Finite Group Databasebeta

← #26226808

Subgroups — #26226808

Order 12032.

834 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245786C211yesmaximal, characteristic
2C2×#15688989C211yesmaximal, characteristic
3#20244158C214yesmaximal, complemented
4#20243583C212yesmaximal
5#20243578C212yesmaximal
6#20243552C212yesmaximal, complemented
7#20243519C212yesmaximal, complemented
8C2×C47×#431C211yesmaximal, characteristic
9C22×C4⋊Dic47C2212yes—
10C2×#13093958C2214yes—
11C2×#13093956C2214yes—
12#15688991—21no—
13#15688989—21no—
14#15688989—21no—
15#15688989C2212yes—
16#15688989C2212yes—
17#15688989C2211yescharacteristic
18#15688989C2211yescharacteristic
19#15688986—21no—
20#15688983C2212yes—
21C4×C4⋊Dic47C2212yes—
22C4×C22⋊Dic47—21no—
23#15688969—21no—
24C2×C47⋊#95—21no—
25C2×C47⋊#95—21no—
26#15688886C2214yes—
27#15688886C2214yes—
28C22×C42×C47C2211yescharacteristic
29C47×#431C418yescomplemented
30C2×C4⋊C8×C47C2212yes—
31C2×C22⋊C8×C47C2211yescharacteristic
32C2×C22⋊C8×C47C2211yescharacteristic
33C2×C4×C8×C47C2212yes—
34C23×Dic47—22no—
35C2×C4⋊Dic47—22no—
36C2×C4⋊Dic47—44no—
37C2×C4⋊Dic47C2312yes—
38C2×C4⋊Dic47C2312yes—
39C2×C4⋊Dic47C2314yes—
40C2×C22⋊Dic47—22no—
41C2×C4×Dic47—22no—
42#13093958—44no—
43#13093956—44no—
44C47⋊#95—22no—
45C47⋊#95—22no—
46C47⋊#95—22no—
47C47⋊#95—22no—
48C47⋊#95—22no—
49C47⋊#95—21no—
50C47⋊#95—21no—
51C47⋊#95—21no—
52C47⋊#95—21no—
53C47⋊#95—22no—
54C23×C4×C47D411yescharacteristic
55C23×C4×C47D411yescharacteristic
56C23×C4×C47C2311yescharacteristic
57C22×C8×C47C2312yes—
58C22×C8×C47C2312yes—
59C2×C42×C47C2×C414yes—
60C2×C42×C47D411yescharacteristic
61C2×C42×C47D411yescharacteristic
62C2×C42×C47Q814yes—
63C2×C42×C47C2311yescharacteristic
64C2×C42×C47C2311yescharacteristic
65C4⋊C8×C47C2×C418yescomplemented
66C22⋊C8×C47C2×C414yes—
67C22⋊C8×C47C2×C414yes—
68C4×C8×C47C2×C418yescomplemented
69C24×C47C2×D411yescharacteristic
70C22×C4×C47—24no—
71C22×C4×C47—22no—
72C22×C4×C47—22no—
73C22×C4×C47—42no—
74C22×C4×C47C4⋊C414yes—
75C22×C4×C47C4⋊C414yes—
76C22×C4×C47C22×C412yes—
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×Q812yes—
83C22×C4×C47C2411yescharacteristic
84C22×Dic47—22no—
85C22×Dic47—22no—
86C22×Dic47—24no—
87C22×Dic47—44no—
88C22×Dic47—44no—
89C22×Dic47—44no—
90C2×C8×C47—22no—
91C2×C8×C47—22no—
92C2×C8×C47—22no—
93C2×C8×C47—22no—
94C2×C8×C47C22×C414yes—
95C2×C8×C47C22×C414yes—
96C42×C47C4⋊C414yes—
97C42×C47C4⋊C414yes—
98C42×C47C22×C412yes—
99C42×C47C22×C412yes—
100C42×C47C2×Q812yes—
101C42×C47C2×Q812yes—
102C22⋊Dic47—44no—
103C4⋊Dic47—24no—
104C4⋊Dic47—24no—
105C4⋊Dic47—24no—
106C4⋊Dic47—24no—
107C4⋊Dic47—44no—
108C4×Dic47—44no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—22no—
