Finite Group Databasebeta

← #26134658

Subgroups — #26134658

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×#15688951C211yesmaximal, characteristic, complemented
3C2×#15688928C211yesmaximal, characteristic, complemented
4C2×#15688925C211yesmaximal, characteristic
5#20244254C211yesmaximal, characteristic
6#20244249C211yesmaximal, characteristic, complemented
7C2×C4×C16×C47C211yesmaximal, characteristic, complemented
8C22×Dic188C2211yescharacteristic
9C22×D376C2211yescharacteristic
10C2×#13093969—22no—
11C2×#13093969—22no—
12C2×#13093969C414yescomplemented
13C2×#13093958C2211yescharacteristic
14C2×#13093958C2211yescharacteristic
15C2×#13093949—21no—
16C2×#13093946—21no—
17#15689029—21no—
18#15689025—21no—
19#15688951—22no—
20#15688928—22no—
21#15688925—22no—
22C22×C16×C47C2211yescharacteristic
23C22×C16×C47C2211yescharacteristic
24C2×C4×C8×C47C2211yescharacteristic
25C4×C16×C47—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—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47—21no—
42C2×(C2×D47)⋊C4—42no—
43#13093969—28no—
44#13093958—21no—
45#13093958—21no—
46#13093958—21no—
47#13093958—21no—
48#13093949—42no—
49#13093946—42no—
50C47⋊#95—42no—
51C22×C8×C47D411yescharacteristic
52C22×C8×C47C2311yescharacteristic
53C2×C42×C47D411yescharacteristic
54C2×C16×C47—21no—
55C2×C16×C47—21no—
56C2×C16×C47C2×C412yes—
57C2×C16×C47D412yes—
58C2×C16×C47D412yes—
59C2×C16×C47D412yes—
60C4×C8×C47—21no—
61C4×C8×C47—21no—
62C23×D47—41no—
63C22×C4×C47D812yes—
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—41no—
66C22×Dic47—41no—
67C22×Dic47—41no—
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×C47—21no—
81C2×C8×C47—21no—
82C2×C8×C47D812yes—
83C2×C8×C47C22×C411yescharacteristic
84C2×C8×C47C2×D411yescharacteristic
85C2×C8×C47C2×D411yescharacteristic
86C2×C8×C47C2×D411yescharacteristic
87C2×C8×C47#4111yescharacteristic
88C2×C8×C47#4111yescharacteristic
89C42×C47—21no—
90C42×C47—21no—
91(C2×D47)⋊C4—84no—
92C4⋊Dic47—41no—
93C4⋊Dic47—41no—
94C4⋊Dic47—41no—
95C4⋊Dic47—41no—
96Dic188—22no—
97Dic188—22no—
98Dic188—41no—
99Dic188—41no—
100D376—22no—
101D376—22no—
102D376—41no—
103D376—41no—
104C16×C47—22no—
105C16×C47C22⋊C414yescomplemented
106C23×C47C2×D811yescharacteristic
107C22×D47—42no—
108C22×D47—81no—
109C22×D47—81no—
110C22×D47—82no—
111C22×D47—82no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47SD3214yescomplemented
115C2×C4×C47C4×D411yescharacteristic
116C2×C4×C47#12111yescharacteristic
117C2×C4×C47#12711yescharacteristic
118C2×C4×C47C2×D811yescharacteristic
119C2×C4×C47#13511yescharacteristic
120C2×C4×C47#13511yescharacteristic
121C2×Dic47—42no—
122C2×Dic47—81no—
123C2×Dic47—81no—
124C2×Dic47—81no—
125C2×Dic47—81no—
126C2×Dic47—82no—
127C2×Dic47—82no—
128D188—42no—
129D188—42no—
130D188—81no—
131D188—81no—
132Dic94—42no—
133Dic94—42no—
134Dic94—81no—
135Dic94—81no—
136C8×C47—22no—
137C8×C47C2×C22⋊C411yescharacteristic
138C8×C47C4×D411yescharacteristic
139C8×C47#12111yescharacteristic
140C8×C47#12411yescharacteristic
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—24no—
153C4×C47#39011yescharacteristic
154C4×C47C4×D811yescharacteristic
155C4×C47#46611yescharacteristic
156C4×C47#49511yescharacteristic
157Dic47—82no—
158Dic47—161no—
159Dic47—161no—
160C22×SD32—471no—
161C2×#367—471no—
162C2×#358—471no—
163C2×#357—471no—
164C2×C4×C16D4711yescharacteristic, complemented
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×C16C2×D4711yescharacteristic
179C22×C16C2×D4711yescharacteristic
180C2×#107—471no—
181C2×#107—471no—
182C2×#103—941no—
