Finite Group Databasebeta

← #26110954

Subgroups — #26110954

Order 12032.

694 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252429C211yesmaximal, characteristic, complemented
2#20241779C212yesmaximal, complemented
3#20241709C211yesmaximal, characteristic, complemented
4#20241221C212yesmaximal, complemented
5C4×#127×C47C211yesmaximal, characteristic
6#15689201—24no—
7#15689176C2212yes—
8#15689176C2212yes—
9#15689087—22no—
10D4⋊(C47⋊C8)—24no—
11#15688847—22no—
12C2×#127×C47C411yescharacteristic
13C2×C4×D4×C47—22no—
14C2×C4×D4×C47C2211yescharacteristic
15C2×C4×D4×C47C2211yescharacteristic
16C47×#390—22no—
17C47×#390C411yescharacteristic
18C47×#390C411yescharacteristic
19C47×#384C411yescharacteristic
20C43×C47C2211yescharacteristic
21C2×#10625329—24no—
22C4⋊(C47⋊C8)—22no—
23C4⋊(C47⋊C8)D414yescomplemented
24C22⋊(C47⋊C8)—24no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27C4×C47⋊C8—22no—
28C4×C47⋊C8—22no—
29C22×D4×C47—22no—
30C22×D4×C47C2×C411yescharacteristic
31C22×D4×C47C2×C411yescharacteristic
32C23×C4×C47—21no—
33C23×C4×C47D412yes—
34#127×C47—22no—
35#127×C47—22no—
36C4×D4×C47—22no—
37C4×D4×C47—22no—
38C4×D4×C47—22no—
39C4×D4×C47—22no—
40C4×D4×C47—48no—
41C2×C4⋊C4×C47—22no—
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47D412yes—
48C2×C42×C47—21no—
49C2×C42×C47—21no—
50C2×C42×C47C2×C411yescharacteristic
51C2×C42×C47C2×C411yescharacteristic
52C2×C42×C47C2311yescharacteristic
53C24×C47—21no—
54C24×C47C22⋊C412yes—
55C2×D4×C47—28no—
56C2×D4×C47—22no—
57C2×D4×C47—22no—
58C2×D4×C47—21no—
59C2×D4×C47—21no—
60C2×D4×C47—21no—
61C2×D4×C47—22no—
62C2×D4×C47—22no—
63C2×D4×C47—44no—
64C2×D4×C47—44no—
65C2×D4×C47M1612yescomplemented
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—21no—
71C22×C4×C47—44no—
72C22×C4×C47—44no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47C22⋊C412yes—
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C4⋊C4×C47—22no—
80C4⋊C4×C47—22no—
81C4⋊C4×C47—44no—
82C22⋊C4×C47—24no—
83C22⋊C4×C47—44no—
84C22⋊C4×C47—44no—
85C22⋊C4×C47—42no—
86C42×C47—24no—
87C42×C47—21no—
88C42×C47—22no—
89C42×C47—42no—
90C42×C47M1612yescomplemented
91C42×C47C2×D412yes—
92C42×C47#4112yes—
93#10625329—84no—
94C2×C47⋊C8—24no—
95C2×C47⋊C8—42no—
96C23×C47—22no—
97C23×C47—22no—
98C23×C47—22no—
99C23×C47—22no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—42no—
103C23×C47—42no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47C2×C22⋊C411yescharacteristic
107D4×C47—24no—
108D4×C47—416no—
109D4×C47—42no—
110D4×C47—42no—
111D4×C47—42no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—44no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—41no—
125C2×C4×C47—82no—
126C2×C4×C47—82no—
127C2×C4×C47—82no—
128C2×C4×C47C4×D412yes—
129C2×C4×C47#12011yescharacteristic
130C2×C4×C47#12112yes—
131C2×C4×C47#12311yescharacteristic
132C2×C4×C47C2×M1611yescharacteristic
133C2×C4×C47C2×M1611yescharacteristic
134C2×C4×C47#13112yes—
135C47⋊C8—44no—
136C47⋊C8—82no—
137#8963—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—44no—
141C22×C47—44no—
142C22×C47—42no—
143C22×C47—42no—
144C22×C47—42no—
145C22×C47—42no—
146C22×C47—82no—
147C22×C47—82no—
148C22×C47—81no—
149C22×C47—81no—
150C22×C47#38611yescharacteristic
151C22×C47#40711yescharacteristic
152C22×C47#40811yescharacteristic
153C4×C47—24no—
154C4×C47—22no—
155C4×C47—21no—
156C4×C47—42no—
157C4×C47—42no—
158C4×C47—81no—
159C4×C47#43512yes—
160C4×C47#43612yes—
161C4×C47#45312yes—
162#2975—471no—
163C4×#127D4711yescharacteristic
164#1478—472no—
165#1460—471no—
