Finite Group Databasebeta

← #26134656

Subgroups — #26134656

Order 12032.

474 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093969C211yesmaximal, characteristic
2C2×#15688926C211yesmaximal, characteristic, complemented
3C2×#15688923C211yesmaximal, characteristic
4C2×C16×Dic47C211yesmaximal, characteristic, complemented
5#20244252C211yesmaximal, characteristic
6#20244247C211yesmaximal, characteristic, complemented
7C2×C47×#367C211yesmaximal, characteristic, complemented
8C22×Dic188C2211yescharacteristic
9C22×D376C2211yescharacteristic
10C2×#13093969—22no—
11C2×#13093969—22no—
12C2×#13093969C414yescomplemented
13C22×C47⋊C16C2211yescharacteristic
14C2×#13093947—21no—
15C2×#13093944—21no—
16C2×C8×Dic47C2211yescharacteristic
17#15689028—21no—
18#15689024—21no—
19#15688926—22no—
20#15688923—22no—
21C16×Dic47—22no—
22C22×C16×C47C2211yescharacteristic
23C2×#107×C47C2211yescharacteristic
24C2×#107×C47C2211yescharacteristic
25C47×#367—22no—
26C22×Dic94—21no—
27C22×D188—21no—
28C2×Dic188—21no—
29C2×Dic188—21no—
30C2×Dic188—21no—
31C2×Dic188—21no—
32C2×Dic188—21no—
33C2×Dic188C2×C412yes—
34C2×D376—21no—
35C2×D376—21no—
36C2×D376—21no—
37C2×D376—21no—
38C2×D376—21no—
39C2×D376C2×C412yes—
40C22×C47⋊C8D411yescharacteristic
41C2×(C2×D47)⋊C4—42no—
42C2×C4×Dic47D411yescharacteristic
43#13093969—28no—
44C2×C47⋊C16—21no—
45C2×C47⋊C16—21no—
46C2×C47⋊C16D412yes—
47#13093947—42no—
48#13093944—42no—
49C8×Dic47—21no—
50C8×Dic47—21no—
51C47⋊#95—42no—
52C22×C8×C47C2311yescharacteristic
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C16×C47C2×C412yes—
56C2×C16×C47D412yes—
57C2×C16×C47D412yes—
58#107×C47—21no—
59#107×C47—21no—
60#107×C47—21no—
61#107×C47—21no—
62C23×D47—41no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47C2×D411yescharacteristic
66C22×Dic47—41no—
67C22×Dic47D812yes—
68C2×D188—22no—
69C2×D188—41no—
70C2×D188—41no—
71C2×D188—41no—
72C2×D188—41no—
73C2×D188—41no—
74C2×Dic94—22no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×Dic94—41no—
80C2×C8×C47C22×C411yescharacteristic
81C2×C8×C47C2×D411yescharacteristic
82C2×C8×C47C2×D411yescharacteristic
83C2×C8×C47C2×D411yescharacteristic
84C2×C8×C47#4111yescharacteristic
85C2×C8×C47#4111yescharacteristic
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90(C2×D47)⋊C4—84no—
91C4×Dic47—21no—
92C4×Dic47—21no—
93C2×C47⋊C8—21no—
94C2×C47⋊C8—21no—
95C2×C47⋊C8D812yes—
96Dic188—22no—
97Dic188—22no—
98Dic188—41no—
99Dic188—41no—
100D376—22no—
101D376—22no—
102D376—41no—
103D376—41no—
104C16×C47C22⋊C414yescomplemented
105C47⋊C16—22no—
106C23×C47C2×D811yescharacteristic
107C22×D47—42no—
108C22×D47—81no—
109C22×D47—81no—
110C22×D47—82no—
111C22×D47—82no—
112C2×C4×C47—81no—
113C2×C4×C47—81no—
114C2×C4×C47—82no—
115C2×C4×C47—82no—
116C2×C4×C47C4×D411yescharacteristic
117C2×C4×C47#12111yescharacteristic
118C2×C4×C47#12711yescharacteristic
119C2×C4×C47C2×D811yescharacteristic
120C2×C4×C47#13511yescharacteristic
121C2×C4×C47#13511yescharacteristic
122C2×Dic47—22no—
123C2×Dic47—22no—
124C2×Dic47—42no—
125C2×Dic47—81no—
126C2×Dic47—81no—
127C2×Dic47SD3214yescomplemented
128D188—42no—
129D188—42no—
130D188—81no—
131D188—81no—
132Dic94—42no—
133Dic94—42no—
134Dic94—81no—
135Dic94—81no—
136C8×C47C2×C22⋊C411yescharacteristic
137C8×C47C4×D411yescharacteristic
138C8×C47#12111yescharacteristic
139C8×C47#12411yescharacteristic
140C47⋊C8—22no—
141#13237—471nomaximal, Sylow
142C22×C47C4×D811yescharacteristic
143C22×C47#46611yescharacteristic
144C22×C47#49311yescharacteristic
145C22×C47C2×SD3212yes—
146C22×C47#50812yes—
147C2×D47—84no—
148C2×D47—82no—
149C2×D47—162no—
150C2×D47—161no—
151C2×D47—161no—
152C4×C47—161no—
153C4×C47—161no—
154C4×C47#39011yescharacteristic
155C4×C47C4×D811yescharacteristic
156C4×C47#46611yescharacteristic
157C4×C47#49511yescharacteristic
158Dic47—24no—
159Dic47—82no—
160C22×SD32—471no—
161C2×#367D4711yescharacteristic, complemented
162C2×#358—471no—
163C2×#357—471no—
164C2×C4×C16—471no—
165#1970—471no—
166#1969—471no—
167C2×C47#197011yescharacteristic
168C2×C47C4×SD3212yes—
169C2×C47#221712yes—
