Finite Group Databasebeta

← #26374684

Subgroups — #26374684

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
1C2×C4×#10625350C211yesmaximal, characteristic
2#20245883C214yesmaximal, complemented
3#20245876C212yesmaximal
4#20245870C212yesmaximal, complemented
5#20245862C214yesmaximal, complemented
6#20245849C214yesmaximal, complemented
7#20245827C218yesmaximal, complemented
8C4×#13093981C214yesmaximal, complemented
9C2×C42×Dic47C211yesmaximal, characteristic, complemented
10C4×#117×C47C211yesmaximal, characteristic, complemented
11C22×#10625350C2211yescharacteristic
12C4×#10625350C4116yescomplemented
13C2×C4×Dic94C2212yes—
14C2×C4×D188C2212yes—
15C2×C4×C22⋊D47C2214yes—
16C2×#13093983C2211yescharacteristic
17C2×#13093981C2218yes—
18C2×#13093981C2212yes—
19C22×C4×Dic47C2214yes—
20C22×C4×Dic47C2212yes—
21C2×C42×D47C2212yes—
22#15689028C2214yes—
23#15689024C2214yes—
24#15689019C2218yes—
25#15689012C22116yes—
26#15689005C22116yes—
27#15689000C2218yes—
28#15688995C2218yes—
29C4×Dic47⋊C4C2218yes—
30C4×Dic47⋊C4C2218yes—
31C4×(C2×D47)⋊C4C2218yes—
32C4×(C2×D47)⋊C4C2218yes—
33#15688969C2218yes—
34C42×Dic47C2214yes—
35C42×Dic47C2212yes—
36C42×Dic47C2212yes—
37C2×#117×C47C2212yes—
38C22×C42×C47C2211yescharacteristic
39C4×C4⋊C4×C47C2214yes—
40C4×C22⋊C4×C47C2214yes—
41C47×#376C2212yes—
42C43×C47C2212yes—
43C2×#10625350—28no—
44C2×#10625350C2×C418yes—
45C22×Dic94C2311yescharacteristic
46C22×D188C2311yescharacteristic
47C22×C22⋊D47C2312yes—
48C23×Dic47C2314yes—
49C22×C4×D47C2312yes—
50C22×C4×D47C2314yes—
51C4×Dic94C2×C4116yescomplemented
52C4×D188C2×C4116yescomplemented
53C4×C22⋊D47C2×C4132yes—
54#13093983C2×C418yes—
55#13093981—232no—
56#13093981C2×C4116yescomplemented
57C2×C4⋊Dic47C2312yes—
58C2×Dic47⋊C4C23116yes—
59C2×Dic47⋊C4C2314yes—
60C2×C22⋊Dic47C2312yes—
61C2×(C2×D47)⋊C4C23116yes—
62C2×(C2×D47)⋊C4C2314yes—
63C2×C4×Dic47—28no—
64C2×C4×Dic47—216no—
65C2×C4×Dic47C2314yes—
66C2×C4×Dic47C2314yes—
67C2×C4×Dic47C2318yes—
68C2×C4×Dic47C2314yes—
69C2×C4×Dic47C2314yes—
70C2×C4×Dic47C2312yes—
71C2×C4×Dic47C2314yes—
72C42×D47C2×C4116yescomplemented
73C47⋊#95C23116yes—
74C47⋊#95C23116yes—
75C23×C4×C47C2312yes—
76C23×C4×C47C2311yescharacteristic
77#117×C47—28no—
78C2×C4⋊C4×C47C2314yes—
79C2×C22⋊C4×C47C2314yes—
80C2×C42×C47C2×C418yes—
81C2×C42×C47C2318yes—
82C2×C42×C47C2314yes—
83C2×C42×C47C2311yescharacteristic
84C2×C42×C47C2312yes—
85C2×C42×C47C2311yescharacteristic
86#95×C47C2318yes—
87C24×C47C2411yescharacteristic
88C23×D47C2412yes—
89C22×C4×C47—28no—
90C22×C4×C47C22×C418yes—
91C22×C4×C47C22×C414yes—
92C22×C4×C47#4118yes—
93C22×C4×C47C2418yes—
94C22×C4×C47C2412yes—
95C22×C4×C47C2411yescharacteristic
96C22×C4×C47C2412yes—
97C22×C4×C47C2411yescharacteristic
98C2×C22⋊D47—28no—
99C2×C22⋊D47C22×C418yes—
100C22×Dic47—28no—
101C22×Dic47—216no—
102C22×Dic47C2412yes—
103C22×Dic47C2414yes—
104C22×Dic47C2418yes—
105#10625350—232no—
106C2×D188—24no—
107C2×D188C22×C414yes—
108C2×C4×D47—216no—
109C2×C4×D47—28no—
110C2×C4×D47C22×C418yes—
111C2×C4×D47C22×C4116yes—
112C2×Dic94—24no—
113C2×Dic94C22×C414yes—
