Finite Group Databasebeta

← #26116212

Subgroups — #26116212

Order 12032.

460 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20244366C211yesmaximal, characteristic, complemented
2#20244217C211yesmaximal, characteristic
3#20243133C211yesmaximal, characteristic, complemented
4#20242147C211yesmaximal, characteristic, complemented
5#20241843C211yesmaximal, characteristic, complemented
6#20241839C211yesmaximal, characteristic
7C47×#3167C211yesmaximal, characteristic
8#15689147C2211yescharacteristic
9#15689147C2211yescharacteristic
10C2×#13093951—22no—
11C2×C4×C47⋊C8C2211yescharacteristic
12C42×Dic47C2211yescharacteristic
13#15688882—22no—
14#15688877—22no—
15#15688876—22no—
16Q8⋊(C47⋊C8)—22no—
17D4⋊(C47⋊C8)—22no—
18C47×#533C2211yescharacteristic
19C47×#532C2211yescharacteristic
20C47×#498C411yescharacteristic, complemented
21C47×#492C411yescharacteristic, complemented
22C47×#487C411yescharacteristic, complemented
23C47×#486C411yescharacteristic, complemented
24C47×#465—22no—
25C4×SD16×C47—22no—
26C2×C4×C8×C47C2211yescharacteristic
27C2×#10625329—21no—
28C2×#10625329—21no—
29C22×C47⋊C8—21no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47D412yes—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)—21no—
35C4⋊(C47⋊C8)—21no—
36C4⋊(C47⋊C8)—21no—
37#13093951—44no—
38C4×C47⋊C8—21no—
39C4×C47⋊C8D412yes—
40C2×#41×C47—21no—
41C2×SD16×C47—22no—
42C22×C8×C47D412yes—
43C4⋊Q8×C47C2×C411yescharacteristic
44C4⋊Q8×C47C2×C411yescharacteristic
45C4⋊Q8×C47C2×C411yescharacteristic
46#127×C47C2×C411yescharacteristic
47#125×C47—21no—
48#124×C47—21no—
49C22⋊Q8×C47—21no—
50C22⋊Q8×C47—21no—
51#121×C47—21no—
52#121×C47—21no—
53C4×Q8×C47—21no—
54C4×Q8×C47—21no—
55C4×D4×C47—21no—
56C4×D4×C47—21no—
57#117×C47—21no—
58C2×C42×C47C2311yescharacteristic
59#106×C47—22no—
60#103×C47—22no—
61#102×C47—22no—
62C4×C8×C47C2×C411yescharacteristic
63C4×C8×C47C2×C411yescharacteristic
64C4×C8×C47D412yes—
65#41×C47—42no—
66#41×C47—42no—
67C2×Q8×C47—21no—
68C2×Q8×C47—21no—
69C2×Q8×C47—21no—
70C2×D4×C47—21no—
71C2×D4×C47—21no—
72C2×D4×C47—21no—
73C2×D4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47#4111yescharacteristic
78C22×Dic47—22no—
79SD16×C47—44no—
80C2×C8×C47—42no—
81C2×C8×C47C22⋊C412yes—
82C2×C8×C47C22⋊C412yes—
83C4⋊C4×C47—21no—
84C4⋊C4×C47—21no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—21no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C22⋊C4×C47—41no—
93C22⋊C4×C47—41no—
94C42×C47—41no—
95C42×C47C22×C411yescharacteristic
96C42×C47C2×D411yescharacteristic
97C42×C47C2×D411yescharacteristic
98C42×C47#4111yescharacteristic
99C4×Dic47—24no—
100C4×Dic47—22no—
101C4×Dic47—22no—
102C4×Dic47—42no—
103#10625329—42no—
104#10625329—42no—
105C2×C47⋊C8—22no—
106C2×C47⋊C8—42no—
107C2×C47⋊C8—41no—
108C2×C47⋊C8—41no—
109C23×C47—41no—
110C23×C47#12111yescharacteristic
111Q8×C47—42no—
112Q8×C47—42no—
113Q8×C47—42no—
114D4×C47—42no—
115D4×C47—42no—
116D4×C47—42no—
117D4×C47—82no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—81no—
124C2×C4×C47—81no—
125C2×C4×C47C2×C22⋊C411yescharacteristic
126C2×C4×C47C4×D411yescharacteristic
127C2×C4×C47C4×D411yescharacteristic
128C2×C4×C47#12111yescharacteristic
129C2×C4×C47#12411yescharacteristic
130C2×C4×C47#12711yescharacteristic
131C2×C4×C47AGL(1,ℤ/8)12yes—
132C2×C4×C47#13712yes—
133C2×Dic47—24no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C8×C47—22no—
137C8×C47—22no—
138C47⋊C8—42no—
139C47⋊C8—81no—
140C47⋊C8—81no—
141#9854—471nomaximal, Sylow
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—81no—
145C22×C47—81no—
146C22×C47—81no—
147C22×C47#39011yescharacteristic
148C22×C47#46812yes—
149C4×C47—21no—
150C4×C47—21no—
151C4×C47—81no—
152C4×C47—81no—
153C4×C47—81no—
154C4×C47#42112yes—
155C4×C47#44012yes—
156C4×C47#44312yes—
157C4×C47#49611yescharacteristic
158C4×C47#49711yescharacteristic
159Dic47—44no—
160#3167D4711yescharacteristic
161#1994—471no—
162#1962—471no—
163#1744—471no—
164#1567—471no—
