Finite Group Databasebeta

← #26376385

Subgroups — #26376385

Order 12032.

3006 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689606C211yesmaximal, characteristic
2C2×#41×Dic47C211yesmaximal, characteristic
3#20246306C214yesmaximal, complemented
4#20246293C2112yesmaximal, complemented
5#20246012C213yesmaximal, complemented
6#20246008C213yesmaximal, complemented
7#20245791C213yesmaximal, complemented
8C2×#15688988C213yesmaximal, complemented
9C2×C47×#520C211yesmaximal, characteristic
10#15689606—28no—
11#41×Dic47C4116yescomplemented
12C2×Q8×Dic47C2212yes—
13C2×Q8×Dic47C2212yes—
14C2×D4×Dic47C2216yes—
15C2×D4×Dic47C2216yes—
16C2×#13093983C2213yes—
17C2×#13093983C2213yes—
18C22×C4⋊Dic47C2213yes—
19C22×C4×Dic47C2213yes—
20#15689021C22112yes—
21#15689019C22112yes—
22#15688996C22112yes—
23#15688995C22112yes—
24#15688992C2213yes—
25#15688992C2213yes—
26#15688988C2213yes—
27#15688988C22112yes—
28#15688988C2216yes—
29#15688988C2211yescharacteristic
30#15688981C22112yes—
31C4×Dic47⋊C4C22112yes—
32C2×C47⋊#95C22112yes—
33C22×#41×C47C2211yescharacteristic
34C47×#520—28no—
35C2×C4×Q8×C47C2212yes—
36C2×C4×D4×C47C2216yes—
37C2×#117×C47C2213yes—
38C22×C4⋊C4×C47C2213yes—
39C23×Dic47C2316yes—
40C23×Dic47C2316yes—
41Q8×Dic47—28no—
42Q8×Dic47C2×C4116yescomplemented
43D4×Dic47—224no—
44D4×Dic47C2×C4148yescomplemented
45#13093983—212no—
46#13093983C2×C4124yescomplemented
47C2×C4⋊Dic47—212no—
48C2×C4⋊Dic47C2313yes—
49C2×C4⋊Dic47C2316yes—
50C2×C4⋊Dic47C2311yescharacteristic
51C2×C4⋊Dic47C2316yes—
52C2×Dic47⋊C4C23112yes—
53C2×Dic47⋊C4C23112yes—
54C2×C22⋊Dic47C2316yes—
55C2×C22⋊Dic47C2316yes—
56C2×C4×Dic47C2×C4124yescomplemented
57C2×C4×Dic47C2316yes—
58C2×C4×Dic47C2316yes—
59C2×C4×Dic47C2311yescharacteristic
60C2×C4×Dic47C2313yes—
61C47⋊#95C23112yes—
62C47⋊#95C23124yes—
63C47⋊#95C2314yes—
64C2×#41×C47C2×C418yes—
65C2×#41×C47D418yes—
66C2×#41×C47Q818yes—
67C22×Q8×C47C2311yescharacteristic
68C22×D4×C47C2313yes—
69C23×C4×C47C2316yes—
70C23×C4×C47C2313yes—
71C4×Q8×C47—28no—
72C4×D4×C47—224no—
73#117×C47—212no—
74C2×C4⋊C4×C47—212no—
75C2×C4⋊C4×C47C2313yes—
76C2×C4⋊C4×C47C2316yes—
77C2×C4⋊C4×C47C2311yescharacteristic
78C2×C22⋊C4×C47C2316yes—
79C2×C42×C47C2316yes—
80C24×C47C2413yes—
81#41×C47C4⋊C4164yescomplemented
82C2×Q8×C47C22×C414yes—
83C2×Q8×C47C2×D414yes—
84C2×Q8×C47C2×Q814yes—
85C2×D4×C47C22×C4112yes—
86C2×D4×C47C2×D4112yes—
87C2×D4×C47C2×Q8112yes—
88C22×C4×C47—212no—
89C22×C4×C47—412no—
90C22×C4×C47C22×C4112yes—
91C22×C4×C47C2×D4112yes—
92C22×C4×C47C2×Q8112yes—
93C22×C4×C47C2412yes—
94C22×C4×C47C2416yes—
95C22×C4×C47C2413yes—
96C22×C4×C47C2411yescharacteristic
97C22×Dic47—212no—
98C22×Dic47—46no—
99C22×Dic47—46no—
100C22×Dic47—412no—
101C22×Dic47C22×C4124yes—
102C22×Dic47C2416yes—
103C22×Dic47C2412yes—
104C22×Dic47C2412yes—
105C22×Dic47C2416yes—
106C4⋊C4×C47—26no—
107C4⋊C4×C47—212no—
108C4⋊C4×C47—22no—
109C22⋊C4×C47—212no—
110C42×C47—212no—
111C22⋊Dic47—212no—
