Finite Group Databasebeta

← #26368880

Subgroups — #26368880

Order 12032.

822 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689531C212yesmaximal
2C2×#15689530C212yesmaximal
3#20251764C211yesmaximal, characteristic
4#20251749C211yesmaximal, characteristic
5#20243557C211yesmaximal, characteristic
6#20243556C211yesmaximal, characteristic
7#20243554C211yesmaximal, characteristic
8#20243551C212yesmaximal
9#20243549C212yesmaximal
10#97×Dic47C211yesmaximal, characteristic
11C47×#1847C211yesmaximal, characteristic
12C2×#13094023C2211yescharacteristic
13C2×#13094023C2211yescharacteristic
14C2×#13094015C2212yes—
15#15689531—28no—
16#15689530—28no—
17C2×#13093958C2212yes—
18C2×#13093957C2212yes—
19C2×#13093957C2212yes—
20C2×#13093956C2212yes—
21C2×#13093955C2212yes—
22C2×#13093955C2212yes—
23C2×#13093935C2211yescharacteristic
24C2×#13093934C2212yes—
25C2×C8×Dic47C2212yes—
26#15689027—22no—
27#15689027—22no—
28#15689022—22no—
29#15689019—22no—
30#15688993C2211yescharacteristic
31#15688989C2211yescharacteristic
32#15688988C2211yescharacteristic
33#15688988C2212yes—
34#15688988C2211yescharacteristic
35C42×Dic47C2211yescharacteristic
36C2×#107×C47C2212yes—
37C2×#106×C47C2212yes—
38C2×#97×C47C2211yescharacteristic
39C47×#384C2211yescharacteristic
40C47×#382C2211yescharacteristic
41C22×Dic94—21no—
42C22×Dic94—21no—
43C22×C47⋊C8C2312yes—
44#13094023—24no—
45#13094023—24no—
46#13094015—24no—
47#13094015—24no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47C2311yescharacteristic
51C2×C4⋊Dic47C2311yescharacteristic
52C2×C4⋊Dic47C2312yes—
53C2×Dic47⋊C4—22no—
54C2×Dic47⋊C4—21no—
55C2×Dic47⋊C4—21no—
56C2×C4×Dic47D412yes—
57C2×C4×Dic47D412yes—
58C2×C4×Dic47C2311yescharacteristic
59C2×C4×Dic47C2311yescharacteristic
60#13093958—24no—
61#13093957—24no—
62#13093957—24no—
63#13093956—24no—
64#13093955—24no—
65#13093955—24no—
66#13093935Q814yescomplemented
67#13093934D418yes—
68C8×Dic47Q818yescomplemented
69C47⋊#95—22no—
70C47⋊#95—24no—
71C47⋊#95—22no—
72C47⋊#95—22no—
73C22×C8×C47C2312yes—
74C2×C4⋊C4×C47—22no—
75C2×C4⋊C4×C47C2312yes—
76C2×C4⋊C4×C47C2311yescharacteristic
77C2×C4⋊C4×C47C2311yescharacteristic
78C2×C42×C47C2311yescharacteristic
79#107×C47—24no—
80#106×C47—24no—
81#97×C47Q814yescomplemented
82#95×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47C2×D412yes—
87C22×C4×C47C2411yescharacteristic
88C22×Dic47—21no—
89C22×Dic47—21no—
90C22×Dic47—22no—
91C22×Dic47C2×D411yescharacteristic
92C22×Dic47C2×D411yescharacteristic
93C22×Dic47C2×D412yes—
94C2×Dic94—44no—
95C2×Dic94—44no—
96C2×Dic94—42no—
97C2×Dic94—42no—
98C2×C8×C47C2×D414yes—
99C2×C8×C47C2×Q814yes—
100C2×C8×C47C2×Q814yes—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—44no—
