Finite Group Databasebeta

← #26374947

Subgroups — #26374947

Order 12032.

2358 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625350C211yesmaximal, characteristic
2#20246040C214yesmaximal, complemented
3#20246032C214yesmaximal, complemented
4#20246022C218yesmaximal, complemented
5#20245883C214yesmaximal, complemented
6#20245878C212yesmaximal
7#20245871C212yesmaximal, complemented
8#20245780C214yesmaximal, complemented
9C4×#13093983C211yesmaximal, characteristic, complemented
10C2×C47×#376C211yesmaximal, characteristic, complemented
11C22×#10625350C2211yescharacteristic
12C4×#10625350C4116yescomplemented
13C2×C4×Dic94C2212yes—
14C2×C4×D188C2212yes—
15C2×C4×C22⋊D47C2214yes—
16C2×#13093983C2212yes—
17C2×#13093983C2211yescharacteristic
18C2×#13093981C2212yes—
19C2×C42×D47C2212yes—
20#15689029C2214yes—
21#15689025C2214yes—
22#15689021C2218yes—
23#15689012C22116yes—
24#15689005C22116yes—
25#15689002C2218yes—
26#15688996C2218yes—
27#15688990C2218yes—
28#15688987C2218yes—
29#15688985C2218yes—
30#15688982C2218yes—
31C4×Dic47⋊C4C2218yes—
32C4×C4⋊Dic47C2214yes—
33C4×(C2×D47)⋊C4C2218yes—
34C4×C22⋊Dic47C2214yes—
35#15688969C2212yes—
36C42×Dic47C2212yes—
37C22×C42×C47C2212yes—
38C22×C42×C47C2211yescharacteristic
39C47×#376C2214yes—
40C47×#376C2212yes—
41C47×#376C2212yes—
42C2×#95×C47C2214yes—
43C2×#10625350—28no—
44C2×#10625350C2×C418yes—
45C22×Dic94C2311yescharacteristic
46C22×D188C2311yescharacteristic
47C22×C22⋊D47C2312yes—
48C22×C4×D47C2312yes—
49C22×C4×D47C2314yes—
50C4×Dic94C2×C4116yescomplemented
51C4×D188C2×C4116yescomplemented
52C4×C22⋊D47C2×C4132yes—
53#13093983—28no—
54#13093983C2×C418yes—
55#13093981C2×C4116yescomplemented
56C2×C4⋊Dic47C2314yes—
57C2×C4⋊Dic47C2312yes—
58C2×Dic47⋊C4C23116yes—
59C2×Dic47⋊C4C2314yes—
60C2×C22⋊Dic47C2314yes—
61C2×C22⋊Dic47C2312yes—
62C2×(C2×D47)⋊C4C23116yes—
63C2×(C2×D47)⋊C4C2314yes—
64C2×C4×Dic47C2314yes—
65C2×C4×Dic47C2318yes—
66C2×C4×Dic47C2312yes—
67C2×C4×Dic47C2314yes—
68C42×D47C2×C4116yescomplemented
69C47⋊#95C2318yes—
70C47⋊#95C23116yes—
71C47⋊#95C23116yes—
72C23×C4×C47C2314yes—
73C23×C4×C47C2312yes—
74C23×C4×C47C2311yescharacteristic
75C2×C42×C47—28no—
76C2×C42×C47C2×C418yes—
77C2×C42×C47C2314yes—
78C2×C42×C47C2314yes—
79C2×C42×C47C2311yescharacteristic
80C2×C42×C47C2311yescharacteristic
81C2×C42×C47C2312yes—
82#95×C47C2314yes—
83#95×C47C2314yes—
84#95×C47C2318yes—
85C24×C47C2411yescharacteristic
86C23×D47C2412yes—
87C22×C4×C47—28no—
88C22×C4×C47—44no—
89C22×C4×C47—48no—
90C22×C4×C47C22×C418yes—
91C22×C4×C47C22×C414yes—
92C22×C4×C47#4118yes—
93C22×C4×C47C2418yes—
94C22×C4×C47C2412yes—
95C22×C4×C47C2412yes—
96C22×C4×C47C2411yescharacteristic
97C22×C4×C47C2411yescharacteristic
98C2×C22⋊D47—28no—
99C2×C22⋊D47C22×C418yes—
100C22×Dic47C2418yes—
101C22×Dic47C2412yes—
102C22×Dic47C2414yes—
103#10625350—232no—
104C2×D188—24no—
105C2×D188C22×C414yes—
106C2×C4×D47—28no—
107C2×C4×D47—48no—
108C2×C4×D47C22×C418yes—
