Finite Group Databasebeta

← #26134075

Subgroups — #26134075

Order 12032.

390 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688948C211yesmaximal, characteristic, complemented
2C2×#15688923C211yesmaximal, characteristic
3C2×#15688923C211yesmaximal, characteristic
4#20244254C211yesmaximal, characteristic
5#20241479C211yesmaximal, characteristic, complemented
6#20241466C211yesmaximal, characteristic, complemented
7C4×#107×C47C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9C22×C47⋊C16C2211yescharacteristic
10C22×C47⋊C16C2211yescharacteristic
11C2×#13093958C2211yescharacteristic
12C2×#13093946—21no—
13#15689025—21no—
14#15688948—22no—
15#15688923—22no—
16#15688923C414yescomplemented
17C2×#107×C47C2211yescharacteristic
18C2×#107×C47C2211yescharacteristic
19C2×C4×C8×C47C2211yescharacteristic
20C4×C4⋊C4×C47—21no—
21C47×#340—21no—
22C22×D188—21no—
23C2×D376—21no—
24C2×D376—21no—
25C2×D376—21no—
26C2×D376—21no—
27C2×D376—21no—
28C2×D376C2×C412yes—
29C2×C4⋊Dic47—21no—
30C2×(C2×D47)⋊C4—41no—
31C2×(C2×D47)⋊C4—41no—
32C2×C47⋊C16—21no—
33C2×C47⋊C16—21no—
34C2×C47⋊C16—21no—
35C2×C47⋊C16—21no—
36C2×C47⋊C16—21no—
37C2×C47⋊C16C2×C412yes—
38#13093958—21no—
39#13093958—21no—
40#13093946—42no—
41C22×C8×C47D411yescharacteristic
42C22×C8×C47C2311yescharacteristic
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C42×C47—41no—
46C2×C42×C47D411yescharacteristic
47#107×C47—21no—
48#107×C47—21no—
49#107×C47C2×C412yes—
50#107×C47D412yes—
51C4×C8×C47—21no—
52C4×C8×C47—21no—
53#95×C47—41no—
54C23×D47—41no—
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47D811yescharacteristic
58C22×C4×C47D811yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×Dic47—41no—
61C2×D188—22no—
62C2×D188—41no—
63C2×D188—41no—
64C2×D188—41no—
65C2×D188—41no—
66C2×D188—41no—
67C2×C8×C47—21no—
68C2×C8×C47—21no—
69C2×C8×C47—21no—
70C2×C8×C47C22×C411yescharacteristic
71C2×C8×C47C2×D411yescharacteristic
72C2×C8×C47#4111yescharacteristic
73C2×C8×C47#4111yescharacteristic
74C2×C8×C47#4111yescharacteristic
75C2×C8×C47#4111yescharacteristic
76C4⋊C4×C47—22no—
77C4⋊C4×C47—22no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47—82no—
83(C2×D47)⋊C4—82no—
84(C2×D47)⋊C4—82no—
85C4⋊Dic47—41no—
86C4⋊Dic47—41no—
87D376—22no—
88D376—22no—
89D376—41no—
90D376—41no—
91C47⋊C16—22no—
92C47⋊C16—41no—
93C23×C47C2×D811yescharacteristic
94C22×D47—42no—
95C22×D47—81no—
96C22×D47—81no—
97C22×D47—81no—
98C22×D47—81no—
99C22×D47—81no—
100C22×D47—81no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—81no—
109C2×C4×C47—81no—
110C2×C4×C47—81no—
111C2×C4×C47—81no—
112C2×C4×C47D1612yes—
113C2×C4×C47C4×D411yescharacteristic
114C2×C4×C47#12111yescharacteristic
115C2×C4×C47#12311yescharacteristic
116C2×C4×C47#13511yescharacteristic
117C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C47#13711yescharacteristic
119C2×Dic47—81no—
120C2×Dic47—81no—
121C2×Dic47—81no—
122D188—42no—
123D188—42no—
124D188—81no—
125D188—81no—
126C8×C47—41no—
127C8×C47#11711yescharacteristic
128C8×C47C4×D411yescharacteristic
129C8×C47#12411yescharacteristic
130C8×C47#12611yescharacteristic
131#13130—471nomaximal, Sylow
132C22×C47C4×D811yescharacteristic
133C22×C47#45911yescharacteristic
134C22×C47#48011yescharacteristic
135C22×C47C2×D1611yescharacteristic
136C22×C47#50811yescharacteristic
137C22×C47#50911yescharacteristic
138C22×C47#51011yescharacteristic
139C2×D47—82no—
140C2×D47—82no—
141C2×D47—82no—
142C2×D47—161no—
143C2×D47—161no—
144C2×D47—161no—
145C2×D47—161no—
146C4×C47—22no—
147C4×C47—41no—
148C4×C47—82no—
149C4×C47—82no—
150C4×C47#38811yescharacteristic
151C4×C47#44011yescharacteristic
152C4×C47#46411yescharacteristic
153C4×C47#48211yescharacteristic
154Dic47—161no—
155C2×#366—471no—
156C2×#357—471no—
157C2×#357—471no—
158#1970—471no—
159C4×#107D4711yescharacteristic, complemented
160#1380—471no—
161#1374—471no—
162C2×C47#192511yescharacteristic
163C2×C47C4×D1611yescharacteristic
164C2×C47#217511yescharacteristic
165C2×C47#220611yescharacteristic