112C23×C47C2×C4⋊C412yes—
113C23×C47#13514yes—
114C23×C47C22×D411yescharacteristic
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—22no—
121C2×C4×C47—28no—
122C2×C4×C47—24no—
123C2×C4×C47—24no—
124C2×C4×C47—44no—
125C2×C4×C47C2×C4⋊C412yes—
126C2×C4×C47C2×C4⋊C412yes—
127C2×C4×C47C2×C4⋊C412yes—
128C2×C4×C47C2×C4⋊C412yes—
129C2×C4×C47C2×C4⋊C412yes—
130C2×C4×C47C23×C411yescharacteristic
131C2×C4×C47C22×Q811yescharacteristic
132C2×C4×C47H(5,𝔽2)12yes—
133C2×C4×C472−1+412yes—
134C2×Dic47—44no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—44no—
140C2×Dic47—42no—
141C2×Dic47—42no—
142C2×Dic47—44no—
143C2×Dic47—88no—
144C8×C47—24no—
145C8×C47—24no—
146#21983—471nomaximal, Sylow
147C22×C47—22no—
148C22×C47—22no—
149C22×C47—24no—
150C22×C47—24no—
151C22×C47#42714yes—
152C22×C47C22×C4⋊C411yescharacteristic
153C22×C47#53511yescharacteristic
154C22×C47#53611yescharacteristic
155C22×C47C2×#13512yes—
156C22×C47#57512yes—
157C4×C47—22no—
158C4×C47—22no—
159C4×C47—22no—
160C4×C47—22no—
161C4×C47—22no—
162C4×C47#52012yes—
163C4×C47#54412yes—
164Dic47—82no—
165Dic47—84no—
166Dic47—82no—
167C2×#431D4711yescharacteristic
168#2283—471no—
169C2×#382—471no—
170#1949—474no—
171#1853—472no—
172#1852—472no—
173#1846—472no—
174#1835—472no—
175C2×C47—24no—
176C2×C47#231711yescharacteristic
177C2×C47C2×#42711yescharacteristic
178C2×C47#291111yescharacteristic
179C2×C47#321112yes—
180C2×C47#323212yes—
181C22×C4⋊C4—472no—
182C22×C42C2×D4711yescharacteristic
183#431Dic4718yescomplemented
184C2×#107—474no—
185C2×#106—474no—
186C2×C4⋊C8C2×D4712yes—
187C2×C22⋊C8C2×D4711yescharacteristic
188C2×C22⋊C8C2×D4711yescharacteristic
189C2×C4×C8C2×D4712yes—
190#383—941no—
191#382—471no—
192#382—472no—
193#382—471no—
194#382—472no—
195#382—941no—
196#382—941no—
197#381—941no—
198#380—472no—
199C4×C4⋊C4—472no—
200C4×C22⋊C4—941no—
201#376—941no—
202C2×#95—941no—
203C2×#95—941no—
204#340—474no—
205#340—474no—
206C47#2198311yescharacteristic, complemented, Sylow
207C23×C4—942no—
208C23×C4D18811yescharacteristic
209C23×C4D18811yescharacteristic
210C23×C4C22×D4711yescharacteristic
211C22×C8C22×D4712yes—
212C22×C8C22×D4712yes—
213C2×C4⋊C4—474no—
214C2×C4⋊C4—472no—
215C2×C4⋊C4—472no—
216C2×C4⋊C4—942no—
217C2×C4⋊C4—1884no—
218C2×C22⋊C4—942no—
219C2×C42—942no—
220C2×C42Dic9414yes—
221C2×C42D18811yescharacteristic
222C2×C42D18811yescharacteristic
223C2×C42C2×Dic4714yes—
224C2×C42C22×D4711yescharacteristic
225C2×C42C22×D4711yescharacteristic
226#107—1884no—
227#106—1884no—
228C4⋊C8C2×Dic4718yescomplemented
229C22⋊C8C2×Dic4714yes—
230C22⋊C8C2×Dic4714yes—
231C4×C8C2×Dic4718yescomplemented
232#95—942no—
233#95—942no—
234#95—942no—
235#95—942no—
236#95—942no—
237#95—941no—
238#95—941no—
