Finite Group Databasebeta

← #27081260

Subgroups — #27081260

Order 12032.

1702 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252494C214yesmaximal, complemented
2#20252490C214yesmaximal, complemented
3#20252483C218yesmaximal, complemented
4Q8×#10625329C218yesmaximal, complemented
5#20252026C214yesmaximal
6#20252021C212yesmaximal
7C2×C47×#533C211yesmaximal, characteristic
8#15689254C22116yescomplemented
9#15689254C22132yescomplemented
10Q8×C47⋊C8C22116yescomplemented
11#15689184C22116yes—
12#15689184C22116yes—
13#15689180C2214yes—
14#15689180C2214yes—
15#15689176C2214yes—
16#15689176C2214yes—
17#15689147C2214yes—
18#15689147C2218yes—
19#15689087C2214yes—
20C4×#10625329C2214yes—
21C4×#10625329C2218yes—
22C47×#533C412yes—
23C47×#533C414yes—
24C47×#533C412yes—
25C47×#533C2214yes—
26C47×#533C2212yes—
27C47×#533C2212yes—
28C2×C4⋊Q8×C47C412yes—
29C2×#125×C47C412yes—
30C2×C22⋊Q8×C47C414yes—
31C2×C4×Q8×C47C2214yes—
32C2×#117×C47C2212yes—
33C22×C42×C47C2211yescharacteristic
34C2×#10625329C2318yes—
35C4⋊(C47⋊C8)C2318yes—
36C4⋊(C47⋊C8)C23132yes—
37C4⋊(C47⋊C8)C2318yes—
38C22⋊(C47⋊C8)C2318yes—
39C22⋊(C47⋊C8)C2318yes—
40#13093935C2318yes—
41#13093935C23116yes—
42C4×C47⋊C8C2318yes—
43C4×C47⋊C8C23116yes—
44C22×Q8×C47C2×C414yes—
45C23×C4×C47C2×C412yes—
46C23×C4×C47C2311yescharacteristic
47C4⋊Q8×C47C2×C414yes—
48C4⋊Q8×C47C2×C414yes—
49C4⋊Q8×C47C2×C414yes—
50C4⋊Q8×C47C2×C414yes—
51#125×C47C2×C418yes—
52#125×C47C2×C414yes—
53#125×C47C2×C414yes—
54C22⋊Q8×C47C2×C418yes—
55C22⋊Q8×C47C2×C418yes—
56C22⋊Q8×C47C2×C418yes—
57C22⋊Q8×C47C2×C418yes—
58C4×Q8×C47C2×C418yes—
59C4×Q8×C47C2×C418yes—
60C4×Q8×C47C2318yes—
61C4×Q8×C47C2318yes—
62#117×C47C2×C414yes—
63#117×C47C2×C414yes—
64#117×C47C2314yes—
65#117×C47C2314yes—
66C2×C4⋊C4×C47C2×C414yes—
67C2×C4⋊C4×C47C2×C418yes—
68C2×C4⋊C4×C47C2×C414yes—
69C2×C22⋊C4×C47C2×C414yes—
70C2×C42×C47C2×C411yescharacteristic
71C2×C42×C47C2×C411yescharacteristic
72C2×C42×C47C2×C411yescharacteristic
73C2×C42×C47C2×C412yes—
74C2×C42×C47C2×C411yescharacteristic
75C2×C42×C47C2312yes—
76C2×C42×C47C2311yescharacteristic
77C2×C42×C47C2312yes—
78C2×C42×C47C2311yescharacteristic
79C2×C42×C47C2314yes—
80C24×C47C22×C411yescharacteristic
81C2×Q8×C47—48no—
82C2×Q8×C47C22×C418yes—
83C2×Q8×C47C22×C418yes—
84C22×C4×C47—24no—
85C22×C4×C47—44no—
86C22×C4×C47C22×C411yescharacteristic
87C22×C4×C47C22×C411yescharacteristic
88C22×C4×C47C22×C411yescharacteristic
89C22×C4×C47C22×C412yes—
90C22×C4×C47C22×C412yes—
91C22×C4×C47C22×C412yes—
92C22×C4×C47C22×C412yes—
93C22×C4×C47C22×C412yes—
94C22×C4×C47C22×C412yes—
95C22×C4×C47C22×C418yes—
96C22×C4×C47C2411yescharacteristic
97C22×C4×C47C2411yescharacteristic
98C22×C4×C47C2411yescharacteristic
99C4⋊C4×C47C22×C418yes—
100C4⋊C4×C47C22×C418yes—
101C4⋊C4×C47C22×C4116yes—
102C4⋊C4×C47C22×C4116yes—
103C4⋊C4×C47C22×C418yes—
104C4⋊C4×C47C22×C418yes—
105C22⋊C4×C47C22×C418yes—
106C22⋊C4×C47C22×C418yes—
107C42×C47C22×C412yes—