166C2×C47#221111yescharacteristic
167C2×C47#223211yescharacteristic
168C2×C47#223411yescharacteristic
169D47—162no—
170C22×D8—471no—
171C22×C16—471no—
172C22×C16—471no—
173C2×#107—471no—
174C2×#107C2×D4711yescharacteristic
175C2×#107C2×D4711yescharacteristic
176C2×#102—941no—
177C2×C4×C8C2×D4711yescharacteristic
178#390—941no—
179C4×C4⋊C4—21no—
180#366—942no—
181#357—474no—
182#357—942no—
183#340—21no—
184C47#1313011yescharacteristic, complemented, Sylow
185C22×D4—941no—
186C2×D8—472no—
187C2×D8—941no—
188C2×D8—941no—
189C2×D8—941no—
190C2×D8—941no—
191C2×D8—941no—
192C22×C8C22⋊D4711yescharacteristic
193C22×C8C22×D4711yescharacteristic
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—941no—
197C2×C22⋊C4—1881no—
198C2×C22⋊C4—1881no—
199C2×C42—41no—
200C2×C42C22⋊D4711yescharacteristic
201C2×C16—472no—
202C2×C16—941no—
203C2×C16—941no—
204C2×C16—941no—
205C2×C16—941no—
206C2×C16—941no—
207#107—21no—
208#107—21no—
209#107—941no—
210#107—941no—
211#107C4×D4712yes—
212#107D18812yes—
213#102—1882no—
214C4×C8—21no—
215C4×C8—21no—
216#95—41no—
217C24—1881no—
218C2×D4—942no—
219C2×D4—1881no—
220C2×D4—1881no—
221C2×D4—1881no—
222C2×D4—1881no—
223C2×D4—1881no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—1881no—
227C22×C4#1062533411yescharacteristic
228C22×C4#1062533411yescharacteristic
229C22×C4C2×C22⋊D4711yescharacteristic
230D8—942no—
231D8—942no—
232D8—1881no—
233D8—1881no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8C2×C4×D4711yescharacteristic
238C2×C8C2×D18811yescharacteristic
239C2×C8#1062535011yescharacteristic
240C2×C8#1062535011yescharacteristic
241C2×C8#1062535011yescharacteristic
242C2×C8#1062535011yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—1881no—
248C4⋊C4—1881no—
249C22⋊C4—3762no—
250C22⋊C4—3762no—
251C42—21no—
252C42—21no—
253C42—82no—
254C16—942no—
255C16—1881no—
256C23—3761no—
257C23—3761no—
258C23—3761no—
259C23—3761no—
260C23—3761no—
261C23—3761no—
262C23—1882no—
263C23C2×#1062533411yescharacteristic
264D4—3761no—
265D4—3761no—
266D4—1882no—
267D4—1882no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—3761no—
280C2×C4—3761no—
281C2×C4—3761no—
282C2×C4#1309396512yes—
283C2×C4C4×C22⋊D4711yescharacteristic
284C2×C4#1309399811yescharacteristic
285C2×C4#1309400711yescharacteristic
286C2×C4#1309404411yescharacteristic
287C2×C4#1309404911yescharacteristic
288C2×C4#1309405511yescharacteristic
289C8—41no—
290C8#1309398111yescharacteristic
291C8C4×D18811yescharacteristic
292C8#1309401211yescharacteristic
293C8#1309401811yescharacteristic
294C22—3762no—
295C22—3762no—
296C22—3762no—
297C22—7521no—
298C22—7521no—
299C22—7521no—
300C22—7521no—
301C22C4×#1062533411yescharacteristic
302C22#1568931811yescharacteristic
303C22#1568943611yescharacteristic
304C22C2×#1309396511yescharacteristic
305C22#1568955311yescharacteristic
306C22#1568955811yescharacteristic
307C22#1568956411yescharacteristic
308C4—22no—
309C4—41no—
310C4—82no—
311C4—82no—
312C4—7521no—
313C4#1568901411yescharacteristic
314C4#1568922111yescharacteristic
315C4#1568934511yescharacteristic
316C4#1568944711yescharacteristic
317C2—7522no—
318C2#2024400811yescharacteristic
319C2C4×#1309396511yescharacteristic
320C2#2024525011yescharacteristic
321C2#2024542311yescharacteristic
322C2#2024545011yescharacteristic
323C2#2024556811yescharacteristic
324C2#2024557911yescharacteristic
subgroupG/H
centerC23C2×#10625334
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC4×#107×C47C2
socleC23×C47C2×D8
prime radicalC22×D376C22
cosocleC22×C8×C47C23
hypercenterC4×#107D47
hypocenterC47#13130
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
1. C2×#15688948C2semidirect
5. #20241479C2semidirect
6. #20241466C2semidirect
7. C4×#107×C47C2semidirect
16. #15688923C4semidirect
159. C4×#107D47semidirect
184. C47#13130semidirect

Maximal subgroups

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

subgroupindex
C2×#156889482
C2×#156889232
#202442542
#202414792
#202414662
C4×#107×C472
#1313047