109C2×C4×D47C22×C4116yes—
110C2×Dic94—24no—
111C2×Dic94C22×C414yes—
112C42×C47—28no—
113C42×C47—28no—
114C42×C47C22×C414yes—
115C42×C47C22×C414yes—
116C42×C47C22×C418yes—
117C22⋊Dic47—28no—
118C22⋊Dic47C22×C418yes—
119(C2×D47)⋊C4—416no—
120(C2×D47)⋊C4C22×C4116yes—
121C4⋊Dic47—28no—
122C4⋊Dic47C22×C418yes—
123Dic47⋊C4—416no—
124Dic47⋊C4C22×C4116yes—
125C4×Dic47—28no—
126C4×Dic47—48no—
127C4×Dic47C22×C418yes—
128C4×Dic47C22×C4116yes—
129C23×C47—24no—
130C23×C47C23×C412yes—
131C23×C47C2×#4114yes—
132C23×C47C2511yescharacteristic
133C22×D47—24no—
134C22×D47—28no—
135C22×D47C23×C414yes—
136C2×C4×C47—24no—
137C2×C4×C47—24no—
138C2×C4×C47—28no—
139C2×C4×C47—28no—
140C2×C4×C47—28no—
141C2×C4×C47—48no—
142C2×C4×C47—416no—
143C2×C4×C47#117116yes—
144C2×C4×C47C23×C414yes—
145C2×C4×C47C23×C414yes—
146C2×C4×C47C23×C412yes—
147C2×C4×C47C23×C412yes—
148C2×C4×C47C2×#4114yes—
149C2×C4×C47C2×#4114yes—
150C22⋊D47—216no—
151C2×Dic47—24no—
152C2×Dic47—28no—
153C2×Dic47—28no—
154C2×Dic47—216no—
155C2×Dic47C23×C414yes—
156C2×Dic47C23×C418yes—
157D188—28no—
158C4×D47—216no—
159C4×D47—416no—
160Dic94—28no—
161#34056—471nomaximal, Sylow
162C22×C47—24no—
163C22×C47—28no—
164C22×C47C2×#11714yes—
165C22×C47C24×C411yescharacteristic
166C22×C47C22×#4112yes—
167C22×C47#58514yes—
168C2×D47—24no—
169C2×D47—28no—
170C2×D47—48no—
171C4×C47—24no—
172C4×C47—24no—
173C4×C47—48no—
174C4×C47C2×#11714yes—
175C4×C47C2×#11714yes—
176Dic47—24no—
177Dic47—28no—
178Dic47—48no—
179C2×C4×#41—471no—
180#2320—474no—
181#2319—474no—
182#2318—478no—
183#2298—474no—
184#2297—472no—
185#2296—472no—
186#2283—474no—
187C4×#117—471no—
188C2×#376D4711yescharacteristic, complemented
189C2×C47—24no—
190C2×C47C22×#11711yescharacteristic
191C2×C47#342812yes—
192C2×C47#345514yes—
193D47—44no—
194C22×#41—471no—
195C4×#41—4716no—
196C2×C4×Q8—472no—
197C2×C4×D4—474no—
198C2×C4×D4—472no—
199C2×#117—471no—
200C2×#117—472no—
201C2×#117—472no—
202C22×C42—472no—
203C22×C42C2×D4712yes—
204C22×C42C2×D4711yescharacteristic
205#391—474no—
206#390—474no—
207#389—478no—
208#388—4716no—
209#387—4716no—
210#386—478no—
211#385—478no—
212#383—478no—
213#382—478no—
214#381—478no—
215#380—478no—
216C4×C4⋊C4—474no—
217C4×C4⋊C4—478no—
218C4×C22⋊C4—474no—
219C4×C22⋊C4—478no—
220#376—472no—
221#376C2×D4714yes—
222#376C2×D4712yes—
223#376C2×D4712yes—
224C2×#95C2×D4714yes—
225C43—472no—
226C47#3405611yescharacteristic, complemented, Sylow
227C2×#41—478no—
228C2×#41—948no—
229C22×Q8—471no—
230C22×D4—472no—
231C22×D4—471no—
232C23×C4—472no—
233C23×C4—474no—
234C23×C4C22×D4714yes—
235C23×C4C22×D4712yes—
236C23×C4C22×D4711yescharacteristic
237C4×Q8—4716no—
238C4×D4—4732no—
239C4×D4—4716no—
240#117—478no—
241#117—4716no—
242#117—948no—
243C2×C4⋊C4—474no—
244C2×C4⋊C4—472no—
245C2×C4⋊C4—474no—
246C2×C4⋊C4—4716no—
247C2×C22⋊C4—472no—
248C2×C22⋊C4—474no—
249C2×C22⋊C4—474no—
250C2×C22⋊C4—4716no—
251C2×C42—28no—
252C2×C42—474no—
253C2×C42—472no—
254C2×C42—4716no—