114C4⋊C4×C47—28no—
115C22⋊C4×C47—28no—
116C42×C47—28no—
117C42×C47—216no—
118C42×C47C22×C414yes—
119C42×C47C22×C414yes—
120C42×C47C22×C418yes—
121C22⋊Dic47C22×C418yes—
122(C2×D47)⋊C4—232no—
123(C2×D47)⋊C4C22×C4116yes—
124C4⋊Dic47C22×C418yes—
125Dic47⋊C4—232no—
126Dic47⋊C4C22×C4116yes—
127C4×Dic47—28no—
128C4×Dic47—28no—
129C4×Dic47—28no—
130C4×Dic47—216no—
131C4×Dic47C22×C418yes—
132C4×Dic47C22×C4116yes—
133C4×Dic47#41116yescomplemented
134C23×C47—24no—
135C23×C47C23×C412yes—
136C23×C47C2×#4114yes—
137C23×C47C2511yescharacteristic
138C22×D47—28no—
139C22×D47—24no—
140C22×D47C23×C414yes—
141C2×C4×C47—24no—
142C2×C4×C47—216no—
143C2×C4×C47—216no—
144C2×C4×C47—28no—
145C2×C4×C47#117116yescomplemented
146C2×C4×C47C23×C414yes—
147C2×C4×C47C23×C412yes—
148C2×C4×C47C23×C412yes—
149C2×C4×C47C23×C414yes—
150C2×C4×C47C2×#4118yes—
151C2×C4×C47C2×#4114yes—
152C2×C4×C47C2×#4114yes—
153C22⋊D47—216no—
154C2×Dic47—232no—
155C2×Dic47—28no—
156C2×Dic47—28no—
157C2×Dic47—28no—
158C2×Dic47—24no—
159C2×Dic47C23×C414yes—
160C2×Dic47C23×C418yes—
161C2×Dic47C2×#41116yes—
162D188—28no—
163C4×D47—216no—
164C4×D47—232no—
165Dic94—28no—
166#34026—471nomaximal, Sylow
167C22×C47—24no—
168C22×C47—28no—
169C22×C47C2×#11714yes—
170C22×C47C24×C411yescharacteristic
171C22×C47C22×#4114yes—
172C22×C47C22×#4112yes—
173C2×D47—24no—
174C2×D47—28no—
175C2×D47—48no—
176C4×C47—24no—
177C4×C47—48no—
178C4×C47C2×#11714yes—
179C4×C47C2×#11714yes—
180C4×C47C4×#4118yes—
181Dic47—24no—
182Dic47—28no—
183Dic47—216no—
184C2×C4×#41—471no—
185#2298—474no—
186#2297—472no—
187#2296—472no—
188#2295—474no—
189#2293—474no—
190#2290—478no—
191C4×#117—474no—
192C4×#117D4711yescharacteristic, complemented
193C2×C43—471no—
194C2×C47—24no—
195C2×C47C22×#11711yescharacteristic
196C2×C47C2×C4×#4112yes—
197C2×C47#344414yes—
198D47—44no—
199C22×#41—471no—
200C4×#41—4716no—
201C2×C4×Q8—472no—
202C2×C4×D4—474no—
203C2×C4×D4—472no—
204C2×#117—471no—
205C2×#117—472no—
206C2×#117—478no—
207C2×#117C2×D4712yes—
208C22×C42—472no—
209C22×C42—474no—
210C22×C42—472no—
211C22×C42C2×D4711yescharacteristic
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—478no—
221C4×C4⋊C4C2×D4714yes—
222C4×C22⋊C4—478no—
223C4×C22⋊C4—478no—
224C4×C22⋊C4C2×D4714yes—
225#376—478no—
226#376C2×D4712yes—
227C43—474no—
228C43—472no—
229C43—472no—
230C43C2×D4712yes—
231C47#3402611yescharacteristic, complemented, Sylow
232C2×#41—478no—
233C2×#41—948no—
234C22×Q8—471no—
235C22×D4—472no—
236C22×D4—471no—
237C23×C4—474no—
238C23×C4—472no—
239C23×C4—474no—
240C23×C4C22×D4712yes—
241C23×C4C22×D4711yescharacteristic
242C4×Q8—4716no—
243C4×D4—4732no—
244C4×D4—4716no—
245#117—28no—
246#117—478no—
247#117—4716no—
248#117—9432no—
249C2×C4⋊C4—474no—
250C2×C4⋊C4—472no—
251C2×C4⋊C4—4716no—
252C2×C4⋊C4C22×D4714yes—
253C2×C22⋊C4—472no—
254C2×C22⋊C4—474no—
255C2×C22⋊C4—4716no—
256C2×C22⋊C4C22×D4714yes—
257C2×C42—474no—
258C2×C42—472no—
259C2×C42—474no—
260C2×C42—4716no—
261C2×C42—474no—
262C2×C42—474no—
263C2×C42—478no—
264C2×C42—474no—
265C2×C42—9416no—
266C2×C42—948no—
267C2×C42C4×D4718yes—
268C2×C42C22×D4718yes—
