Finite Group Databasebeta

← #26316221

Subgroups — #26316221

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20249510C211yesmaximal, characteristic
2#20246108C211yesmaximal, characteristic
3#20244537C212yesmaximal
4#20244527C212yesmaximal
5#20244209C211yesmaximal, characteristic
6#20243523C211yesmaximal, characteristic
7#20243519C211yesmaximal, characteristic
8#20243401C212yesmaximal
9#20243391C212yesmaximal
10#20243246C211yesmaximal, characteristic
11C47×#1956C211yesmaximal, characteristic
12C2×#13094023C2211yescharacteristic
13C2×#13094015C2211yescharacteristic
14C2×Q8×Dic47C2212yes—
15C2×Dic47⋊C8C2212yes—
16C2×#13093948C2212yes—
17C2×#13093948C2212yes—
18C2×#13093935C2211yescharacteristic
19#15689056—22no—
20#15689043—22no—
21#15689030C2212yes—
22#15689009—22no—
23#15689009—22no—
24#15688992C2211yescharacteristic
25#15688989C2211yescharacteristic
26#15688989C2211yescharacteristic
27#15688988C2211yescharacteristic
28#15688969C2211yescharacteristic
29#15688886C2212yes—
30#15688886C2212yes—
31#15688885C2212yes—
32#15688885C2212yes—
33#15688877C2212yes—
34C2×C4⋊Q8×C47C2211yescharacteristic
35C2×#103×C47C2212yes—
36C2×#103×C47C2212yes—
37C2×C4×C8×C47C2211yescharacteristic
38C47×#384C2211yescharacteristic
39C22×Dic94—21no—
40C22×C47⋊C8C2312yes—
41#13094023—42no—
42#13094015—42no—
43Q8×Dic47—24no—
44Q8×Dic47—24no—
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—21no—
47C2×C4⋊Dic47C2311yescharacteristic
48C2×C4⋊Dic47C2311yescharacteristic
49C2×C4⋊Dic47C2312yes—
50C2×Dic47⋊C4—21no—
51C2×Dic47⋊C4—21no—
52C2×Dic47⋊C4—22no—
53C2×C4×Dic47—22no—
54C2×C4×Dic47C2311yescharacteristic
55C2×C4×Dic47C2311yescharacteristic
56Dic47⋊C8—24no—
57#13093948—42no—
58#13093948—42no—
59#13093935—22no—
60C47⋊#95—22no—
61C47⋊#95—22no—
62C47⋊#95—22no—
63C47⋊#95—22no—
64C47⋊#95—22no—
65C47⋊#95D412yes—
66C47⋊#95D412yes—
67C22×Q8×C47C2312yes—
68C22×C8×C47C2312yes—
69C4⋊Q8×C47—22no—
70C4⋊Q8×C47Q814yescomplemented
71C2×C4⋊C4×C47—22no—
72C2×C4⋊C4×C47—22no—
73C2×C4⋊C4×C47C2311yescharacteristic
74C2×C4⋊C4×C47C2311yescharacteristic
75C2×C42×C47C2311yescharacteristic
76#103×C47—42no—
77#103×C47—42no—
78C4×C8×C47—22no—
79C2×Q8×C47—22no—
80C2×Q8×C47—22no—
81C2×Q8×C47—22no—
82C2×Q8×C47—22no—
83C2×Q8×C47—22no—
84C2×Q8×C47C2×Q814yes—
85C22×C4×C47—22no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47C2×D412yes—
89C22×C4×C47C2411yescharacteristic
90C22×Dic47—22no—
91C22×Dic47—21no—
92C22×Dic47—21no—
93C22×Dic47C2×D411yescharacteristic
94C22×Dic47C2×D411yescharacteristic
95C22×Dic47C2×D412yes—
96C2×Dic94—42no—
97C2×Dic94—82no—
98C2×C8×C47—22no—
99C2×C8×C47—22no—
100C2×C8×C47—22no—
101C4⋊C4×C47—28no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—82no—
105C42×C47C2×Q812yes—
106C42×C47#4112yes—
107C4⋊Dic47—22no—
108C4⋊Dic47—22no—
109C4⋊Dic47—22no—
110C4⋊Dic47—22no—
111C4⋊Dic47—42no—
112C4⋊Dic47—44no—
113Dic47⋊C4—42no—
114Dic47⋊C4—42no—
115Dic47⋊C4—82no—
116C4×Dic47—22no—
117C4×Dic47—44no—
118C4×Dic47#4114yes—
119C2×C47⋊C8—22no—
120C2×C47⋊C8—22no—
121C2×C47⋊C8—22no—
122C23×C47C22×D411yescharacteristic
123Q8×C47—24no—
124Q8×C47—24no—
125Q8×C47—42no—
126Q8×C47—42no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—24no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47C22⋊Q814yes—
137C2×C4×C47C22×Q811yescharacteristic
138C2×C4×C47C2×#4112yes—
139C2×C4×C47C2×#4111yescharacteristic
140C2×C4×C47H(5,𝔽2)12yes—
141C2×Dic47—22no—
142C2×Dic47—22no—
143C2×Dic47—21no—
144C2×Dic47—21no—
145C2×Dic47—22no—
146C2×Dic47—22no—
147C2×Dic47—42no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47#12414yes—
155Dic94—84no—
156C8×C47—42no—
157C47⋊C8—42no—
158#29153—471nomaximal, Sylow
159C22×C47C2×C22⋊Q811yescharacteristic
160C22×C47C2×#12411yescharacteristic
161C22×C47#53511yescharacteristic
162C22×C47#57712yes—
163C22×C47#57812yes—
164C4×C47—24no—
165C4×C47—44no—
166C4×C47—81no—
167C4×C47—81no—
168C4×C47#55412yes—
169C4×C47#55512yes—
170Dic47—24no—
171Dic47—41no—
