Finite Group Databasebeta

← #26105232

Subgroups — #26105232

Order 12032.

446 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20244721C211yesmaximal, characteristic, complemented
2#20244634C211yesmaximal, characteristic, complemented
3#20244524C211yesmaximal, characteristic
4#20241211C211yesmaximal, characteristic, complemented
5#20241211C211yesmaximal, characteristic, complemented
6#20241209C211yesmaximal, characteristic
7C47×#2727C211yesmaximal, characteristic
8C2×#13094024C2211yescharacteristic
9C2×#13093955—21no—
10C2×#13093947—21no—
11C2×#13093947—21no—
12C2×#13093945—21no—
13#15688885—21no—
14#15688885—21no—
15#15688878C2211yescharacteristic
16#15688878C2211yescharacteristic
17#15688878C2211yescharacteristic
18C2×#126×C47—21no—
19C47×#399—21no—
20C47×#396—21no—
21C47×#394—21no—
22C47×#390C2211yescharacteristic
23C47×#389—21no—
24C47×#389C2211yescharacteristic
25C47×#389C2211yescharacteristic
26C47×#388—21no—
27C22×Dic94—21no—
28C22×C47⋊C8—21no—
29C22×C47⋊C8—21no—
30C22×C47⋊C8—21no—
31#13094024—42no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34#13093955—42no—
35#13093947—42no—
36#13093947—42no—
37#13093945—42no—
38C22×D4×C47D411yescharacteristic
39#126×C47—42no—
40#126×C47—42no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—21no—
55C2×C22⋊C4×C47—21no—
56C2×C42×C47—21no—
57C2×C42×C47C2311yescharacteristic
58#95×C47—21no—
59#95×C47—21no—
60#95×C47—21no—
61C24×C47—21no—
62C2×D4×C47—21no—
63C2×D4×C47—21no—
64C2×D4×C47—41no—
65C2×D4×C47—41no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×Dic47—41no—
75C2×Dic94—22no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×Dic94—41no—
80C2×Dic94—41no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—21no—
83C4⋊C4×C47—21no—
84C4⋊C4×C47—42no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—81no—
93C4⋊C4×C47SD1612yes—
94C22⋊C4×C47—42no—
95C22⋊C4×C47—42no—
96C22⋊C4×C47—41no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—41no—
99C22⋊C4×C47—41no—
100C22⋊C4×C47—81no—
101C22⋊C4×C47—81no—
102C42×C47—41no—
103C42×C47—42no—
104C4⋊Dic47—22no—
105C4⋊Dic47—22no—
106C4⋊Dic47—41no—
107C4⋊Dic47—41no—
108C2×C47⋊C8—22no—
109C2×C47⋊C8—41no—
110C2×C47⋊C8—41no—
111C2×C47⋊C8—41no—
112C2×C47⋊C8—41no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8—41no—
115C2×C47⋊C8—41no—
116C2×C47⋊C8—41no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47—41no—
121C23×C47—41no—
122C23×C47—41no—
123C23×C47—41no—
124C23×C47#12011yescharacteristic
125D4×C47—42no—
126D4×C47—81no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—22no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47C2×SD1611yescharacteristic
149C2×C4×C47#13511yescharacteristic
150C2×C4×C47#13511yescharacteristic
151C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C47#13711yescharacteristic
154C2×Dic47—42no—
155C2×Dic47—81no—
156C2×Dic47—81no—
157Dic94—42no—
158Dic94—42no—
159Dic94—81no—
160Dic94—81no—
161C47⋊C8—42no—
162C47⋊C8—81no—
163C47⋊C8—81no—
164#7776—471nomaximal, Sylow
165C22×C47—81no—
166C22×C47—81no—
167C22×C47—81no—
168C22×C47—81no—
169C22×C47—81no—
170C22×C47—81no—
171C22×C47—81no—
172C22×C47#44911yescharacteristic
173C22×C47#45011yescharacteristic
174C22×C47#45211yescharacteristic
175C22×C47#45411yescharacteristic
176C22×C47#45511yescharacteristic
177C22×C47#45611yescharacteristic
178C22×C47B(4,𝔽2)11yescharacteristic
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—21no—
183C4×C47—21no—
184C4×C47—21no—
185C4×C47—42no—
186C4×C47—81no—
187C4×C47—81no—
188C4×C47—81no—
189C4×C47—81no—
190Dic47—82no—
191#2727D4711yescharacteristic
192#2058—471no—
