Finite Group Databasebeta

← #26115252

Subgroups — #26115252

Order 12032.

720 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094021C211yesmaximal, characteristic
2C4×C4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
3C2×#15688850C214yesmaximal, complemented
4C47×#1767C211yesmaximal, characteristic, complemented
5C2×#13094021C411yescharacteristic
6C2×C4×D188—21no—
7C2×C4×D188—21no—
8C2×C4×D188C2212yes—
9C2×C4⋊(C47⋊C8)—21no—
10C2×C4×C47⋊C8C2212yes—
11#15689025—21no—
12#15689025—21no—
13#15689025C412yes—
14#15688993C411yescharacteristic
15#15688878—21no—
16#15688850—28no—
17C2×C4⋊C8×C47C2212yes—
18C2×#97×C47—21no—
19C43×C47C2211yescharacteristic
20C47×#336—21no—
21C22×D188—21no—
22C22×D188—21no—
23C22×D188C2×C412yes—
24C22×C4×D47—22no—
25C22×C47⋊C8—22no—
26#13094021—24no—
27C4×D188—24no—
28C4×D188—24no—
29C4×D188—44no—
30C4×D188—42no—
31C4×D188—42no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×C4⋊Dic47C2×C412yes—
35C2×(C2×D47)⋊C4—22no—
36C2×(C2×D47)⋊C4—22no—
37C2×(C2×D47)⋊C4—22no—
38C4⋊(C47⋊C8)—24no—
39C4×C47⋊C8—24no—
40C22×C8×C47—22no—
41C2×C42×C47C2×C411yescharacteristic
42C2×C42×C47C2×C411yescharacteristic
43C2×C42×C47D411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C2×C42×C47C2311yescharacteristic
48C4⋊C8×C47—22no—
49C4⋊C8×C47D414yescomplemented
50#97×C47—42no—
51C23×D47—22no—
52C22×C4×C47C22⋊C411yescharacteristic
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22×C411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×Dic47—22no—
60C2×D188—24no—
61C2×D188—24no—
62C2×D188—24no—
63C2×D188—24no—
64C2×D188—22no—
65C2×D188—42no—
66C2×D188—42no—
67C2×D188—42no—
68C2×D188—42no—
69C2×D188M1614yes—
70C2×C4×D47—44no—
71C2×C4×D47—42no—
72C2×C4×D47—42no—
73C2×C4×D47—44no—
74C2×C8×C47—24no—
75C2×C8×C47—42no—
76C2×C8×C47—42no—
77C42×C47—22no—
78C42×C47—22no—
79C42×C47—22no—
80C42×C47—22no—
81C42×C47—41no—
82C42×C47D812yes—
83C42×C47SD1612yes—
84C42×C47C2×D412yes—
85C42×C47#4112yes—
86(C2×D47)⋊C4—44no—
87(C2×D47)⋊C4—44no—
88(C2×D47)⋊C4—42no—
89(C2×D47)⋊C4—42no—
90C4⋊Dic47—24no—
91C4⋊Dic47—42no—
92C4⋊Dic47—42no—
93C2×C47⋊C8—24no—
94C2×C47⋊C8—42no—
95C2×C47⋊C8—42no—
96C23×C47C2×C22⋊C411yescharacteristic
97C22×D47—24no—
98C22×D47—42no—
99C22×D47—42no—
100C22×D47—42no—
101C22×D47—42no—
102C22×D47—42no—
103C22×D47—42no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47#10212yes—
116C2×C4×C47#10212yes—
117C2×C4×C47#10412yes—
118C2×C4×C47C4×D412yes—
119C2×C4×C47#12112yes—
120C2×C4×C47#12411yescharacteristic
121C2×C4×C47#12711yescharacteristic
122C2×C4×C47C2×M1612yes—
123C2×C4×C47#13112yes—
124C2×C4×C47C2×D811yescharacteristic
125C2×C4×C47C2×SD1611yescharacteristic
126C2×C4×C47AGL(1,ℤ/8)12yes—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—42no—
130D188—28no—
131D188—48no—
132D188—48no—
133D188—44no—
134C4×D47—84no—
135C4×D47—84no—
136C8×C47—44no—
137C47⋊C8—44no—
138#9678—471nomaximal, Sylow
139C22×C47#39011yescharacteristic
140C22×C47#40711yescharacteristic
141C22×C47#40811yescharacteristic
142C22×C47C2×#10211yescharacteristic
143C22×C47#41811yescharacteristic
144C22×C47C2×#10411yescharacteristic
145C22×C47#42111yescharacteristic
146C2×D47—44no—
147C2×D47—44no—
148C2×D47—44no—
149C2×D47—82no—
150C2×D47—82no—
151C2×D47—82no—
152C2×D47—82no—
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—22no—
158C4×C47—41no—
159C4×C47#43514yes—
160C4×C47#45912yes—
161C4×C47#46012yes—
162Dic47—82no—
163C4×#127—471no—
164#1767D4711yescharacteristic, complemented
165C4×C4⋊C8—471no—
166C2×D4⋊C8—474no—
167C2×C47#148611yescharacteristic
168C2×C47#149011yescharacteristic
169C2×C47#149211yescharacteristic
170C2×C47#149611yescharacteristic
171C2×C47#195511yescharacteristic
172C2×C47#198011yescharacteristic
173C2×C47#198211yescharacteristic
174D47—84no—
175C2×#127—471no—
176C2×C4×D4—472no—
177C2×C4×D4—941no—
178C2×C4×D4—941no—
179C2×C4⋊C8—941no—
