Finite Group Databasebeta

← #26113210

Subgroups — #26113210

Order 12032.

444 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#20246039C211yesmaximal, characteristic
2#20244333C211yesmaximal, characteristic
3#20241900C211yesmaximal, characteristic
4#20241835C211yesmaximal, characteristic
5#20241280C211yesmaximal, characteristic
6#20241245C211yesmaximal, characteristic
7C47×#3113C211yesmaximal, characteristic
8#15689180C2211yescharacteristic
9C4×#10625329C2211yescharacteristic
10#15689030—21no—
11#15689022—21no—
12#15689009—22no—
13#15688991—21no—
14#15688989—21no—
15C4×C4⋊Dic47C2211yescharacteristic
16#15688969C2211yescharacteristic
17#15688882—22no—
18#15688860—21no—
19#15688860—21no—
20Q8⋊(C47⋊C8)—22no—
21C47×#533C2211yescharacteristic
22C2×C4×Q8×C47C2211yescharacteristic
23C47×#483—21no—
24C47×#475C412yescomplemented
25C47×#462C412yescomplemented
26C47×#451—21no—
27C47×#441—22no—
28C47×#432C2211yescharacteristic
29C2×#10625329—21no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—22no—
32C2×C4×Dic47—21no—
33C2×C4×Dic47D412yes—
34C4⋊(C47⋊C8)—21no—
35C22⋊(C47⋊C8)—21no—
36#13093935—21no—
37#13093935—21no—
38C4×C47⋊C8—21no—
39C47⋊#95—22no—
40C47⋊#95—22no—
41C47⋊#95—22no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95D411yescharacteristic
45C47⋊#95D411yescharacteristic
46C22×Q8×C47D411yescharacteristic
47C2×Q16×C47—22no—
48C4⋊Q8×C47C2×C411yescharacteristic
49C4⋊Q8×C47C2×C411yescharacteristic
50#125×C47—21no—
51C22⋊Q8×C47—21no—
52C22⋊Q8×C47—21no—
53C4×Q8×C47—21no—
54C4×Q8×C47—21no—
55C4×Q8×C47—21no—
56C4×Q8×C47—21no—
57C4×Q8×C47C2×C412yes—
58C4×Q8×C47D412yes—
59#117×C47—21no—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47D411yescharacteristic
62C2×C42×C47—21no—
63C2×C42×C47C2311yescharacteristic
64#106×C47—22no—
65C4⋊C8×C47C2×C412yes—
66#103×C47—22no—
67#103×C47—22no—
68C22⋊C8×C47—21no—
69#97×C47—21no—
70C2×Q8×C47—21no—
71C2×Q8×C47—21no—
72C2×Q8×C47—21no—
73C2×Q8×C47—42no—
74C2×Q8×C47C22⋊C412yes—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×Dic47—21no—
81C22×Dic47—22no—
82C22×Dic47—21no—
83Q16×C47—44no—
84C2×C8×C47—22no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—22no—
87C4⋊C4×C47—21no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—21no—
90C4⋊C4×C47—21no—
91C4⋊C4×C47—21no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—41no—
94C4⋊C4×C47C22⋊C412yes—
95C22⋊C4×C47—41no—
96C42×C47—22no—
97C42×C47—22no—
98C42×C47—41no—
99C42×C47C22×C411yescharacteristic
100C42×C47C2×D411yescharacteristic
101C42×C47#4111yescharacteristic
102C42×C47#4111yescharacteristic
103C4⋊Dic47—42no—
104C4⋊Dic47—82no—
105C4×Dic47—42no—
106C4×Dic47—42no—
107#10625329—22no—
108#10625329—41no—
109C2×C47⋊C8—41no—
110C2×C47⋊C8—41no—
111C23×C47#12011yescharacteristic
112Q8×C47—22no—
113Q8×C47—22no—
114Q8×C47—42no—
115Q8×C47—42no—
116Q8×C47—81no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47C2×C22⋊C411yescharacteristic
129C2×C4×C47C4×D411yescharacteristic
130C2×C4×C47C4×D411yescharacteristic
131C2×C4×C47#12111yescharacteristic
132C2×C4×C47#12111yescharacteristic
133C2×C4×C47#12311yescharacteristic
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—42no—
137C2×Dic47—41no—
138C2×Dic47—42no—
139C2×Dic47—41no—
140C2×Dic47—42no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C8×C47—42no—
144C47⋊C8—42no—
145C47⋊C8—81no—
146#9328—471nomaximal, Sylow
147C22×C47—21no—
148C22×C47—21no—
149C22×C47#38611yescharacteristic
150C22×C47#45511yescharacteristic
151C22×C47#45611yescharacteristic
152C4×C47—21no—
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—42no—
158C4×C47—42no—
159C4×C47—81no—
160C4×C47—81no—
