Finite Group Databasebeta

← #26374687

Subgroups — #26374687

Order 12032.

2782 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245886C214yesmaximal, complemented
2#20245879C212yesmaximal
3#20245872C212yesmaximal, complemented
4#20245866C214yesmaximal, complemented
5#20245853C214yesmaximal, complemented
6#20245833C218yesmaximal, complemented
7C4×#13093983C214yesmaximal, complemented
8C4×#13093983C211yesmaximal, characteristic, complemented
9C2×C42×Dic47C211yesmaximal, characteristic, complemented
10C2×C4×#41×C47C211yesmaximal, characteristic
11C2×#13093983C2218yes—
12C2×#13093983C2212yes—
13C22×C4×Dic47C2214yes—
14C22×C4×Dic47C2212yes—
15#15689030C2214yes—
16#15689026C2214yes—
17#15689022C2218yes—
18#15689016C22116yes—
19#15689009C22116yes—
20#15689003C2218yes—
21#15688998C2218yes—
22C4×C4⋊Dic47C2218yes—
23C4×C4⋊Dic47C2214yes—
24C4×C4⋊Dic47C2218yes—
25C4×C22⋊Dic47C2214yes—
26C4×C22⋊Dic47C2218yes—
27C4×C22⋊Dic47C2218yes—
28#15688969C2218yes—
29#15688969C2212yes—
30C42×Dic47C2214yes—
31C42×Dic47C2212yes—
32C42×Dic47C2212yes—
33C42×Dic47C2212yes—
34C22×#41×C47C2211yescharacteristic
35C4×#41×C47C4116yescomplemented
36C2×C4×Q8×C47C2212yes—
37C2×C4×D4×C47C2214yes—
38C2×C4×D4×C47C2212yes—
39C2×#117×C47C2211yescharacteristic
40C2×#117×C47C2212yes—
41C22×C42×C47C2211yescharacteristic
42C22×C42×C47C2212yes—
43C23×Dic47C2314yes—
44#13093983—28no—
45#13093983—232no—
46C2×C4⋊Dic47C23116yes—
47C2×C4⋊Dic47C2314yes—
48C2×C22⋊Dic47C2314yes—
49C2×C22⋊Dic47C23116yes—
50C2×C4×Dic47—28no—
51C2×C4×Dic47—216no—
52C2×C4×Dic47C2314yes—
53C2×C4×Dic47C2314yes—
54C2×C4×Dic47C2318yes—
55C2×C4×Dic47C2314yes—
56C2×C4×Dic47C2318yes—
57C2×C4×Dic47C2314yes—
58C2×C4×Dic47C2314yes—
59C47⋊#95C23116yes—
60C47⋊#95C23116yes—
61C47⋊#95C2318yes—
62C2×#41×C47—28no—
63C2×#41×C47C2×C418yes—
64C22×Q8×C47C2311yescharacteristic
65C22×D4×C47C2312yes—
66C22×D4×C47C2311yescharacteristic
67C23×C4×C47C2312yes—
68C23×C4×C47C2312yes—
69C23×C4×C47C2314yes—
70C23×C4×C47C2311yescharacteristic
71C4×Q8×C47C2×C4116yescomplemented
72C4×D4×C47C2×C4132yes—
73C4×D4×C47C2×C4116yescomplemented
74#117×C47C2×C418yes—
75#117×C47C2×C4116yescomplemented
76C2×C4⋊C4×C47C2314yes—
77C2×C4⋊C4×C47C2312yes—
78C2×C22⋊C4×C47C2312yes—
79C2×C22⋊C4×C47C2314yes—
80C2×C42×C47C2×C418yes—
81C2×C42×C47C2×C4116yescomplemented
82C2×C42×C47C2311yescharacteristic
83C2×C42×C47C2312yes—
84C2×C42×C47C2314yes—
85C2×C42×C47C2312yes—
86C2×C42×C47C2311yescharacteristic