108C42×C47C22×C412yes—
109C42×C47C22×C411yescharacteristic
110C42×C47C22×C412yes—
111C42×C47C22×C411yescharacteristic
112C42×C47C22×C411yescharacteristic
113C42×C47C22×C412yes—
114C42×C47C22×C411yescharacteristic
115C42×C47C22×C418yes—
116C42×C47C2412yes—
117C42×C47C2412yes—
118C42×C47C2418yes—
119#10625329—216no—
120C2×C47⋊C8C24116yes—
121C23×C47—22no—
122C23×C47—42no—
123C23×C47C23×C411yescharacteristic
124C23×C47C23×C411yescharacteristic
125C23×C47C23×C411yescharacteristic
126Q8×C47—216no—
127Q8×C47—48no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47—22no—
131C2×C4×C47—24no—
132C2×C4×C47—24no—
133C2×C4×C47—24no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47—48no—
140C2×C4×C47C23×C411yescharacteristic
141C2×C4×C47C23×C411yescharacteristic
142C2×C4×C47C23×C412yes—
143C2×C4×C47C23×C412yes—
144C2×C4×C47C23×C412yes—
145C2×C4×C47C23×C4116yes—
146C2×C4×C47C23×C411yescharacteristic
147C2×C4×C47C23×C412yes—
148C2×C4×C472−1+414yes—
149C2×C4×C47C2511yescharacteristic
150C47⋊C8—216no—
151#60249—471nomaximal, Sylow
152C22×C47—22no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—22no—
156C22×C47—22no—
157C22×C47—44no—
158C22×C47#51912yes—
159C22×C47C24×C411yescharacteristic
160C4×C47—21no—
161C4×C47—21no—
162C4×C47—22no—
163C4×C47—22no—
164C4×C47—22no—
165C4×C47—216no—
166C4×C47#51912yes—
167C4×C47#56814yes—
168C4×C47C2×2−1+412yes—
169C2×#533D4711yescharacteristic
170#2987—474no—
171#2986—474no—
172#2985—478no—
173Q8×M16—478no—
174#2921—474no—
175#2920—472no—
176C2×C47—22no—
177C2×C47—41no—
178C2×C47—41no—
179C2×C47C2×#51911yescharacteristic
180C2×C47C2×#56812yes—
181#533Dic4712yes—
182#533Dic4714yes—
183#533Dic4712yes—
184#533C2×D4714yes—
185#533C2×D4712yes—
186#533C2×D4712yes—
187C2×C4⋊Q8Dic4712yes—
188C2×#125Dic4712yes—
189C2×C22⋊Q8Dic4714yes—
190C2×C4×Q8C2×D4714yes—
191C2×#117C2×D4712yes—
192C22×C42C2×D4711yescharacteristic
193#446—4716no—
194#446—4732no—
195C8×Q8—4716no—
196#433—4716no—
197#433—4716no—
198#432—474no—
199#432—474no—
200#431—474no—
201#431—474no—
202#423—474no—
203#423—478no—
204#407—474no—
205C4×M16—474no—
206C4×M16—478no—
207C47#6024911yescharacteristic, complemented, Sylow
208C22×Q8C2×Dic4714yes—
209C23×C4C2×Dic4712yes—
210C23×C4C22×D4711yescharacteristic
211C2×M16—478no—
212C4⋊Q8C2×Dic4714yes—
213C4⋊Q8C2×Dic4714yes—
214C4⋊Q8C2×Dic4714yes—
215C4⋊Q8C2×Dic4714yes—
216#125C2×Dic4718yes—
217#125C2×Dic4714yes—
218#125C2×Dic4714yes—
219C22⋊Q8C2×Dic4718yes—
220C22⋊Q8C2×Dic4718yes—
221C22⋊Q8C2×Dic4718yes—
222C22⋊Q8C2×Dic4718yes—
223C4×Q8C2×Dic4718yes—
224C4×Q8C2×Dic4718yes—
225C4×Q8C22×D4718yes—
226C4×Q8C22×D4718yes—
227#117C2×Dic4714yes—
228#117C2×Dic4714yes—
229#117C22×D4714yes—
230#117C22×D4714yes—
231C2×C4⋊C4C2×Dic4714yes—
232C2×C4⋊C4C2×Dic4718yes—
233C2×C4⋊C4C2×Dic4714yes—
234C2×C22⋊C4C2×Dic4714yes—
235C2×C42C2×Dic4711yescharacteristic
236C2×C42C2×Dic4711yescharacteristic
237C2×C42C2×Dic4711yescharacteristic