165#1493—471no—
166#1492—471no—
167C2×C47—22no—
168C2×C47—161no—
169C2×C47#196311yescharacteristic
170C2×C47#197311yescharacteristic
171C2×C47#197411yescharacteristic
172#533C2×D4711yescharacteristic
173#532C2×D4711yescharacteristic
174#498Dic4711yescharacteristic, complemented
175#492Dic4711yescharacteristic, complemented
176#487Dic4711yescharacteristic, complemented
177#486Dic4711yescharacteristic, complemented
178#465—22no—
179C4×SD16—22no—
180#423—471no—
181#423—471no—
182C2×#104—942no—
183C2×C4×C8—471no—
184C2×C4×C8C2×D4711yescharacteristic
185C43—471no—
186#339—942no—
187#336—942no—
188#335—942no—
189Q8⋊C8—942no—
190D4⋊C8—942no—
191C47#985411yescharacteristic, complemented, Sylow
192C2×#41—21no—
193C2×SD16—22no—
194C2×M16—941no—
195C2×M16—941no—
196C22×C8—941no—
197C22×C8C22⋊D4712yes—
198C4⋊Q8C2×Dic4711yescharacteristic
199C4⋊Q8C2×Dic4711yescharacteristic
200C4⋊Q8C2×Dic4711yescharacteristic
201#127C2×Dic4711yescharacteristic
202#125—21no—
203#124—21no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206#121—21no—
207#121—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×D4—21no—
211C4×D4—21no—
212#117—21no—
213C2×C42—472no—
214C2×C42—941no—
215C2×C42—941no—
216C2×C42C22×D4711yescharacteristic
217#106—22no—
218C4⋊C8—941no—
219C4⋊C8—941no—
220C4⋊C8—941no—
221C4⋊C8—941no—
222#104—1884no—
223#103—22no—
224#102—22no—
225C4×C8—472no—
226C4×C8—941no—
227C4×C8C2×Dic4711yescharacteristic
228C4×C8C2×Dic4711yescharacteristic
229C4×C8C22⋊D4712yes—
230#41—42no—
231#41—42no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—41no—
239C22×C4—41no—
240C22×C4—942no—
241C22×C4D4×D4711yescharacteristic
242C22×C4#1062535211yescharacteristic
243C22×C4C2×C22⋊D4711yescharacteristic
244SD16—44no—
245M16—1882no—
246M16—1882no—
247C2×C8—42no—
248C2×C8—942no—
249C2×C8—1881no—
250C2×C8—1881no—
251C2×C8—1882no—
252C2×C8C22⋊Dic4712yes—
253C2×C8C22⋊Dic4712yes—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C42—41no—
266C42—944no—
267C42—942no—
268C42—942no—
269C42—1882no—
270C42D4×D4711yescharacteristic
271C42#1062535211yescharacteristic
272C42C22×Dic4711yescharacteristic
273C42C2×C22⋊D4711yescharacteristic
274C23—41no—
275C23#1309399511yescharacteristic
276Q8—42no—
277Q8—42no—
278Q8—42no—
279D4—42no—
280D4—42no—
281D4—42no—
282D4—82no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—944no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4C2×C22⋊Dic4711yescharacteristic
294C2×C4D4×Dic4711yescharacteristic
295C2×C4D4×Dic4711yescharacteristic
296C2×C4#1309399511yescharacteristic
297C2×C4#1309401111yescharacteristic
298C2×C4#1309402011yescharacteristic
299C2×C4#1309405112yes—
300C2×C4#1309405312yes—
301C8—22no—
302C8—22no—
303C8—3761no—
304C8—3761no—
305C8—1882no—
306C22—21no—
307C22—21no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22#1568902611yescharacteristic
312C22#1568936612yes—
313C4—21no—
314C4—21no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—1884no—
319C4#1568914012yes—
320C4#1568922412yes—
321C4#1568924012yes—
322C4#1568950911yescharacteristic
323C4#1568951411yescharacteristic
324C2—22no—
325C2—161no—
326C2#2024422111yescharacteristic
327C2#2024427111yescharacteristic
328C2#2024427711yescharacteristic
subgroupG/H
centerC2×C4#13094020
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3167C2
socleC22×C47#390
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#3167D47
hypocenterC47#9854
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #20244366C2semidirect
3. #20243133C2semidirect
4. #20242147C2semidirect
5. #20241843C2semidirect
20. C47×#498C4semidirect
21. C47×#492C4semidirect
22. C47×#487C4semidirect
23. C47×#486C4semidirect
174. #498Dic47semidirect
175. #492Dic47semidirect
176. #487Dic47semidirect
177. #486Dic47semidirect
191. C47#9854semidirect

Maximal subgroups

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

subgroupindex
#202443662
#202442172
#202431332
#202421472
#202418432
#202418392
C47×#31672
#985447