87C24×C47C2412yes—
88C24×C47C2411yescharacteristic
89#41×C47—232no—
90C2×Q8×C47—24no—
91C2×Q8×C47C22×C414yes—
92C2×D4×C47—28no—
93C2×D4×C47—24no—
94C2×D4×C47C22×C418yes—
95C2×D4×C47C22×C414yes—
96C22×C4×C47—28no—
97C22×C4×C47—28no—
98C22×C4×C47—216no—
99C22×C4×C47C22×C418yes—
100C22×C4×C47C22×C418yes—
101C22×C4×C47C22×C4116yes—
102C22×C4×C47C22×C414yes—
103C22×C4×C47#4118yes—
104C22×C4×C47C2412yes—
105C22×C4×C47C2414yes—
106C22×C4×C47C2412yes—
107C22×C4×C47C2411yescharacteristic
108C22×C4×C47C2412yes—
109C22×C4×C47C2411yescharacteristic
110C22×Dic47—28no—
111C22×Dic47—216no—
112C22×Dic47C2418yes—
113C22×Dic47C2418yes—
114C4⋊C4×C47C22×C418yes—
115C4⋊C4×C47C22×C4116yes—
116C22⋊C4×C47C22×C418yes—
117C22⋊C4×C47C22×C4116yes—
118C42×C47C22×C414yes—
119C42×C47C22×C414yes—
120C42×C47C22×C418yes—
121C42×C47C22×C418yes—
122C42×C47C22×C4116yes—
123C22⋊Dic47—28no—
124C22⋊Dic47—232no—
125C4⋊Dic47—28no—
126C4⋊Dic47—232no—
127C4×Dic47—28no—
128C4×Dic47—28no—
129C4×Dic47—28no—
130C4×Dic47—216no—
131C4×Dic47—28no—
132C4×Dic47—216no—
133C4×Dic47#41116yescomplemented
134C23×C47—24no—
135C23×C47—28no—
136C23×C47—24no—
137C23×C47C23×C414yes—
138C23×C47C23×C412yes—
139C23×C47C2×#4114yes—
140C23×C47C2511yescharacteristic
141Q8×C47—28no—
142D4×C47—216no—
143D4×C47—28no—
144C2×C4×C47—24no—
145C2×C4×C47—28no—
146C2×C4×C47—24no—
147C2×C4×C47—216no—
148C2×C4×C47—216no—
149C2×C4×C47—232no—
150C2×C4×C47#117116yescomplemented
151C2×C4×C47C23×C414yes—
152C2×C4×C47C23×C418yes—
153C2×C4×C47C23×C414yes—
154C2×C4×C47C23×C412yes—
155C2×C4×C47C23×C412yes—
156C2×C4×C47C23×C414yes—
157C2×C4×C47C2×#4118yes—
158C2×C4×C47C2×#4114yes—
159C2×C4×C47C2×#4114yes—
160C2×Dic47—232no—
161C2×Dic47—28no—
162C2×Dic47—28no—
163C2×Dic47—216no—
164C2×Dic47—28no—
165C2×Dic47C2×#41116yes—
166#34026—471nomaximal, Sylow
167C22×C47—24no—
168C22×C47—28no—
169C22×C47—24no—
170C22×C47—28no—
171C22×C47—48no—
172C22×C47C2×#11714yes—
173C22×C47C24×C411yescharacteristic
174C22×C47C22×#4114yes—
175C22×C47C22×#4112yes—
176C4×C47—24no—
177C4×C47—24no—
178C4×C47—28no—
179C4×C47C2×#11714yes—
180C4×C47C2×#11714yes—
181C4×C47C4×#4118yes—
182Dic47—216no—
183Dic47—48no—
184C2×C4×#41D4711yescharacteristic
185#2298—474no—
186#2297—472no—
187#2296—472no—
188#2295—474no—
189#2293—474no—
190#2290—478no—
191C4×#117—474no—
192C4×#117—471no—
193C2×C43—471no—
194C2×C47—24no—
195C2×C47—44no—
196C2×C47C22×#11711yescharacteristic
197C2×C47C2×C4×#4112yes—
198C2×C47#344414yes—