239#95—941no—
240#95—941no—
241#95—942no—
242C24C2×D18811yescharacteristic
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—942no—
248C22×C4—942no—
249C22×C4—944no—
250C22×C4—1884no—
251C22×C4—1884no—
252C22×C4—1884no—
253C22×C4C4⋊Dic4714yes—
254C22×C4C4⋊Dic4714yes—
255C22×C4C2×Dic9412yes—
256C22×C4C2×D18811yescharacteristic
257C22×C4C2×D18811yescharacteristic
258C22×C4C2×D18811yescharacteristic
259C22×C4C2×D18811yescharacteristic
260C22×C4C2×D18811yescharacteristic
261C22×C4C22×Dic4712yes—
262C22×C4C23×D4711yescharacteristic
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8C22×Dic4714yes—
268C2×C8C22×Dic4714yes—
269C4⋊C4—944no—
270C4⋊C4—944no—
271C4⋊C4—944no—
272C4⋊C4—944no—
273C4⋊C4—1884no—
274C22⋊C4—1884no—
275C42—1884no—
276C42C4⋊Dic4714yes—
277C42C4⋊Dic4714yes—
278C42C2×Dic9412yes—
279C42C2×Dic9412yes—
280C42C22×Dic4712yes—
281C42C22×Dic4712yes—
282C23—22no—
283C23—21no—
284C23—21no—
285C23C2×C4⋊Dic4712yes—
286C23#1309404614yes—
287C23C22×D18811yescharacteristic
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—28no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—44no—
298C2×C4—3768no—
299C2×C4—1882no—
300C2×C4—1882no—
301C2×C4—1884no—
302C2×C4—1882no—
303C2×C4—1882no—
304C2×C4—1884no—
305C2×C4—1884no—
306C2×C4—1882no—
307C2×C4—1882no—
308C2×C4C2×C4⋊Dic4712yes—
309C2×C4C2×C4⋊Dic4712yes—
310C2×C4C2×C4⋊Dic4712yes—
311C2×C4C2×C4⋊Dic4712yes—
312C2×C4C2×C4⋊Dic4712yes—
313C2×C4C23×Dic4711yescharacteristic
314C2×C4C22×Dic9411yescharacteristic
315C2×C4#1309407012yes—
316C2×C4#1309407112yes—
317C8—24no—
318C8—24no—
319C22—22no—
320C22—24no—
321C22—24no—
322C22—22no—
323C22#1568916114yes—
324C22C22×C4⋊Dic4711yescharacteristic
325C22#1568969211yescharacteristic
326C22#1568969911yescharacteristic
327C22C2×#1309404612yes—
328C22#1568992312yes—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—3762no—
335C4—3764no—
336C4—3762no—
337C4#1568960612yes—
338C4#1568975112yes—
339C2—24no—
340C2#2024601111yescharacteristic
341C2C2×#1568916111yescharacteristic
342C2#2025197811yescharacteristic
343C2#2025435512yes—
344C2#2025455812yes—
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#431C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenterC2×#431D47
hypocenterC47#21983
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
3. #20244158C2semidirect
6. #20243552C2semidirect
7. #20243519C2semidirect
29. C47×#431C4semidirect
65. C4⋊C8×C47C2×C4semidirect
68. C4×C8×C47C2×C4semidirect
183. #431Dic47semidirect
206. C47#21983semidirect
228. C4⋊C8C2×Dic47semidirect
231. C4×C8C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202457862
C2×#156889892
#202441582
#202435832
#202435782
#202435522
#202435192
C2×C47×#4312
#2198347