170C2×C47#224811yescharacteristic
171C2×C47#224911yescharacteristic
172D47—162no—
173C22×Q16—471no—
174C22×D8—471no—
175C2×SD32—474no—
176C2×SD32—942no—
177C2×SD32—942no—
178C22×C16—471no—
179C22×C16C2×D4711yescharacteristic
180C2×#107C2×D4711yescharacteristic
181C2×#107C2×D4711yescharacteristic
182C2×#103—941no—
183C2×#102—941no—
184C2×C4×C8—471no—
185#391—941no—
186#390—941no—
187#367—22no—
188#358—942no—
189#357—942no—
190C4×C16—942no—
191C47#1323711yescharacteristic, complemented, Sylow
192C22×Q8—941no—
193C22×D4—941no—
194C2×Q16—472no—
195C2×Q16—941no—
196C2×Q16—941no—
197C2×Q16—941no—
198C2×Q16—941no—
199C2×Q16—941no—
200C2×D8—472no—
201C2×D8—941no—
202C2×D8—941no—
203C2×D8—941no—
204C2×D8—941no—
205C2×D8—941no—
206C22×C8—471no—
207C22×C8C22×D4711yescharacteristic
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C22⋊C4—1882no—
211C2×C42—471no—
212SD32—948no—
213C2×C16—472no—
214C2×C16—941no—
215C2×C16—941no—
216C2×C16C4×D4712yes—
217C2×C16D18812yes—
218C2×C16C22⋊D4712yes—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#103—1882no—
224#102—1882no—
225C4×C8—941no—
226C4×C8—941no—
227#95—1882no—
228C24—1881no—
229C2×Q8—942no—
230C2×Q8—1881no—
231C2×Q8—1881no—
232C2×Q8—1881no—
233C2×Q8—1881no—
234C2×Q8—1881no—
235C2×D4—942no—
236C2×D4—1881no—
237C2×D4—1881no—
238C2×D4—1881no—
239C2×D4—1881no—
240C2×D4—1881no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—472no—
244C22×C4—1881no—
245C22×C4D4×D4711yescharacteristic
246Q16—942no—
247Q16—942no—
248Q16—1881no—
249Q16—1881no—
250D8—942no—
251D8—942no—
252D8—1881no—
253D8—1881no—
254C2×C8—472no—
255C2×C8—941no—
256C2×C8—941no—
257C2×C8C2×C4×D4711yescharacteristic
258C2×C8C2×D18811yescharacteristic
259C2×C8D4×D4711yescharacteristic
260C2×C8#1062535411yescharacteristic
261C2×C8#1062535411yescharacteristic
262C2×C8C2×C22⋊D4711yescharacteristic
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C22⋊C4—3764no—
268C42—941no—
269C42—941no—
270C16—942no—
271C16(C2×D47)⋊C414yescomplemented
272C23—3761no—
273C23—3762no—
274C23—3762no—
275C23—3761no—
276C23—1882no—
277C23D8×D4711yescharacteristic
278Q8—3761no—
279Q8—3761no—
280Q8—1882no—
281Q8—1882no—
282D4—3761no—
283D4—3761no—
284D4—1882no—
285D4—1882no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—474no—
291C2×C4—942no—
292C2×C4—942no—
293C2×C4—3761no—
294C2×C4—3761no—
295C2×C4—1882no—
296C2×C4#1309398611yescharacteristic
297C2×C4#1309399911yescharacteristic
298C2×C4#1309402011yescharacteristic
299C2×C4D8×D4711yescharacteristic
300C2×C4#1309404711yescharacteristic
301C2×C4#1309404711yescharacteristic
302C8—942no—
303C8C2×(C2×D47)⋊C411yescharacteristic
304C8#1309398611yescharacteristic
305C8#1309399911yescharacteristic
306C8#1309401311yescharacteristic
307C22—3762no—
308C22—3764no—
309C22—7521no—
310C22—7521no—
311C22—7522no—
312C22#1568920511yescharacteristic
313C22#1568936011yescharacteristic
314C22#1568949811yescharacteristic
315C22SD32×D4712yes—
316C22#1568955412yes—
317C4—161no—
318C4—161no—
319C4—944no—
320C4—3762no—
321C4#1568902411yescharacteristic
322C4#1568920511yescharacteristic
323C4#1568936011yescharacteristic
324C4#1568950811yescharacteristic
325C2—7522no—
326C2#2024425211yescharacteristic
327C2#2024523912yes—
328C2#2024548512yes—
329C2#2024564211yescharacteristic
330C2#2024564811yescharacteristic
subgroupG/H
centerC23D8×D47
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#367C2
socleC23×C47C2×D8
prime radicalC22×D376C22
cosocleC22×C8×C47C23
hypercenterC2×#367D47
hypocenterC47#13237
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#367

Complements

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

NH ≅ G/Ntype
2. C2×#15688926C2semidirect
4. C2×C16×Dic47C2semidirect
6. #20244247C2semidirect
7. C2×C47×#367C2semidirect
12. C2×#13093969C4semidirect
104. C16×C47C22⋊C4semidirect
127. C2×Dic47SD32semidirect
161. C2×#367D47semidirect
191. C47#13237semidirect
271. C16(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#130939692
C2×#156889262
C2×#156889232
C2×C16×Dic472
#202442522
#202442472
C2×C47×#3672
#1323747