199C22×#41C2×D4711yescharacteristic
200C4×#41Dic47116yescomplemented
201C2×C4×Q8C2×D4712yes—
202C2×C4×D4C2×D4714yes—
203C2×C4×D4C2×D4712yes—
204C2×#117—478no—
205C2×#117—472no—
206C2×#117C2×D4711yescharacteristic
207C2×#117C2×D4712yes—
208C22×C42—474no—
209C22×C42—472no—
210C22×C42C2×D4711yescharacteristic
211C22×C42C2×D4712yes—
212#391—474no—
213#390—474no—
214#389—478no—
215#388—4716no—
216#387—4716no—
217#386—478no—
218#385—478no—
219C4×C4⋊C4—478no—
220C4×C4⋊C4—474no—
221C4×C4⋊C4—478no—
222C4×C22⋊C4—474no—
223C4×C22⋊C4—478no—
224C4×C22⋊C4—478no—
225#376—478no—
226#376—472no—
227C43—474no—
228C43—472no—
229C43—472no—
230C43—472no—
231C47#3402611yescharacteristic, complemented, Sylow
232C2×#41—28no—
233C2×#41C2×Dic4718yes—
234C22×Q8C22×D4711yescharacteristic
235C22×D4C22×D4712yes—
236C22×D4C22×D4711yescharacteristic
237C23×C4—474no—
238C23×C4C22×D4714yes—
239C23×C4C22×D4712yes—
240C23×C4C22×D4712yes—
241C23×C4C22×D4711yescharacteristic
242C4×Q8C2×Dic47116yescomplemented
243C4×D4C2×Dic47132yes—
244C4×D4C2×Dic47116yescomplemented
245#117—9432no—
246#117—948no—
247#117C2×Dic4718yes—
248#117C2×Dic47116yescomplemented
249C2×C4⋊C4—4716no—
250C2×C4⋊C4—474no—
251C2×C4⋊C4C22×D4714yes—
252C2×C4⋊C4C22×D4712yes—
253C2×C22⋊C4—474no—
254C2×C22⋊C4—4716no—
255C2×C22⋊C4C22×D4712yes—
256C2×C22⋊C4C22×D4714yes—
257C2×C42—474no—
258C2×C42—478no—
259C2×C42—474no—
260C2×C42—474no—
261C2×C42—474no—
262C2×C42—478no—
263C2×C42—474no—
264C2×C42—9416no—
265C2×C42—948no—
266C2×C42C2×Dic4718yes—
267C2×C42C2×Dic47116yescomplemented
268C2×C42C22×D4711yescharacteristic
269C2×C42C22×D4712yes—
270C2×C42C22×D4712yes—
271C2×C42C22×D4714yes—
272C2×C42C22×D4711yescharacteristic
273#95—4716no—
274#95—4716no—
275#95—478no—
276C24C23×D4712yes—
277C24C23×D4711yescharacteristic
278#41—232no—
279C2×Q8—24no—
280C2×Q8C22×Dic4714yes—
281C2×D4—28no—
282C2×D4—24no—
283C2×D4C22×Dic4718yes—
284C2×D4C22×Dic4714yes—
285C22×C4—28no—
286C22×C4—216no—
287C22×C4—28no—
288C22×C4—478no—
289C22×C4—478no—
290C22×C4—9416no—
291C22×C4—948no—
292C22×C4#1062535018yes—
293C22×C4C22×Dic4718yes—
294C22×C4C22×Dic4718yes—
295C22×C4C22×Dic47116yes—
296C22×C4C22×Dic4714yes—
297C22×C4C23×D4712yes—
298C22×C4C23×D4712yes—
299C22×C4C23×D4714yes—
300C22×C4C23×D4712yes—
301C22×C4C23×D4711yescharacteristic
302C22×C4C23×D4711yescharacteristic
303C4⋊C4—9432no—
304C4⋊C4—948no—
305C4⋊C4C22×Dic4718yes—
306C4⋊C4C22×Dic47116yes—
307C22⋊C4—9432no—
308C22⋊C4—948no—