193#2043—471no—
194#2023—471no—
195#1271—471no—
196#1271—471no—
197#1270—471no—
198C2×C47—161no—
199C2×C47#159711yescharacteristic
200C2×C47#160411yescharacteristic
201C2×C47#160611yescharacteristic
202C2×C47#163511yescharacteristic
203C2×C47#164211yescharacteristic
204C2×C47#165211yescharacteristic
205C2×C47#166411yescharacteristic
206C2×C4⋊Q8—471no—
207C2×#126—21no—
208C2×#106—941no—
209C2×#103—941no—
210C2×#103—941no—
211C2×#102—941no—
212#399—21no—
213#396—21no—
214#394—21no—
215#390C2×D4711yescharacteristic
216#389—21no—
217#389C2×D4711yescharacteristic
218#389C2×D4711yescharacteristic
219#388—21no—
220#340—941no—
221#340—941no—
222#336—471no—
223#336—471no—
224#336—471no—
225C47#777611yescharacteristic, complemented, Sylow
226C22×Q8—941no—
227C22×D4C22⋊D4711yescharacteristic
228C22×C8—941no—
229C22×C8—941no—
230C22×C8—941no—
231C4⋊Q8—1882no—
232#126—42no—
233#126—42no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—941no—
239C2×C4⋊C4—941no—
240C2×C4⋊C4C22⋊D4711yescharacteristic
241C2×C4⋊C4C22⋊D4711yescharacteristic
242C2×C4⋊C4C22⋊D4711yescharacteristic
243C2×C4⋊C4C22⋊D4711yescharacteristic
244C2×C4⋊C4C22⋊D4711yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—21no—
252C2×C42C22×D4711yescharacteristic
253#106—1882no—
254#103—1882no—
255#103—1882no—
256#102—1882no—
257#95—21no—
258#95—21no—
259#95—21no—
260C24—21no—
261C2×Q8—942no—
262C2×Q8—1881no—
263C2×Q8—1881no—
264C2×Q8—1881no—
265C2×Q8—1881no—
266C2×Q8—1881no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—1881no—
277C22×C4C2×C22⋊D4711yescharacteristic
278C22×C4C2×C22⋊D4711yescharacteristic
279C22×C4C2×C22⋊D4711yescharacteristic
280C2×C8—942no—
281C2×C8—1881no—
282C2×C8—1881no—
283C2×C8—1881no—
284C2×C8—1881no—
285C2×C8—1881no—
286C2×C8—1881no—
287C2×C8—1881no—
288C2×C8—1881no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C4⋊C4—942no—
302C4⋊C4—942no—
303C4⋊C4—1881no—
304C4⋊C4—1881no—
305C4⋊C4#1062533512yes—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C42—41no—
315C42—42no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#1309399311yescharacteristic
324Q8—3761no—
325Q8—3761no—
326Q8—1882no—
327Q8—1882no—
328D4—42no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—3761no—
352C2×C4—3761no—
353C2×C4—1882no—
354C2×C4C2×#1062533511yescharacteristic
355C2×C4#1309404411yescharacteristic
356C2×C4#1309404411yescharacteristic
357C2×C4#1309404911yescharacteristic
358C2×C4#1309405011yescharacteristic
359C2×C4#1309405411yescharacteristic
360C8—3761no—
361C8—3761no—
362C8—1882no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1568926811yescharacteristic
371C22#1568927411yescharacteristic
372C22#1568928511yescharacteristic
373C22#1568929611yescharacteristic
374C22#1568930111yescharacteristic
375C22#1568930611yescharacteristic
376C22#1568931011yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—3762no—
389C2—161no—
390C2#2024231711yescharacteristic
391C2#2024235711yescharacteristic
392C2#2024236811yescharacteristic
393C2#2024253911yescharacteristic
394C2#2024258811yescharacteristic
395C2#2024265811yescharacteristic
396C2#2024272211yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2727C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2727D47
hypocenterC47#7776
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
1. #20244721C2semidirect
2. #20244634C2semidirect
4. #20241211C2semidirect
5. #20241211C2semidirect
225. C47#7776semidirect

Maximal subgroups

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

subgroupindex
#202447212
#202446342
#202445242
#202412112
#202412092
C47×#27272
#777647