269C2×C42C22×D4714yes—
270C2×C42C22×D4711yescharacteristic
271C2×C42C22×D4712yes—
272C2×C42C22×D4711yescharacteristic
273#95—4716no—
274#95—4716no—
275#95C22×D4718yes—
276C24—472no—
277C24C23×D4711yescharacteristic
278#41—9432no—
279C2×Q8—474no—
280C2×Q8—944no—
281C2×D4—478no—
282C2×D4—474no—
283C2×D4—948no—
284C2×D4—944no—
285C22×C4—28no—
286C22×C4—472no—
287C22×C4—478no—
288C22×C4—474no—
289C22×C4—478no—
290C22×C4—4716no—
291C22×C4—9416no—
292C22×C4—948no—
293C22×C4—9416no—
294C22×C4—948no—
295C22×C4C2×C4×D4718yes—
296C22×C4C2×C4×D4714yes—
297C22×C4#1062535018yes—
298C22×C4C23×D4718yes—
299C22×C4C23×D4712yes—
300C22×C4C23×D4712yes—
301C22×C4C23×D4711yescharacteristic
302C22×C4C23×D4711yescharacteristic
303C4⋊C4—28no—
304C4⋊C4—4716no—
305C4⋊C4—478no—
306C4⋊C4—9432no—
307C22⋊C4—28no—
308C22⋊C4—478no—
309C22⋊C4—4716no—
310C22⋊C4—9432no—
311C42—216no—
312C42—28no—
313C42—478no—
314C42—4716no—
315C42—4716no—
316C42—9416no—
317C42—948no—
318C42—948no—
319C42—948no—
320C42C2×C4×D4714yes—
321C42C2×C4×D4714yes—
322C42C2×C4×D4718yes—
323C23—24no—
324C23—474no—
325C23—948no—
326C23—944no—
327C23C22×C4×D4712yes—
328C23C2×#1062535014yes—
329C23C24×D4711yescharacteristic
330Q8—948no—
331D4—9416no—
332D4—948no—
333C2×C4—24no—
334C2×C4—216no—
335C2×C4—216no—
336C2×C4—28no—
337C2×C4—478no—
338C2×C4—474no—
339C2×C4—4716no—
340C2×C4—9432no—
341C2×C4—948no—
342C2×C4—948no—
343C2×C4—948no—
344C2×C4—944no—
345C2×C4—9416no—
346C2×C4—9432no—
347C2×C4#13093981116yescomplemented
348C2×C4C22×C4×D4714yes—
349C2×C4C22×C4×D4712yes—
350C2×C4C22×C4×D4712yes—
351C2×C4C22×C4×D4714yes—
352C2×C4#41×D4718yes—
353C2×C4C2×#1062535014yes—
354C2×C4C2×#1062535014yes—
355C22—24no—
356C22—28no—
357C22—944no—
358C22—948no—
359C22—1888no—
360C22C2×#1309398114yes—
361C22C23×C4×D4711yescharacteristic
362C22C2×#41×D4714yes—
363C22C22×#1062535012yes—
364C4—24no—
365C4—48no—
366C4—944no—
367C4—948no—
368C4—9416no—
369C4C2×#1309398114yes—
370C4C2×#1309398114yes—
371C4#1568959118yes—
372C2—24no—
373C2—1884no—
374C2C22×#1309398111yescharacteristic
375C2C2×#1568959112yes—
376C2#2025668114yes—
subgroupG/H
centerC2×C42C22×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC4×#117×C47C2
socle, cosocleC23×C47C25
prime radicalC22×#10625350C22
hypercenterC4×#117D47
hypocenterC47#34026
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C42 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
2. #20245883C2semidirect
4. #20245870C2semidirect
5. #20245862C2semidirect
6. #20245849C2semidirect
7. #20245827C2semidirect
8. C4×#13093981C2semidirect
9. C2×C42×Dic47C2semidirect
10. C4×#117×C47C2semidirect
12. C4×#10625350C4semidirect
51. C4×Dic94C2×C4semidirect
52. C4×D188C2×C4semidirect
56. #13093981C2×C4semidirect
72. C42×D47C2×C4semidirect
133. C4×Dic47#41semidirect
145. C2×C4×C47#117semidirect
192. C4×#117D47semidirect
231. C47#34026semidirect
347. C2×C4#13093981semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253502
#202458832
#202458762
#202458702
#202458622
#202458492
#202458272
C4×#130939812
C2×C42×Dic472
C4×#117×C472
#3402647