309C22⋊C4C22×Dic4718yes—
310C22⋊C4C22×Dic47116yes—
311C42—4716no—
312C42—9416no—
313C42—948no—
314C42—9416no—
315C42—948no—
316C42—948no—
317C42—948no—
318C42C22×Dic4714yes—
319C42C22×Dic4714yes—
320C42C22×Dic4718yes—
321C42C22×Dic4718yes—
322C42C22×Dic47116yes—
323C23—24no—
324C23—28no—
325C23—24no—
326C23C23×Dic4714yes—
327C23C23×Dic4712yes—
328C23C2×#1062535014yes—
329C23C24×D4711yescharacteristic
330Q8—28no—
331D4—216no—
332D4—28no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—24no—
336C2×C4—216no—
337C2×C4—216no—
338C2×C4—232no—
339C2×C4—4716no—
340C2×C4—9432no—
341C2×C4—948no—
342C2×C4—948no—
343C2×C4—9416no—
344C2×C4—948no—
345C2×C4#13093983116yescomplemented
346C2×C4C23×Dic4714yes—
347C2×C4C23×Dic4718yes—
348C2×C4C23×Dic4714yes—
349C2×C4C23×Dic4712yes—
350C2×C4C23×Dic4712yes—
351C2×C4C23×Dic4714yes—
352C2×C4#41×D4718yes—
353C2×C4C2×#1062535014yes—
354C2×C4C2×#1062535014yes—
355C22—24no—
356C22—28no—
357C22—24no—
358C22—28no—
359C22—48no—
360C22C2×#1309398314yes—
361C22C24×Dic4711yescharacteristic
362C22C2×#41×D4714yes—
363C22C22×#1062535012yes—
364C4—24no—
365C4—24no—
366C4—28no—
367C4—9416no—
368C4—1888no—
369C4C2×#1309398314yes—
370C4C2×#1309398314yes—
371C4#41×Dic4718yes—
372C2—24no—
373C2—44no—
374C2C22×#1309398311yescharacteristic
375C2C2×#41×Dic4712yes—
376C2#2025668114yes—
subgroupG/H
centerC2×C42C22×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC2×C4×#41×C47C2
socle, cosocleC23×C47C25
prime radicalC22×#41×C47C22
hypercenterC2×C4×#41D47
hypocenterC47#34026
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C42 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #20245886C2semidirect
3. #20245872C2semidirect
4. #20245866C2semidirect
5. #20245853C2semidirect
6. #20245833C2semidirect
7. C4×#13093983C2semidirect
8. C4×#13093983C2semidirect
9. C2×C42×Dic47C2semidirect
35. C4×#41×C47C4semidirect
71. C4×Q8×C47C2×C4semidirect
73. C4×D4×C47C2×C4semidirect
75. #117×C47C2×C4semidirect
81. C2×C42×C47C2×C4semidirect
133. C4×Dic47#41semidirect
150. C2×C4×C47#117semidirect
200. C4×#41Dic47semidirect
231. C47#34026semidirect
242. C4×Q8C2×Dic47semidirect
244. C4×D4C2×Dic47semidirect
248. #117C2×Dic47semidirect
267. C2×C42C2×Dic47semidirect
345. C2×C4#13093983semidirect

Maximal subgroups

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

subgroupindex
#202458862
#202458792
#202458722
#202458662
#202458532
#202458332
C4×#130939832
C2×C42×Dic472
C2×C4×#41×C472
#3402647