172Dic47—41no—
173Dic47—82no—
174Dic47—81no—
175Dic47—81no—
176#2660—471no—
177#2327—471no—
178#2026—472no—
179#2024—472no—
180#1960—471no—
181#1956D4711yescharacteristic
182#1836—471no—
183#1835—471no—
184#1805—472no—
185#1803—472no—
186#1768—471no—
187C2×C47#256311yescharacteristic
188C2×C47#308212yes—
189C2×C47#313712yes—
190C2×C47#322211yescharacteristic
191C2×C47#322311yescharacteristic
192C2×C4⋊Q8—471no—
193C2×C4⋊Q8C2×D4711yescharacteristic
194C2×#125—471no—
195C2×C4×Q8—472no—
196C2×C4⋊C8—472no—
197C2×#103—472no—
198C2×#103—472no—
199C2×#103C2×D4712yes—
200C2×#103C2×D4712yes—
201C2×#97—471no—
202C2×C4×C8C2×D4711yescharacteristic
203#398—942no—
204#395—942no—
205#391—472no—
206#387—942no—
207#387—942no—
208#384—471no—
209#384C2×D4711yescharacteristic
210#382—471no—
211#382—471no—
212#382—471no—
213#376—471no—
214#340—472no—
215#340—472no—
216#340—472no—
217#340—472no—
218#336—472no—
219C47#2915311yescharacteristic, complemented, Sylow
220C22×Q8—941no—
221C22×Q8C22×D4712yes—
222C22×C8—472no—
223C22×C8C22×D4712yes—
224C4⋊Q8—22no—
225C4⋊Q8—1882no—
226C4⋊Q8Dic9414yescomplemented
227#125—1882no—
228C4×Q8—944no—
229C4×Q8—944no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—472no—
233C2×C4⋊C4—471no—
234C2×C4⋊C4—471no—
235C2×C4⋊C4—942no—
236C2×C4⋊C4—942no—
237C2×C4⋊C4—941no—
238C2×C4⋊C4—941no—
239C2×C4⋊C4—941no—
240C2×C4⋊C4C22×D4711yescharacteristic
241C2×C4⋊C4C22×D4711yescharacteristic
242C2×C42—471no—
243C2×C42—471no—
244C2×C42—942no—
245C2×C42C22×D4711yescharacteristic
246C4⋊C8—944no—
247#103—42no—
248#103—42no—
249#103—1882no—
250#103—1882no—
251#97—942no—
252C4×C8—22no—
253#95—472no—
254#95—472no—
255#95—942no—
256#95—942no—
257#95—942no—
258#95—942no—
259#95—942no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—3762no—
266C2×Q8—1882no—
267C2×Q8C2×Dic9414yes—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—472no—
272C22×C4—471no—
273C22×C4—471no—
274C22×C4—942no—
275C22×C4—941no—
276C22×C4—941no—
277C22×C4D4×D4712yes—
278C22×C4C23×D4711yescharacteristic
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—942no—
283C2×C8—942no—
284C2×C8—942no—
285C4⋊C4—28no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—82no—
289C4⋊C4—942no—
290C4⋊C4—942no—
291C4⋊C4—942no—
292C4⋊C4—942no—
293C4⋊C4—3762no—
294C4⋊C4—1884no—
295C4⋊C4—1882no—
296C4⋊C4—1882no—
297C4⋊C4—1882no—
298C42—474no—
299C42—942no—
300C42—1884no—
301C42C2×Dic9412yes—
302C42#1062535212yes—
303C23C2×D4×D4711yescharacteristic
304Q8—24no—
305Q8—24no—
306Q8—42no—
307Q8—42no—
308Q8—3764no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—474no—
319C2×C4—942no—
320C2×C4—942no—
321C2×C4—942no—
322C2×C4—942no—
323C2×C4—941no—
324C2×C4—941no—
325C2×C4—1882no—
326C2×C4—1882no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1882no—
330C2×C4—1881no—
331C2×C4—1881no—
332C2×C4#1309400114yes—
333C2×C4C22×Dic9411yescharacteristic
334C2×C4#41×D4712yes—
335C2×C4C2×#1062535211yescharacteristic
336C2×C4#1309406912yes—
337C8—42no—
338C8—1882no—
339C22C2×#1309400111yescharacteristic
340C22#124×D4711yescharacteristic
341C22#1568968611yescharacteristic
342C22#1568992912yes—
343C22#1568993612yes—
344C4—24no—
345C4—44no—
346C4—81no—
347C4—81no—
348C4—944no—
349C4—3761no—
350C4—3762no—
351C4—3761no—
352C4—1881no—
353C4—1881no—
354C4#1568983212yes—
355C4#1568983712yes—
356C2#2024855911yescharacteristic
357C2#2025322612yes—
358C2#2025374812yes—
359C2#2025445311yescharacteristic
360C2#2025446411yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1956C2
socle, prime radicalC23×C47C22×D4
hypercenter#1956D47
hypocenterC47#29153
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
70. C4⋊Q8×C47Q8semidirect
219. C47#29153semidirect
226. C4⋊Q8Dic94semidirect

Maximal subgroups

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

subgroupindex
#202495102
#202461082
#202445372
#202445272
#202442092
#202435232
#202435192
#202434012
#202433912
#202432462
C47×#19562
#2915347