161C4×C47#42112yes—
162Dic47—81no—
163Dic47—82no—
164Dic47—81no—
165#3113D4711yescharacteristic
166#2320—471no—
167#1986—471no—
168#1506—471no—
169#1491—471no—
170#1300—471no—
171#1285—471no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47#188211yescharacteristic
175C2×C47#189711yescharacteristic
176C2×C47#190511yescharacteristic
177#533C2×D4711yescharacteristic
178C2×C4×Q8C2×D4711yescharacteristic
179#483—21no—
180#475Dic4712yescomplemented
181#462Dic4712yescomplemented
182#451—21no—
183#441—22no—
184#432—471no—
185#432C2×D4711yescharacteristic
186C4×M16—471no—
187#391—941no—
188#389—941no—
189#387—942no—
190#383—941no—
191#382—941no—
192C4×C4⋊C4—471no—
193#376—471no—
194#339—942no—
195#328—941no—
196#328—941no—
197Q8⋊C8—942no—
198C47#932811yescharacteristic, complemented, Sylow
199C22×Q8C22⋊D4711yescharacteristic
200C2×Q16—22no—
201C2×M16—941no—
202C4⋊Q8C2×Dic4711yescharacteristic
203C4⋊Q8C2×Dic4711yescharacteristic
204#125—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8C2×Dic4712yes—
212C4×Q8C22⋊D4712yes—
213#117—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—942no—
216C2×C4⋊C4—941no—
217C2×C4⋊C4C22⋊D4711yescharacteristic
218C2×C42—21no—
219C2×C42—472no—
220C2×C42—941no—
221C2×C42C22×D4711yescharacteristic
222#106—22no—
223C4⋊C8—941no—
224C4⋊C8C2×Dic4712yes—
225#103—22no—
226#103—22no—
227C22⋊C8—21no—
228C22⋊C8—941no—
229#97—21no—
230#97—941no—
231#97—941no—
232C4×C8—941no—
233#95—471no—
234#95—471no—
235#95—942no—
236#95—942no—
237#95—942no—
238#95—941no—
239#95—941no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8C22⋊Dic4712yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—941no—
248C22×C4—941no—
249C22×C4—942no—
250C22×C4D4×D4711yescharacteristic
251C22×C4D4×D4711yescharacteristic
252C22×C4C2×C22⋊D4711yescharacteristic
253Q16—44no—
254M16—942no—
255M16—1881no—
256C2×C8—22no—
257C2×C8—1881no—
258C2×C8—1881no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—3762no—
269C4⋊C4—1882no—
270C4⋊C4C22⋊Dic4712yes—
271C22⋊C4—41no—
272C42—22no—
273C42—22no—
274C42—41no—
275C42—1882no—
276C42—1882no—
277C42#1062535211yescharacteristic
278C42#1062535211yescharacteristic
279C42C22×Dic4711yescharacteristic
280C42C2×C22⋊D4711yescharacteristic
281C23#1309399211yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—42no—
285Q8—42no—
286Q8—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—1881no—
299C2×C4—1881no—
300C2×C4—1882no—
301C2×C4—1881no—
302C2×C4—1881no—
303C2×C4—1882no—
304C2×C4—1882no—
305C2×C4—1881no—
306C2×C4—1881no—
307C2×C4C2×C22⋊Dic4711yescharacteristic
308C2×C4D4×Dic4711yescharacteristic
309C2×C4D4×Dic4711yescharacteristic
310C2×C4#1309399611yescharacteristic
311C2×C4#1309399611yescharacteristic
312C2×C4#1309400511yescharacteristic
313C8—42no—
314C8—3761no—
315C8—1882no—
316C22—21no—
317C22—21no—
318C22#1568900311yescharacteristic
319C22#1568929911yescharacteristic
320C22#1568930411yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4—3761no—
331C4—3762no—
332C4—3761no—
333C4#1568914012yes—
334C2—21no—
335C2—21no—
336C2#2024375911yescharacteristic
337C2#2024383711yescharacteristic
338C2#2024388411yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3113C2
socleC22×C47#386
prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#3113D47
hypocenterC47#9328
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
24. C47×#475C4semidirect
25. C47×#462C4semidirect
180. #475Dic47semidirect
181. #462Dic47semidirect
198. C47#9328semidirect

Maximal subgroups

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

subgroupindex
#202460392
#202443332
#202419002
#202418352
#202412802
#202412452
C47×#31132
#932847