166#1276—472no—
167C2×C47—21no—
168C2×C47—21no—
169C2×C47—82no—
170C2×C47—82no—
171C2×C47#187011yescharacteristic
172C2×C47#188411yescharacteristic
173C2×C47#188711yescharacteristic
174C2×#127Dic4711yescharacteristic
175C2×C4×D4—22no—
176C2×C4×D4C2×D4711yescharacteristic
177C2×C4×D4C2×D4711yescharacteristic
178#436—944no—
179#431—472no—
180#431—472no—
181#407—942no—
182#390—22no—
183#390Dic4711yescharacteristic
184#390Dic4711yescharacteristic
185#384Dic4711yescharacteristic
186C43C2×D4711yescharacteristic
187D4⋊C8—944no—
188#323—942no—
189C47#896311yescharacteristic, complemented, Sylow
190C22×D4—22no—
191C22×D4C2×Dic4711yescharacteristic
192C22×D4C2×Dic4711yescharacteristic
193C23×C4—21no—
194C23×C4C22⋊D4712yes—
195C2×M16—944no—
196#127—22no—
197#127—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—48no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C2×Dic4711yescharacteristic
205C2×C4⋊C4C2×Dic4711yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4C22⋊D4712yes—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42C2×Dic4711yescharacteristic
213C2×C42C2×Dic4711yescharacteristic
214C2×C42C22×D4711yescharacteristic
215C4⋊C8—474no—
216C4⋊C8—942no—
217C22⋊C8—944no—
218C22⋊C8—942no—
219C22⋊C8—942no—
220C4×C8—942no—
221C4×C8—942no—
222C24—21no—
223C24C22⋊Dic4712yes—
224C2×D4—28no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4#1062532912yescomplemented
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4C22⋊Dic4712yes—
245C22×C4C22×Dic4711yescharacteristic
246C22×C4C2×C22⋊D4711yescharacteristic
247C22×C4C2×C22⋊D4711yescharacteristic
248M16—3764no—
249C2×C8—944no—
250C2×C8—1882no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—44no—
254C22⋊C4—24no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—42no—
258C42—24no—
259C42—21no—
260C42—22no—
261C42—42no—
262C42#1062532912yescomplemented
263C42D4×D4712yes—
264C42#1062535212yes—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—41no—
274C23—41no—
275C23C2×C22⋊Dic4711yescharacteristic
276D4—24no—
277D4—416no—
278D4—42no—
279D4—42no—
280D4—42no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4D4×Dic4712yes—
298C2×C4#1309399211yescharacteristic
299C2×C4#1309399612yes—
300C2×C4#1309400511yescharacteristic
301C2×C4C2×#1062532911yescharacteristic
302C2×C4C2×#1062532911yescharacteristic
303C2×C4#1309403112yes—
304C8—3762no—
305C8—1884no—
306C22—21no—
307C22—21no—
308C22—44no—
309C22—44no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—82no—
316C22—81no—
317C22—81no—
318C22#1568900311yescharacteristic
319C22#1568908711yescharacteristic
320C22#1568909011yescharacteristic
321C4—24no—
322C4—22no—
323C4—21no—
324C4—42no—
325C4—42no—
326C4—81no—
327C4#1568919512yes—
328C4#1568920112yes—
329C4#1568929012yes—
330C2—21no—
331C2—21no—
332C2—82no—
333C2—82no—
334C2#2024368611yescharacteristic
335C2#2024377211yescharacteristic
336C2#2024378711yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC4×#127×C47C2
socleC22×C47#386
prime radicalC2×#127×C47C4
cosocleC2×C42×C47C23
hypercenterC4×#127D47
hypocenterC47#8963
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. #20252429C2semidirect
2. #20241779C2semidirect
3. #20241709C2semidirect
4. #20241221C2semidirect
23. C4⋊(C47⋊C8)D4semidirect
65. C2×D4×C47M16semidirect
90. C42×C47M16semidirect
189. C47#8963semidirect
234. C2×D4#10625329semidirect
262. C42#10625329semidirect

Maximal subgroups

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

subgroupindex
#202524292
#202417792
#202417092
#202412212
C4×#127×C472
#896347