238C2×C42C2×Dic4712yes—
239C2×C42C2×Dic4711yescharacteristic
240C2×C42C22×D4714yes—
241C2×C42C22×D4712yes—
242C2×C42C22×D4711yescharacteristic
243C2×C42C22×D4712yes—
244C2×C42C22×D4711yescharacteristic
245C4⋊C8—478no—
246C4⋊C8—478no—
247C4⋊C8—4732no—
248C22⋊C8—478no—
249C22⋊C8—478no—
250#97—478no—
251#97—4716no—
252C4×C8—478no—
253C4×C8—4716no—
254C24C22×Dic4711yescharacteristic
255C2×Q8—48no—
256C2×Q8C22×Dic4718yes—
257C2×Q8C22×Dic4718yes—
258C22×C4—24no—
259C22×C4—44no—
260C22×C4C22×Dic4711yescharacteristic
261C22×C4C22×Dic4711yescharacteristic
262C22×C4C22×Dic4711yescharacteristic
263C22×C4C22×Dic4712yes—
264C22×C4C22×Dic4712yes—
265C22×C4C22×Dic4712yes—
266C22×C4C22×Dic4712yes—
267C22×C4C22×Dic4712yes—
268C22×C4C22×Dic4712yes—
269C22×C4C22×Dic4718yes—
270C22×C4C23×D4711yescharacteristic
271C22×C4C23×D4711yescharacteristic
272C22×C4C23×D4711yescharacteristic
273M16—9416no—
274C2×C8—4716no—
275C4⋊C4C22×Dic4718yes—
276C4⋊C4C22×Dic4718yes—
277C4⋊C4C22×Dic47116yes—
278C4⋊C4C22×Dic47116yes—
279C4⋊C4C22×Dic4718yes—
280C4⋊C4C22×Dic4718yes—
281C22⋊C4C22×Dic4718yes—
282C22⋊C4C22×Dic4718yes—
283C42C22×Dic4712yes—
284C42C22×Dic4712yes—
285C42C22×Dic4711yescharacteristic
286C42C22×Dic4712yes—
287C42C22×Dic4711yescharacteristic
288C42C22×Dic4711yescharacteristic
289C42C22×Dic4712yes—
290C42C22×Dic4711yescharacteristic
291C42C22×Dic4718yes—
292C42C23×D4712yes—
293C42C23×D4718yes—
294C42C23×D4712yes—
295C23—22no—
296C23—42no—
297C23C23×Dic4711yescharacteristic
298C23C23×Dic4711yescharacteristic
299C23C23×Dic4711yescharacteristic
300Q8—216no—
301Q8—48no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—48no—
314C2×C4C23×Dic4711yescharacteristic
315C2×C4C23×Dic4711yescharacteristic
316C2×C4C23×Dic4712yes—
317C2×C4C23×Dic4712yes—
318C2×C4C23×Dic4712yes—
319C2×C4C23×Dic47116yes—
320C2×C4C23×Dic4711yescharacteristic
321C2×C4C23×Dic4712yes—
322C2×C4#1309407214yes—
323C2×C4C24×D4711yescharacteristic
324C8—9416no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—21no—
329C22—21no—
330C22—44no—
331C22#1568960112yes—
332C22C24×Dic4711yescharacteristic
333C4—22no—
334C4—22no—
335C4—22no—
336C4—216no—
337C4—21no—
338C4—21no—
339C4#1568960112yes—
340C4#1568989014yes—
341C4C2×#1309407212yes—
342C2—22no—
343C2—41no—
344C2—41no—
345C2C2×#1568960111yescharacteristic
346C2C2×#1568989012yes—
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC2×C47×#533C2
prime radicalC24×C47C22×C4
cosocleC2×C4×C47C25
hypercenterC2×#533D47
hypocenterC47#60249
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. #20252494C2semidirect
2. #20252490C2semidirect
3. #20252483C2semidirect
4. Q8×#10625329C2semidirect
8. #15689254C22semidirect
9. #15689254C22semidirect
10. Q8×C47⋊C8C22semidirect
207. C47#60249semidirect

Maximal subgroups

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

subgroupindex
#202524942
#202524902
#202524832
Q8×#106253292
#202520262
#202520212
C2×C47×#5332
#6024947