106C42×C47C2×Q812yes—
107C42×C47C2×Q812yes—
108C4⋊Dic47—22no—
109C4⋊Dic47—22no—
110C4⋊Dic47—22no—
111C4⋊Dic47—22no—
112C4⋊Dic47—44no—
113C4⋊Dic47—44no—
114Dic47⋊C4—44no—
115Dic47⋊C4—42no—
116Dic47⋊C4—42no—
117C4×Dic47—42no—
118C4×Dic47—42no—
119C4×Dic47C2×D414yes—
120C4×Dic47C2×Q814yes—
121C2×C47⋊C8C2×D414yes—
122C2×C47⋊C8C2×Q814yes—
123C2×C47⋊C8C2×Q814yes—
124C23×C47C22×D411yescharacteristic
125C2×C4×C47—22no—
126C2×C4×C47—22no—
127C2×C4×C47—42no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—44no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47C4⋊Q814yes—
134C2×C4×C47C22×D412yes—
135C2×C4×C47C22×Q811yescharacteristic
136C2×C4×C47C22×Q812yes—
137C2×C4×C47C22×Q811yescharacteristic
138C2×Dic47—22no—
139C2×Dic47—22no—
140C2×Dic47—22no—
141C2×Dic47—22no—
142C2×Dic47—42no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—44no—
146C2×Dic47—42no—
147C2×Dic47—42no—
148C2×Dic47#12712yes—
149C2×Dic47#12712yes—
150C2×Dic47C4⋊Q814yes—
151Dic94—84no—
152Dic94—84no—
153C8×C47C4⋊Q818yes—
154C47⋊C8C4⋊Q818yes—
155#33317—471nomaximal, Sylow
156C22×C47C2×#12711yescharacteristic
157C22×C47C2×C4⋊Q811yescharacteristic
158C22×C47C2×C4⋊Q811yescharacteristic
159C22×C47#57512yes—
160C22×C47#57512yes—
161C4×C47—24no—
162C4×C47—81no—
163C4×C47—81no—
164C4×C47—82no—
165C4×C47C2×C4⋊Q812yes—
166C4×C47C2×C4⋊Q812yes—
167Dic47—22no—
168Dic47—22no—
169Dic47—24no—
170Dic47—81no—
171Dic47—81no—
172Dic47—82no—
173C2×#501—472no—
174C2×#501—472no—
175#2862—471no—
176#2859—471no—
177#1847—471no—
178#1847—471no—
179#1847—471no—
180#1847D4711yescharacteristic
181#1846—472no—
182#1846—472no—
183C4×#97—471no—
184C2×C47#286811yescharacteristic
185C2×C47#315311yescharacteristic
186C2×C47#315311yescharacteristic
187C2×C47#316212yes—
188C2×C47#316612yes—
189C2×C4⋊Q8—471no—
190C2×C4⋊Q8—471no—
191C2×#125—472no—
192#501—948no—
193#501—948no—
194C2×#107—472no—
195C2×#107—472no—
196C2×#107—472no—
197C2×#107C2×D4712yes—
198C2×#106—472no—
199C2×#106—472no—
200C2×#106—472no—
201C2×#106C2×D4712yes—
202C2×#97—472no—
203C2×#97—471no—
204C2×#97C2×D4711yescharacteristic
205C2×C4×C8—472no—
206#391—942no—
207#391—942no—
208#389—942no—
209#389—942no—
210#384—471no—
211#384C2×D4711yescharacteristic
212#382—472no—
213#382—471no—
214#382—471no—
215#382—471no—
216#382C2×D4711yescharacteristic
217C43—471no—
218C47#3331711yescharacteristic, complemented, Sylow
219C22×Q8—941no—
220C22×Q8—941no—
221C22×C8—472no—
222C22×C8C22×D4712yes—
223C4⋊Q8—944no—
224C4⋊Q8—944no—
225#125—944no—
226#125—944no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—472no—
229C2×C4⋊C4—471no—
230C2×C4⋊C4—471no—
231C2×C4⋊C4—942no—
232C2×C4⋊C4—942no—
