Finite Group Databasebeta

← #26376729

Subgroups — #26376729

Order 12032.

2310 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
1C2×#15689680C211yesmaximal, characteristic
2#20246391C218yesmaximal
3#20246343C214yesmaximal, complemented
4C4⋊C4×Dic94C214yesmaximal, complemented
5#20246129C214yesmaximal, complemented
6#20246008C212yesmaximal
7C4×#13094024C214yesmaximal, complemented
8C2×#15688993C211yesmaximal, characteristic
9#20245744C212yesmaximal
10C4×#117×C47C211yesmaximal, characteristic
11#15689680C4116yescomplemented
12C2×#13094024C2212yes—
13C2×#13094016C2212yes—
14C2×#13094003C2214yes—
15C2×C4×Dic94C2218yes—
16C2×C4×Dic94C2214yes—
17C2×#13093983C2212yes—
18C22×C4⋊Dic47C2212yes—
19C22×C4⋊Dic47C2214yes—
20#15689029C2218yes—
21#15689021C2218yes—
22#15689021C2218yes—
23#15689020C22116yes—
24#15689010C22116yes—
25#15688995C2218yes—
26#15688993C2214yes—
27#15688993C2212yes—
28#15688993C2212yes—
29#15688992C2214yes—
30#15688988C2214yes—
31C4⋊C4×Dic47C2218yes—
32C4×C4⋊Dic47C2218yes—
33C4×C4⋊Dic47C2218yes—
34C22⋊C4×Dic47C2218yes—
35C2×#117×C47C2211yescharacteristic
36C2×#117×C47C2211yescharacteristic
37C22×C42×C47C2211yescharacteristic
38C4×C4⋊C4×C47C2212yes—
39C4×C4⋊C4×C47C2212yes—
40C4×C22⋊C4×C47C2214yes—
41C47×#376C2212yes—
42C43×C47C2212yes—
43C22×Dic94C2314yes—
44C23×Dic47C2314yes—
45#13094024C2×C4116yescomplemented
46#13094016C2×C4116yescomplemented
47#13094003C2×C4132yes—
48C4×Dic94—416no—
49C4×Dic94C2×C4132yescomplemented
50#13093983C2×C4116yes—
51C2×C4⋊Dic47—28no—
52C2×C4⋊Dic47—48no—
53C2×C4⋊Dic47C2314yes—
54C2×C4⋊Dic47C2318yes—
55C2×C4⋊Dic47C2314yes—
56C2×C4⋊Dic47C2318yes—
57C2×C4⋊Dic47C2314yes—
58C2×C4⋊Dic47C2314yes—
59C2×Dic47⋊C4C23116yes—
60C2×Dic47⋊C4C2318yes—
61C2×C22⋊Dic47C2314yes—
62C2×C4×Dic47C23116yes—
63C2×C4×Dic47C2314yes—
64C47⋊#95C23116yes—
65C47⋊#95C23116yes—
66C23×C4×C47C2312yes—
67C23×C4×C47C2311yescharacteristic
68#117×C47—24no—
69#117×C47Q818yescomplemented
70C2×C4⋊C4×C47C2311yescharacteristic
71C2×C4⋊C4×C47C2311yescharacteristic
72C2×C4⋊C4×C47C2312yes—
73C2×C22⋊C4×C47C2312yes—
74C2×C22⋊C4×C47C2312yes—
75C2×C42×C47C2×C418yes—
76C2×C42×C47C2314yes—
77C2×C42×C47C2312yes—
78C2×C42×C47C2314yes—
79C2×C42×C47C2312yes—
80C2×C42×C47C2311yescharacteristic
81C2×C42×C47C2311yescharacteristic
82C2×C42×C47C2312yes—
83#95×C47C2314yes—
84#95×C47C2314yes—
85C24×C47C2411yescharacteristic
86C22×C4×C47—24no—
87C22×C4×C47—24no—
88C22×C4×C47C22×C418yes—
89C22×C4×C47C22×C414yes—
90C22×C4×C47C2×Q814yes—
91C22×C4×C47#4114yes—
92C22×C4×C47C2412yes—
93C22×C4×C47C2412yes—
94C22×C4×C47C2412yes—
95C22×C4×C47C2414yes—
96C22×C4×C47C2411yescharacteristic
97C22×C4×C47C2411yescharacteristic
98C22×C4×C47C2412yes—
99C22×Dic47—28no—
100C22×Dic47—48no—
101C22×Dic47C2418yes—
102C22×Dic47C2418yes—
103C2×Dic94—28no—
104C2×Dic94—28no—
105C2×Dic94C22×C4116yes—
106C4⋊C4×C47—22no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47C2×Q818yes—
109C22⋊C4×C47—24no—
110C22⋊C4×C47C2×Q818yes—
111C42×C47—24no—
112C42×C47—28no—
113C42×C47—28no—
114C42×C47C22×C418yes—
115C42×C47C22×C414yes—
116C42×C47C22×C414yes—
117C42×C47C2×Q818yes—
118C22⋊Dic47C22×C4116yes—
