Finite Group Databasebeta

← #26132158

Subgroups — #26132158

Order 12032.

740 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 131 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×#15688922C212yesmaximal, complemented
2C2×#15688920C211yesmaximal, characteristic
3#20243792C211yesmaximal, characteristic
4#20243782C212yesmaximal, complemented
5C2×C47×#563C211yesmaximal, characteristic
6#15689601C2211yescharacteristic
7#15689132—22no—
8#15689128—21no—
9C2×#13093943C2212yes—
10C2×#13093943C2211yescharacteristic
11#15688922—24no—
12#15688920—22no—
13#15688883—22no—
14#15688883—21no—
15C47×#563—24no—
16C47×#563—22no—
17C47×#563C414yescomplemented
18C2×C4⋊Q8×C47C2212yes—
19C2×#126×C47—22no—
20C2×#125×C47—21no—
21C2×#124×C47C2211yescharacteristic
22C2×#123×C47—21no—
23C2×C22⋊Q8×C47—22no—
24C2×#10625329—21no—
25C2×#10625329—22no—
26Q8×Dic47—42no—
27#13093983—21no—
28#13093983—22no—
29#13093948—44no—
30#13093945—42no—
31#13093943—22no—
32#13093943—22no—
33#13093943—22no—
34C22×Q8×C47D412yes—
35C22×Q8×C47C2311yescharacteristic
36C22×D4×C47D411yescharacteristic
37C23×C4×C47—21no—
38C4⋊Q8×C47—22no—
39C4⋊Q8×C47—22no—
40C4⋊Q8×C47C2×C414yes—
41C4⋊Q8×C47D414yes—
42#126×C47—28no—
43#126×C47—28no—
44#125×C47—24no—
45#125×C47—24no—
46#124×C47—22no—
47#124×C47C2×C412yes—
48#124×C47D412yes—
49#123×C47—24no—
50#123×C47—24no—
51C22⋊Q8×C47—28no—
52C22⋊Q8×C47—28no—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—22no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—22no—
57C2×C4⋊C4×C47—22no—
58C2×C22⋊C4×C47—22no—
59C2×C22⋊C4×C47—21no—
60C2×C22⋊C4×C47—22no—
61C2×C42×C47D411yescharacteristic
62C2×C42×C47D412yes—
63C24×C47—21no—
64C2×Q8×C47—22no—
65C2×Q8×C47—41no—
66C2×Q8×C47—41no—
67C2×Q8×C47—42no—
68C2×Q8×C47—42no—
69C2×Q8×C47C22⋊C414yes—
70C2×Q8×C47C22×C411yescharacteristic
71C2×Q8×C47C2×D412yes—
72C2×Q8×C47C2×D411yescharacteristic
73C2×D4×C47—21no—
74C2×D4×C47—41no—
75C2×D4×C47—41no—
76C2×D4×C47C22⋊C412yes—
77C22×C4×C47—21no—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—24no—
81C22×C4×C47—81no—
82C22×C4×C47—81no—
83C22×C4×C47C2×D412yes—
84C22×C4×C47C2×D411yescharacteristic
85C4⋊C4×C47—24no—
86C4⋊C4×C47—28no—
87C4⋊C4×C47—24no—
88C4⋊C4×C47—24no—
89C4⋊C4×C47—24no—
90C4⋊C4×C47—24no—
91C4⋊C4×C47—24no—
92C22⋊C4×C47—28no—
93C22⋊C4×C47—24no—
94C22⋊C4×C47—24no—
95C22⋊C4×C47—24no—
96C42×C47—21no—
97C42×C47—22no—
98C42×C47C22⋊C412yes—
99C42×C47C22⋊C414yes—
100C22⋊Dic47—41no—
101C4⋊Dic47—22no—
102C4⋊Dic47—24no—
103C4×Dic47—41no—
104C4×Dic47—42no—
105#10625329—41no—
106#10625329—42no—
107#10625329—42no—
108#10625329—41no—
109C2×C47⋊C8—41no—
110C2×C47⋊C8—42no—
111C23×C47—22no—
112C23×C47—44no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47#12011yescharacteristic
116Q8×C47—41no—
117Q8×C47—42no—
118Q8×C47—41no—
119Q8×C47—44no—
120Q8×C47—82no—
121D4×C47—42no—
122D4×C47—81no—
123C2×C4×C47—21no—
124C2×C4×C47—22no—
125C2×C4×C47—21no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—24no—
129C2×C4×C47—24no—
130C2×C4×C47—42no—
131C2×C4×C47—44no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—84no—
135C2×C4×C47C2×C22⋊C412yes—
136C2×C4×C47C2×C22⋊C411yescharacteristic
137C2×C4×C47#12012yes—
138C2×C4×C47#12011yescharacteristic
139C2×Dic47—42no—
140C47⋊C8—81no—
141C47⋊C8—82no—
142#12775—471nomaximal, Sylow
143C22×C47—22no—
144C22×C47—44no—
145C22×C47—42no—
146C22×C47—82no—
147C22×C47—82no—
148C22×C47#37911yescharacteristic
149C22×C47B(4,𝔽2)11yescharacteristic
150C22×C47#45811yescharacteristic