233C2×C4⋊C4—941no—
234C2×C4⋊C4—941no—
235C2×C4⋊C4—942no—
236C2×C4⋊C4C22×D4712yes—
237C2×C4⋊C4C22×D4711yescharacteristic
238C2×C4⋊C4C22×D4711yescharacteristic
239C2×C42—471no—
240C2×C42—471no—
241C2×C42—472no—
242C2×C42—472no—
243C2×C42C22×D4711yescharacteristic
244#107—24no—
245#107—944no—
246#107—944no—
247#107—944no—
248#106—24no—
249#106—944no—
250#106—944no—
251#106—944no—
252#97—478no—
253#97—474no—
254#97Dic9414yescomplemented
255C4×C8—478no—
256#95—22no—
257#95—944no—
258#95—942no—
259#95—942no—
260#95—942no—
261C2×Q8—1882no—
262C2×Q8—1882no—
263C2×Q8—1884no—
264C2×Q8—1884no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—472no—
269C22×C4—471no—
270C22×C4—471no—
271C22×C4—941no—
272C22×C4—941no—
273C22×C4—942no—
274C22×C4D4×D4712yes—
275C22×C4C23×D4711yescharacteristic
276C2×C8—474no—
277C2×C8—474no—
278C2×C8—474no—
279C2×C8C2×Dic9414yes—
280C2×C8D4×D4714yes—
281C2×C8Q8×D4714yes—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—44no—
287C4⋊C4—942no—
288C4⋊C4—942no—
289C4⋊C4—942no—
290C4⋊C4—942no—
291C4⋊C4—1884no—
292C4⋊C4—1884no—
293C4⋊C4—1882no—
294C4⋊C4—1882no—
295C4⋊C4—1884no—
296C42—474no—
297C42—474no—
298C42—1882no—
299C42—1882no—
300C42C2×Dic9412yes—
301C42Q8×D4712yes—
302C23C2×D4×D4711yescharacteristic
303Q8—3764no—
304Q8—3764no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—472no—
314C2×C4—472no—
315C2×C4—474no—
316C2×C4—942no—
317C2×C4—942no—
318C2×C4—942no—
319C2×C4—942no—
320C2×C4—1882no—
321C2×C4—1881no—
322C2×C4—1881no—
323C2×C4—1882no—
324C2×C4—1884no—
325C2×C4—1882no—
326C2×C4#1309402314yes—
327C2×C4C2×D4×D4712yes—
328C2×C4C2×Q8×D4711yescharacteristic
329C2×C4C2×Q8×D4712yes—
330C2×C4C22×Dic9411yescharacteristic
331C8—478no—
332C8#1309402318yes—
333C22#127×D4711yescharacteristic
334C22C4⋊Q8×D4711yescharacteristic
335C22C2×#1309402311yescharacteristic
336C22#1568991912yes—
337C22#1568992212yes—
338C4—24no—
339C4—81no—
340C4—81no—
341C4—82no—
342C4—942no—
343C4—942no—
344C4—944no—
345C4—3761no—
346C4—3762no—
347C4—3761no—
348C4C4⋊Q8×D4712yes—
349C4C2×#1309402312yes—
350C2#2025178511yescharacteristic
351C2#2025385111yescharacteristic
352C2#2025385311yescharacteristic
353C2#2025390912yes—
354C2#2025393812yes—
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1847C2
socle, prime radicalC23×C47C22×D4
hypercenter#1847D47
hypocenterC47#33317
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
66. #13093935Q8semidirect
68. C8×Dic47Q8semidirect
81. #97×C47Q8semidirect
218. C47#33317semidirect
254. #97Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156895312
C2×#156895302
#202517642
#202517492
#202435572
#202435562
#202435542
#202435512
#202435492
#97×Dic472
C47×#18472
#3331747