119C4⋊Dic47—28no—
120C4⋊Dic47—28no—
121C4⋊Dic47—216no—
122C4⋊Dic47—48no—
123C4⋊Dic47C22×C4116yes—
124C4⋊Dic47C22×C4116yes—
125Dic47⋊C4—416no—
126Dic47⋊C4C22×C4132yes—
127C4×Dic47—416no—
128C4×Dic47C22×C4116yes—
129C23×C47—22no—
130C23×C47—22no—
131C23×C47C23×C412yes—
132C23×C47C22×Q812yes—
133C23×C47C2×#4112yes—
134C23×C47C2511yescharacteristic
135C2×C4×C47—24no—
136C2×C4×C47—22no—
137C2×C4×C47—24no—
138C2×C4×C47—22no—
139C2×C4×C47—216no—
140C2×C4×C47—22no—
141C2×C4×C47—22no—
142C2×C4×C47—24no—
143C2×C4×C47—24no—
144C2×C4×C47C4×Q8116yescomplemented
145C2×C4×C47C23×C414yes—
146C2×C4×C47C23×C412yes—
147C2×C4×C47C23×C412yes—
148C2×C4×C47C23×C414yes—
149C2×C4×C47C22×Q818yes—
150C2×C4×C47C22×Q812yes—
151C2×C4×C47C22×Q812yes—
152C2×C4×C47C2×#4112yes—
153C2×C4×C47C2×#4112yes—
154C2×C4×C47H(5,𝔽2)14yes—
155C2×C4×C472−1+414yes—
156C2×Dic47—28no—
157C2×Dic47—28no—
158C2×Dic47—28no—
159C2×Dic47—216no—
160C2×Dic47—88no—
161C2×Dic47C23×C4116yes—
162Dic94—232no—
163#34214—471nomaximal, Sylow
164C22×C47—24no—
165C22×C47—24no—
166C22×C47—22no—
167C22×C47—22no—
168C22×C47C2×C4×Q814yes—
169C22×C47C24×C411yescharacteristic
170C22×C47C23×Q811yescharacteristic
171C22×C47C22×#4111yescharacteristic
172C22×C47C2×H(5,𝔽2)12yes—
173C22×C47C2×2−1+412yes—
174C4×C47—24no—
175C4×C47—28no—
176C4×C47—42no—
177C4×C47—42no—
178C4×C47C2×C4×Q814yes—
179C4×C47C2×C4×Q814yes—
180C4×C47#52018yes—
181Dic47—216no—
182Dic47—48no—
183C2×#533—471no—
184#2360—478no—
185#2355—474no—
186Q8×C4⋊C4—474no—
187#2330—474no—
188#2317—472no—
189C4×C4⋊Q8—474no—
190C2×#384—471no—
191#2274—472no—
192C4×#117D4711yescharacteristic
193C2×C47—24no—
194C2×C47C22×C4×Q811yescharacteristic
195C2×C47C2×#52012yes—
196C2×C47#350312yes—
197C2×C47#350811yescharacteristic
198C2×C47#350911yescharacteristic
199#533—4716no—
200C2×C4⋊Q8—472no—
201C2×#125—472no—
202C2×C22⋊Q8—474no—
203C2×C4×Q8—474no—
204C2×C4×Q8—478no—
205C2×#117—472no—
206C2×#117C2×D4711yescharacteristic
207C2×#117C2×D4711yescharacteristic
208C22×C4⋊C4—472no—
209C22×C4⋊C4—474no—
210C22×C42C2×D4711yescharacteristic
211#391—478no—
212#389—478no—
213#389—478no—
214#389—4716no—
215#387—4716no—
216#385—478no—
217#384—474no—
218#384—472no—
219#384—472no—
220#384—474no—
221#382—474no—
222C4×C4⋊C4—478no—
223C4×C4⋊C4—478no—
224C4×C4⋊C4—478no—
225C4×C4⋊C4C2×D4712yes—
226C4×C4⋊C4C2×D4712yes—
227C4×C22⋊C4—478no—
228C4×C22⋊C4C2×D4714yes—
229#376C2×D4712yes—
230C43C2×D4712yes—
231C47#3421411yescharacteristic, complemented, Sylow
232C22×Q8—474no—
233C23×C4—474no—
234C23×C4C22×D4712yes—
235C23×C4C22×D4711yescharacteristic
236C4⋊Q8—4716no—
237#125—4716no—
238C22⋊Q8—4732no—
239C4×Q8—4732no—
240C4×Q8—18816no—
241#117—24no—
242#117—4716no—
243#117Dic9418yescomplemented
244C2×C4⋊C4—474no—
245C2×C4⋊C4—478no—
246C2×C4⋊C4—474no—
247C2×C4⋊C4—478no—
248C2×C4⋊C4—474no—
249C2×C4⋊C4—474no—
250C2×C4⋊C4—478no—
251C2×C4⋊C4—4716no—
252C2×C4⋊C4—948no—
253C2×C4⋊C4—1888no—
254C2×C4⋊C4C22×D4711yescharacteristic
255C2×C4⋊C4C22×D4711yescharacteristic
256C2×C4⋊C4C22×D4712yes—
257C2×C22⋊C4—474no—
258C2×C22⋊C4C22×D4712yes—
259C2×C22⋊C4C22×D4712yes—
260C2×C42—474no—
261C2×C42—4716no—
262C2×C42C4×D4718yes—