255C2×C42—474no—
256C2×C42—478no—
257C2×C42C4×D4718yes—
258C2×C42C22×D4714yes—
259C2×C42C22×D4714yes—
260C2×C42C22×D4712yes—
261C2×C42C22×D4711yescharacteristic
262C2×C42C22×D4711yescharacteristic
263#95—478no—
264#95—4716no—
265#95—4716no—
266#95C22×D4714yes—
267#95C22×D4714yes—
268#95C22×D4718yes—
269C24—472no—
270C24C23×D4711yescharacteristic
271#41—9432no—
272C2×Q8—474no—
273C2×Q8—944no—
274C2×D4—478no—
275C2×D4—474no—
276C2×D4—948no—
277C2×D4—944no—
278C22×C4—28no—
279C22×C4—44no—
280C22×C4—48no—
281C22×C4—472no—
282C22×C4—474no—
283C22×C4—478no—
284C22×C4—4716no—
285C22×C4—478no—
286C22×C4—948no—
287C22×C4—1888no—
288C22×C4C2×C4×D4718yes—
289C22×C4C2×C4×D4714yes—
290C22×C4#1062535018yes—
291C22×C4C23×D4718yes—
292C22×C4C23×D4712yes—
293C22×C4C23×D4712yes—
294C22×C4C23×D4711yescharacteristic
295C22×C4C23×D4711yescharacteristic
296C4⋊C4—4716no—
297C4⋊C4—478no—
298C4⋊C4—948no—
299C4⋊C4—18816no—
300C22⋊C4—478no—
301C22⋊C4—4716no—
302C22⋊C4—948no—
303C22⋊C4—18816no—
304C42—28no—
305C42—28no—
306C42—478no—
307C42—4716no—
308C42—948no—
309C42—1888no—
310C42C2×C4×D4714yes—
311C42C2×C4×D4714yes—
312C42C2×C4×D4718yes—
313C23—24no—
314C23—474no—
315C23—944no—
316C23—948no—
317C23C22×C4×D4712yes—
318C23C2×#1062535014yes—
319C23C24×D4711yescharacteristic
320Q8—948no—
321D4—9416no—
322D4—948no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—48no—
329C2×C4—416no—
330C2×C4—478no—
331C2×C4—474no—
332C2×C4—944no—
333C2×C4—948no—
334C2×C4—9416no—
335C2×C4—948no—
336C2×C4—9416no—
337C2×C4—18816no—
338C2×C4#13093981116yes—
339C2×C4C22×C4×D4714yes—
340C2×C4C22×C4×D4714yes—
341C2×C4C22×C4×D4712yes—
342C2×C4C22×C4×D4712yes—
343C2×C4C2×#1062535014yes—
344C2×C4C2×#1062535014yes—
345C22—28no—
346C22—24no—
347C22—944no—
348C22—948no—
349C22—1888no—
350C22C2×#1309398114yes—
351C22C23×C4×D4711yescharacteristic
352C22C22×#1062535012yes—
353C22#1568995914yes—
354C4—24no—
355C4—24no—
356C4—48no—
357C4—944no—
358C4—948no—
359C4—1888no—
360C4C2×#1309398114yes—
361C4C2×#1309398114yes—
362C2—24no—
363C2—1884no—
364C2C22×#1309398111yescharacteristic
365C2#2025658912yes—
366C2#2025677814yes—
subgroupG/H
centerC22×C4C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC2×C47×#376C2
socle, cosocleC23×C47C25
prime radicalC22×#10625350C22
hypercenterC2×#376D47
hypocenterC47#34056
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
2. #20246040C2semidirect
3. #20246032C2semidirect
4. #20246022C2semidirect
5. #20245883C2semidirect
7. #20245871C2semidirect
8. #20245780C2semidirect
9. C4×#13093983C2semidirect
10. C2×C47×#376C2semidirect
12. C4×#10625350C4semidirect
50. C4×Dic94C2×C4semidirect
51. C4×D188C2×C4semidirect
55. #13093981C2×C4semidirect
68. C42×D47C2×C4semidirect
188. C2×#376D47semidirect
226. C47#34056semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253502
#202460402
#202460322
#202460222
#202458832
#202458782
#202458712
#202457802
C4×#130939832
C2×C47×#3762
#3405647