151C4×C47—21no—
152C4×C47—22no—
153C4×C47—21no—
154C4×C47—22no—
155C4×C47—42no—
156C4×C47—44no—
157C4×C47—44no—
158Dic47—82no—
159C2×#563D4711yescharacteristic
160C2×#356—472no—
161C2×#355—471no—
162#1888—471no—
163#1886—472no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—82no—
167C2×C47#179011yescharacteristic
168C2×C47#220312yes—
169#563—24no—
170#563—22no—
171#563Dic4714yescomplemented
172C2×C4⋊Q8C2×D4712yes—
173C2×#126—22no—
174C2×#125—21no—
175C2×#124C2×D4711yescharacteristic
176C2×#123—21no—
177C2×C22⋊Q8—22no—
178#519—471no—
179#419—942no—
180#418—941no—
181C2×#101—472no—
182C2×#101—471no—
183#356—944no—
184#355—942no—
185#339—942no—
186#339—941no—
187C47#1277511yescharacteristic, complemented, Sylow
188C22×Q8C22⋊D4712yes—
189C22×Q8C22×D4711yescharacteristic
190C22×D4C22⋊D4711yescharacteristic
191C23×C4—21no—
192C2×M16—941no—
193C2×M16—942no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8C2×Dic4714yes—
197C4⋊Q8C22⋊D4714yes—
198#126—28no—
199#126—28no—
200#125—24no—
201#125—24no—
202#124—22no—
203#124C2×Dic4712yes—
204#124C22⋊D4712yes—
205#123—24no—
206#123—24no—
207C22⋊Q8—28no—
208C22⋊Q8—28no—
209C4×Q8—1882no—
210#117—941no—
211#117—942no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C42C22⋊D4711yescharacteristic
221C2×C42C22⋊D4712yes—
222#103—1884no—
223#102—1882no—
224#101—942no—
225#101—942no—
226#101—942no—
227C24—21no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8C22⋊Dic4714yes—
234C2×Q8C22×Dic4711yescharacteristic
235C2×Q8C2×C22⋊D4712yes—
236C2×Q8C2×C22⋊D4711yescharacteristic
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4C22⋊Dic4712yes—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4C2×C22⋊D4712yes—
248C22×C4C2×C22⋊D4711yescharacteristic
249M16—1881no—
250M16—1882no—
251M16—1882no—
252M16—1881no—
253C2×C8—1881no—
254C2×C8—1882no—
255C4⋊C4—24no—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—942no—
263C4⋊C4—944no—
264C22⋊C4—28no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—1881no—
269C42—21no—
270C42—22no—
271C42—1881no—
272C42—1882no—
273C42C22⋊Dic4712yes—
274C42C22⋊Dic4714yes—
275C23—22no—
276C23—44no—
277C23—41no—
278C23—41no—
279C23#1309399311yescharacteristic
280Q8—41no—
281Q8—42no—
282Q8—41no—
283Q8—44no—
284Q8—82no—
285D4—42no—
286D4—81no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—84no—
299C2×C4—1882no—
300C2×C4C2×C22⋊Dic4712yes—
301C2×C4C2×C22⋊Dic4711yescharacteristic
302C2×C4#1309399312yes—
303C2×C4#1309399311yescharacteristic
304C8—3761no—
305C8—3762no—
306C22—22no—
307C22—44no—
308C22—42no—
309C22—82no—
310C22—82no—
311C22#1568897911yescharacteristic
312C22#1568931011yescharacteristic
313C22#1568931311yescharacteristic
314C4—21no—
315C4—22no—
316C4—21no—
317C4—22no—
318C4—42no—
319C4—44no—
320C4—44no—
321C4—3762no—
322C2—21no—
323C2—21no—
324C2—82no—
325C2#2024333611yescharacteristic
326C2#2024540912yes—
subgroupG/H
centerC22#15689310
commutatorC2×Q8×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC2×C47×#563C2
socleC22×C47B(4,𝔽2)
prime radicalC22×D4×C47D4
cosocleC22×Q8×C47C23
hypercenterC2×#563D47
hypocenterC47#12775
LCSG ▷ C2×Q8×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. C2×#15688922C2semidirect
4. #20243782C2semidirect
17. C47×#563C4semidirect
171. #563Dic47semidirect
187. C47#12775semidirect

Maximal subgroups

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

subgroupindex
C2×#156889222
C2×#156889202
#202437922
#202437822
C2×C47×#5632
#1277547