183C2×#102—941no—
184C2×C4×C8C2×D4711yescharacteristic
185#391—941no—
186#390—941no—
187#367—942no—
188#358—942no—
189#357—942no—
190C4×C16—22no—
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×C8D18811yescharacteristic
207C22×C8C22×D4711yescharacteristic
208C2×C4⋊C4—941no—
209C2×C4⋊C4—941no—
210C2×C22⋊C4—1882no—
211C2×C42D18811yescharacteristic
212SD32—948no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16C4×D4712yes—
216C2×C16D18812yes—
217C2×C16D18812yes—
218C2×C16C22⋊D4712yes—
219#107—941no—
220#107—941no—
221#107—941no—
222#107—941no—
223#103—1882no—
224#102—1882no—
225C4×C8—21no—
226C4×C8—21no—
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—1881no—
242C22×C4—1881no—
243C22×C4—1881no—
244C22×C4D37612yes—
245C22×C4C2×D18811yescharacteristic
246Q16—942no—
247Q16—942no—
248Q16—1881no—
249Q16—1881no—
250D8—942no—
251D8—942no—
252D8—1881no—
253D8—1881no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8D37612yes—
257C2×C8C2×C4×D4711yescharacteristic
258C2×C8C2×D18811yescharacteristic
259C2×C8C2×D18811yescharacteristic
260C2×C8#1062535011yescharacteristic
261C2×C8#1062535011yescharacteristic
262C2×C8C2×C22⋊D4711yescharacteristic
263C4⋊C4—1881no—
264C4⋊C4—1881no—
265C4⋊C4—1881no—
266C4⋊C4—1881no—
267C22⋊C4—3764no—
268C42—21no—
269C42—21no—
270C16—22no—
271C16(C2×D47)⋊C414yescomplemented
272C23—3761no—
273C23—3762no—
274C23—3762no—
275C23—3761no—
276C23—1882no—
277C23C2×D37611yescharacteristic
278Q8—3761no—
279Q8—3761no—
280Q8—1882no—
281Q8—1882no—
282D4—3761no—
283D4—3761no—
284D4—1882no—
285D4—1882no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—3761no—
289C2×C4—3761no—
290C2×C4—3761no—
291C2×C4—3761no—
292C2×C4—3762no—
293C2×C4—3762no—
294C2×C4—1882no—
295C2×C4#1309396914yescomplemented
296C2×C4C4×D18811yescharacteristic
297C2×C4#1309399811yescharacteristic
298C2×C4#1309402111yescharacteristic
299C2×C4C2×D37611yescharacteristic
300C2×C4#1309404611yescharacteristic
301C2×C4#1309404611yescharacteristic
302C8—22no—
303C8C2×(C2×D47)⋊C411yescharacteristic
304C8C4×D18811yescharacteristic
305C8#1309399811yescharacteristic
306C8#1309401211yescharacteristic
307C22—3762no—
308C22—3764no—
309C22—7521no—
310C22—7521no—
311C22—7522no—
312C22C4×D37611yescharacteristic
313C22#1568935911yescharacteristic
314C22#1568950011yescharacteristic
315C22C2×#1309396912yes—
316C22#1568955512yes—
317C4—24no—
318C4—3762no—
319C4—7521no—
320C4—7521no—
321C4#1568902511yescharacteristic
322C4C4×D37611yescharacteristic
323C4#1568935911yescharacteristic
324C4#1568950711yescharacteristic
325C2—7522no—
326C2#2024425411yescharacteristic
327C2C4×#1309396912yes—
328C2#2024548412yes—
329C2#2024564411yescharacteristic
330C2#2024564911yescharacteristic
subgroupG/H
centerC23C2×D376
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C4×C16×C47C2
socleC23×C47C2×D8
prime radicalC22×D376C22
cosocleC22×C8×C47C23
hypercenterC2×C4×C16D47
hypocenterC47#13237
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16

Complements

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

NH ≅ G/Ntype
2. C2×#15688951C2semidirect
3. C2×#15688928C2semidirect
6. #20244249C2semidirect
7. C2×C4×C16×C47C2semidirect
12. C2×#13093969C4semidirect
105. C16×C47C22⋊C4semidirect
114. C2×C4×C47SD32semidirect
164. C2×C4×C16D47semidirect
191. C47#13237semidirect
271. C16(C2×D47)⋊C4semidirect
295. C2×C4#13093969semidirect

Maximal subgroups

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

subgroupindex
C22×#130939692
C2×#156889512
C2×#156889282
C2×#156889252
#202442542
#202442492
C2×C4×C16×C472
#1323747