180C2×C4⋊C8C2×D4712yes—
181C2×#97—21no—
182C2×C4×C8—472no—
183#390—472no—
184#390—941no—
185#390—941no—
186#384—471no—
187C43C2×D4711yescharacteristic
188#336—21no—
189#336—941no—
190D4⋊C8—948no—
191C47#967811yescharacteristic, complemented, Sylow
192C22×D4—472no—
193C22×D4—941no—
194C22×D4—941no—
195C23×C4—942no—
196C22×C8—22no—
197C22×C8—942no—
198#127—944no—
199C4×D4—944no—
200C4×D4—944no—
201C4×D4—1884no—
202C4×D4—1882no—
203C4×D4—1882no—
204C2×C4⋊C4—472no—
205C2×C4⋊C4—941no—
206C2×C4⋊C4—941no—
207C2×C22⋊C4—942no—
208C2×C22⋊C4—942no—
209C2×C22⋊C4—942no—
210C2×C42C4×D4711yescharacteristic
211C2×C42C4×D4711yescharacteristic
212C2×C42D18811yescharacteristic
213C2×C42D18811yescharacteristic
214C2×C42C22⋊D4711yescharacteristic
215C2×C42C22⋊D4711yescharacteristic
216C2×C42C22×D4711yescharacteristic
217C4⋊C8—22no—
218C4⋊C8—944no—
219C4⋊C8D18814yescomplemented
220#97—42no—
221C4×C8—944no—
222C24—942no—
223C2×D4—474no—
224C2×D4—944no—
225C2×D4—944no—
226C2×D4—944no—
227C2×D4—942no—
228C2×D4—944no—
229C2×D4—1882no—
230C2×D4—1882no—
231C2×D4—1882no—
232C2×D4—1882no—
233C22×C4—942no—
234C22×C4—1884no—
235C22×C4—1882no—
236C22×C4—1882no—
237C22×C4—1884no—
238C22×C4(C2×D47)⋊C411yescharacteristic
239C22×C4(C2×D47)⋊C411yescharacteristic
240C22×C4(C2×D47)⋊C411yescharacteristic
241C22×C4(C2×D47)⋊C411yescharacteristic
242C22×C4C2×C4×D4711yescharacteristic
243C22×C4C2×D18811yescharacteristic
244C22×C4C2×C22⋊D4711yescharacteristic
245C2×C8—24no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—944no—
249C2×C8—1882no—
250C2×C8—1882no—
251C4⋊C4—944no—
252C4⋊C4—1882no—
253C4⋊C4—1882no—
254C22⋊C4—1884no—
255C22⋊C4—1884no—
256C22⋊C4—1882no—
257C22⋊C4—1882no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—41no—
263C42#1062533412yes—
264C42#1062533612yes—
265C42C2×D18812yes—
266C42#1062535012yes—
267C23—944no—
268C23—1882no—
269C23—1882no—
270C23—1882no—
271C23—1882no—
272C23—1882no—
273C23—1882no—
274C23C2×(C2×D47)⋊C411yescharacteristic
275D4—948no—
276D4—1888no—
277D4—1888no—
278D4—1884no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—3764no—
291C2×C4—3764no—
292C2×C4—1882no—
293C2×C4—1882no—
294C2×C4—1882no—
295C2×C4#1309394412yes—
296C2×C4#1309394412yes—
297C2×C4#1309395012yes—
298C2×C4C4×D18812yes—
299C2×C4#1309399812yes—
300C2×C4#1309401211yescharacteristic
301C2×C4#1309402111yescharacteristic
302C2×C4M16×D4712yes—
303C2×C4#1309403212yes—
304C2×C4C2×#1062533411yescharacteristic
305C2×C4C2×#1062533611yescharacteristic
306C2×C4#1309404912yes—
307C8—44no—
308C8—1884no—
309C22—3762no—
310C22—3762no—
311C22—3762no—
312C22—3762no—
313C22—1884no—
314C22—1884no—
315C22—1884no—
316C22#1568902511yescharacteristic
317C22#1568908611yescharacteristic
318C22#1568909111yescharacteristic
319C22C2×#1309394411yescharacteristic
320C22#1568912711yescharacteristic
321C22C2×#1309395011yescharacteristic
322C22#1568913911yescharacteristic
323C4—21no—
324C4—21no—
325C4—22no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—3762no—
330C4#1568919614yes—
331C4#1568931812yes—
332C4#1568932312yes—
333C2—3764no—
334C2#2024181211yescharacteristic
335C2#2024183011yescharacteristic
336C2#2024183811yescharacteristic
337C2#2024185611yescharacteristic
338C2#2024418611yescharacteristic
339C2#2024430711yescharacteristic
340C2#2024431511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1767C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13094021C4
cosocleC2×C42×C47C23
hypercenter#1767D47
hypocenterC47#9678
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊(C47⋊C8)C2semidirect
3. C2×#15688850C2semidirect
4. C47×#1767C2semidirect
49. C4⋊C8×C47D4semidirect
164. #1767D47semidirect
191. C47#9678semidirect
219. C4⋊C8D188semidirect

Maximal subgroups

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

subgroupindex
C4×#130940212
C4×C4⋊(C47⋊C8)2
C2×#156888502
C47×#17672
#967847