112C22⋊Dic47C22×C4124yes—
113C4⋊Dic47—26no—
114C4⋊Dic47—212no—
115C4⋊Dic47—22no—
116C4⋊Dic47C22×C4124yes—
117Dic47⋊C4—412no—
118Dic47⋊C4—412no—
119C4×Dic47—212no—
120C4×Dic47C22×C414yes—
121C4×Dic47C22×C4112yes—
122C4×Dic47C22×C4124yes—
123C23×C47—26no—
124C23×C47—26no—
125C23×C47C23×C416yes—
126C23×C47C22×D416yes—
127C23×C47C22×Q816yes—
128C23×C47C2511yescharacteristic
129Q8×C47C2×C4⋊C4116yes—
130D4×C47C2×C4⋊C4148yes—
131C2×C4×C47—22no—
132C2×C4×C47—26no—
133C2×C4×C47—24no—
134C2×C4×C47—212no—
135C2×C4×C47—812no—
136C2×C4×C47C2×C4⋊C4148yes—
137C2×C4×C47C23×C412yes—
138C2×C4×C47C23×C416yes—
139C2×C4×C47C22×D416yes—
140C2×C4×C47C22×D412yes—
141C2×C4×C47C22×Q812yes—
142C2×C4×C47C22×Q816yes—
143C2×Dic47—26no—
144C2×Dic47—26no—
145C2×Dic47—22no—
146C2×Dic47—22no—
147C2×Dic47—22no—
148C2×Dic47—26no—
149C2×Dic47—24no—
150C2×Dic47—212no—
151C2×Dic47—424no—
152C2×Dic47—812no—
153C2×Dic47C23×C414yes—
154C2×Dic47C23×C4112yes—
155#34184—471nomaximal, Sylow
156C22×C47—212no—
157C22×C47—26no—
158C22×C47—26no—
159C22×C47—212no—
160C22×C47C22×C4⋊C4112yes—
161C22×C47C24×C411yescharacteristic
162C22×C47C23×D411yescharacteristic
163C22×C47C23×Q811yescharacteristic
164C22×C47#58512yes—
165C22×C47#58512yes—
166C4×C47—42no—
167C4×C47—46no—
168C4×C47C22×C4⋊C414yes—
169C4×C47C22×C4⋊C4112yes—
170Dic47—24no—
171Dic47—212no—
172Dic47—42no—
173Dic47—46no—
174C2×#520—471no—
175C2×#520D4711yescharacteristic
176C2×C4×#41—471no—
177#2351—474no—
178#2349—4712no—
179#2317—473no—
180#2317—473no—
181#2285—473no—
182C2×#382—473no—
183C2×C47—212no—
184C2×C47C23×C4⋊C411yescharacteristic
185C2×C47#342812yes—
186C2×C47#350011yescharacteristic
187C2×C47#350011yescharacteristic
188C2×C47#350412yes—
189C22×#41C2×D4711yescharacteristic
190#520—28no—
191#520—948no—
192C4×#41—4716no—
193C2×C4×Q8—472no—
194C2×C4×Q8—472no—
195C2×C4×Q8C2×D4712yes—
196C2×C4×D4—476no—
197C2×C4×D4—476no—
198C2×C4×D4C2×D4716yes—
199C2×#117—473no—
200C2×#117—473no—
201C2×#117C2×D4713yes—
202C22×C4⋊C4—473no—
203C22×C4⋊C4C2×D4713yes—
204C22×C42—473no—
205#389—4712no—
206#389—4712no—
207#385—4712no—
208#385—4712no—
209#384—473no—
210#384—473no—
211#382—476no—
212#382—4712no—
213#382—473no—
214#382—471no—
215#380—4712no—
216C4×C4⋊C4—4712no—
217C2×#95—4712no—
218C47#3418411yescharacteristic, complemented, Sylow
219C2×#41Dic9418yes—
220C2×#41C4×D4718yes—
221C2×#41C22⋊D4718yes—
222C22×Q8C22×D4711yescharacteristic
223C22×D4C22×D4713yes—
224C23×C4—476no—
225C23×C4—476no—
226C23×C4C22×D4716yes—
227C23×C4C22×D4713yes—
228C4×Q8—28no—
229C4×Q8—4716no—
230C4×Q8—948no—
231C4×D4—224no—
232C4×D4—4748no—
233C4×D4—9424no—
234#117—212no—
235#117—4724no—
236#117—9412no—
237C2×C4⋊C4—212no—
238C2×C4⋊C4—476no—
239C2×C4⋊C4—473no—
240C2×C4⋊C4—476no—
241C2×C4⋊C4—471no—
242C2×C4⋊C4—4712no—
243C2×C4⋊C4—4712no—
244C2×C4⋊C4—9412no—
245C2×C4⋊C4C22×D4713yes—
246C2×C4⋊C4C22×D4716yes—
247C2×C4⋊C4C22×D4711yescharacteristic
248C2×C22⋊C4—476no—
249C2×C22⋊C4—476no—