263C2×C42C22×D4712yes—
264C2×C42C22×D4714yes—
265C2×C42C22×D4714yes—
266C2×C42C22×D4712yes—
267C2×C42C22×D4712yes—
268C2×C42C22×D4711yescharacteristic
269C2×C42C22×D4711yescharacteristic
270#95—4716no—
271#95—4716no—
272#95C22×D4714yes—
273#95C22×D4714yes—
274C24C23×D4711yescharacteristic
275C2×Q8—4716no—
276C2×Q8—948no—
277C2×Q8—948no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—478no—
281C22×C4—478no—
282C22×C4—948no—
283C22×C4—1888no—
284C22×C4C2×Dic9414yes—
285C22×C4C2×C4×D4718yes—
286C22×C4C2×C4×D4714yes—
287C22×C4#1062535014yes—
288C22×C4C23×D4712yes—
289C22×C4C23×D4712yes—
290C22×C4C23×D4712yes—
291C22×C4C23×D4714yes—
292C22×C4C23×D4711yescharacteristic
293C22×C4C23×D4711yescharacteristic
294C22×C4C23×D4712yes—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—4732no—
298C4⋊C4—4716no—
299C4⋊C4—4716no—
300C4⋊C4—948no—
301C4⋊C4—948no—
302C4⋊C4—9416no—
303C4⋊C4—18816no—
304C4⋊C4—1888no—
305C4⋊C4C2×Dic9418yes—
306C22⋊C4—24no—
307C22⋊C4—4716no—
308C22⋊C4C2×Dic9418yes—
309C42—24no—
310C42—28no—
311C42—28no—
312C42—4716no—
313C42—18816no—
314C42C2×Dic9418yes—
315C42C2×C4×D4718yes—
316C42C2×C4×D4714yes—
317C42C2×C4×D4714yes—
318C23—22no—
319C23—22no—
320C23C22×C4×D4712yes—
321C23C22×Dic9412yes—
322C23C2×#1062535012yes—
323C23C24×D4711yescharacteristic
324Q8—9432no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—216no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—4716no—
335C2×C4—948no—
336C2×C4—948no—
337C2×C4—948no—
338C2×C4—9416no—
339C2×C4—3768no—
340C2×C4C4×Dic94116yescomplemented
341C2×C4C22×C4×D4714yes—
342C2×C4C22×C4×D4712yes—
343C2×C4C22×C4×D4712yes—
344C2×C4C22×C4×D4714yes—
345C2×C4C22×Dic9418yes—
346C2×C4C22×Dic9412yes—
347C2×C4C22×Dic9412yes—
348C2×C4C2×#1062535012yes—
349C2×C4C2×#1062535012yes—
350C2×C4#1309407014yes—
351C2×C4#1309407114yes—
352C22—24no—
353C22—24no—
354C22—22no—
355C22—22no—
356C22C2×C4×Dic9414yes—
357C22C23×C4×D4711yescharacteristic
358C22C23×Dic9411yescharacteristic
359C22C22×#1062535011yescharacteristic
360C22C2×#1309407012yes—
361C22C2×#1309407112yes—
362C4—24no—
363C4—28no—
364C4—42no—
365C4—42no—
366C4—9416no—
367C4—1888no—
368C4C2×C4×Dic9414yes—
369C4C2×C4×Dic9414yes—
370C4#1568960318yes—
371C2—24no—
372C2C22×C4×Dic9411yescharacteristic
373C2C2×#1568960312yes—
374C2#2025720712yes—
375C2#2025726411yescharacteristic
376C2#2025727411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC4×#117×C47C2
socle, cosocleC23×C47C25
prime radicalC24×C47C24
hypercenterC4×#117D47
hypocenterC47#34214
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
3. #20246343C2semidirect
4. C4⋊C4×Dic94C2semidirect
5. #20246129C2semidirect
7. C4×#13094024C2semidirect
11. #15689680C4semidirect
45. #13094024C2×C4semidirect
46. #13094016C2×C4semidirect
49. C4×Dic94C2×C4semidirect
69. #117×C47Q8semidirect
144. C2×C4×C47C4×Q8semidirect
231. C47#34214semidirect
243. #117Dic94semidirect
340. C2×C4C4×Dic94semidirect

Maximal subgroups

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

subgroupindex
C2×#156896802
#202463912
#202463432
C4⋊C4×Dic942
#202461292
#202460082
C4×#130940242
C2×#156889932
#202457442
C4×#117×C472
#3421447