#1309393912yes—
323C2×C4C2×C22⋊Dic4712yes—
324C22—24no—
325C22—24no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—24no—
330C22—44no—
331C22—44no—
332C22—48no—
333C22—44no—
334C22—44no—
335C22—42no—
336C22—44no—
337C22—82no—
338C22—816no—
339C22—82no—
340C22—84no—
341C22—816no—
342C22—84no—
343C22#1568891614yescomplemented
344C22#1568897911yescharacteristic
345C22C2×#1309393911yescharacteristic
346C22C2×#1309393911yescharacteristic
347C4—28no—
348C4—22no—
349C4—22no—
350C4—7524no—
351C2—21no—
352C2—21no—
353C2—24no—
354C2—44no—
355C2—82no—
356C2—84no—
357C2#2024330311yescharacteristic
358C2C2×#1568891612yes—
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC22×D4C22×D47
Fitting, prime radicalC22×#127×C47C2
socleC22×C47C2×#99
cosocleC22×D4×C47C23
hypercenterC22×#127D47
hypocenterC47#12693
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C22×#127

Complements

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

NH ≅ G/Ntype
1. C2×#15688916C2semidirect
2. #20243303C2semidirect
5. C2×#13093939C22semidirect
9. C23×D4×C47C4semidirect
13. C2×#127×C47C4semidirect
15. C22×C42×C47C4semidirect
80. C2×D4×C47C22⋊C4semidirect
93. C42×C47C22⋊C4semidirect
164. C22×C47#353semidirect
184. C23×D4Dic47semidirect
188. C2×#127Dic47semidirect
190. C22×C42Dic47semidirect
195. C47#12693semidirect
260. C2×D4C22⋊Dic47semidirect
276. C42C22⋊Dic47semidirect
343. C22#15688916semidirect

Maximal subgroups

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

subgroupindex
C2×#156889162
#202433032
C22×#127×C472
#1269347