250C2×C22⋊C4C22×D4716yes—
251C2×C42—4724no—
252C2×C42—473no—
253C2×C42—476no—
254C2×C42—471no—
255C2×C42—476no—
256C2×C42C22×D4716yes—
257#95—4712no—
258#95—4724no—
259#95—474no—
260C24C23×D4713yes—
261#41Dic47⋊C4164yescomplemented
262C2×Q8C2×Dic9414yes—
263C2×Q8C2×C4×D4714yes—
264C2×Q8C2×C22⋊D4714yes—
265C2×D4C2×Dic94112yes—
266C2×D4C2×C4×D47112yes—
267C2×D4C2×C22⋊D47112yes—
268C22×C4—212no—
269C22×C4—412no—
270C22×C4—472no—
271C22×C4—476no—
272C22×C4—4724no—
273C22×C4—476no—
274C22×C4—472no—
275C22×C4—9412no—
276C22×C4—18812no—
277C22×C4—1886no—
278C22×C4—1886no—
279C22×C4C2×Dic94112yes—
280C22×C4C2×C4×D47112yes—
281C22×C4C2×C22⋊D47112yes—
282C22×C4C23×D4716yes—
283C22×C4C23×D4712yes—
284C22×C4C23×D4713yes—
285C22×C4C23×D4711yescharacteristic
286C4⋊C4—26no—
287C4⋊C4—212no—
288C4⋊C4—22no—
289C4⋊C4—4724no—
290C4⋊C4—946no—
291C4⋊C4—9412no—
292C4⋊C4—942no—
293C4⋊C4—18812no—
294C4⋊C4—18812no—
295C22⋊C4—212no—
296C22⋊C4—4724no—
297C22⋊C4—9412no—
298C42—212no—
299C42—474no—
300C42—4712no—
301C42—4724no—
302C42—9412no—
303C23—26no—
304C23—26no—
305C23C22×C4×D4716yes—
306C23C22×C22⋊D4716yes—
307C23C22×Dic9416yes—
308C23C24×D4711yescharacteristic
309Q8C2×Dic47⋊C4116yes—
310D4C2×Dic47⋊C4148yes—
311C2×C4—22no—
312C2×C4—26no—
313C2×C4—212no—
314C2×C4—24no—
315C2×C4—812no—
316C2×C4—474no—
317C2×C4—4712no—
318C2×C4—946no—
319C2×C4—942no—
320C2×C4—946no—
321C2×C4—942no—
322C2×C4—942no—
323C2×C4—946no—
324C2×C4—944no—
325C2×C4—9412no—
326C2×C4—37612no—
327C2×C4—18824no—
328C2×C4C2×Dic47⋊C4148yes—
329C2×C4C22×C4×D4712yes—
330C2×C4C22×C4×D4716yes—
331C2×C4C22×C22⋊D4716yes—
332C2×C4C22×C22⋊D4712yes—
333C2×C4C22×Dic9412yes—
334C2×C4C22×Dic9416yes—
335C22—212no—
336C22—26no—
337C22—26no—
338C22—212no—
339C22C22×Dic47⋊C4112yes—
340C22C23×C4×D4711yescharacteristic
341C22C23×C22⋊D4711yescharacteristic
342C22C23×Dic9411yescharacteristic
343C22#1568995812yes—
344C22#1568996012yes—
345C4—46no—
346C4—42no—
347C4—944no—
348C4—9412no—
349C4—1882no—
350C4—1886no—
351C4C22×Dic47⋊C414yes—
352C4C22×Dic47⋊C4112yes—
353C2—212no—
354C2C23×Dic47⋊C411yescharacteristic
355C2#2025658812yes—
356C2#2025717611yescharacteristic
357C2#2025718211yescharacteristic
358C2#2025721012yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC2×C47×#520C2
socle, cosocleC23×C47C25
prime radicalC22×#41×C47C22
hypercenterC2×#520D47
hypocenterC47#34184
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
3. #20246306C2semidirect
4. #20246293C2semidirect
5. #20246012C2semidirect
6. #20246008C2semidirect
7. #20245791C2semidirect
8. C2×#15688988C2semidirect
11. #41×Dic47C4semidirect
42. Q8×Dic47C2×C4semidirect
44. D4×Dic47C2×C4semidirect
46. #13093983C2×C4semidirect
56. C2×C4×Dic47C2×C4semidirect
81. #41×C47C4⋊C4semidirect
218. C47#34184semidirect
261. #41Dic47⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156896062
C2×#41×Dic472
#202463062
#202462932
#202460122
#202460082
#202457912
C2×#156889882
C2×C47×#5202
#3418447