Finite Group Databasebeta

← #26107991

Subgroups — #26107991

Order 12032.

1050 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 159 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#20243380C211yesmaximal, characteristic, complemented
2#20242056C212yesmaximal, complemented
3#20241817C212yesmaximal, complemented
4#20241709C211yesmaximal, characteristic, complemented
5C2×C47×#550C211yesmaximal, characteristic
6#15689176C2212yes—
7#15689147C2212yes—
8C2×#13093941—21no—
9#15688892—21no—
10#15688867—24no—
11Q8⋊(C47⋊C8)—24no—
12#15688847—22no—
13C22×#41×C47C412yes—
14C47×#550—24no—
15C47×#550—24no—
16C2×#127×C47—21no—
17C2×#127×C47C2211yescharacteristic
18C2×#121×C47—22no—
19C2×#121×C47C412yes—
20C2×C4×Q8×C47C2211yescharacteristic
21C2×C4×D4×C47—21no—
22C2×C4×D4×C47C2211yescharacteristic
23C2×#10625329—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)—22no—
27#13093941—42no—
28C22⋊(C47⋊C8)—22no—
29C4×C47⋊C8—22no—
30C2×#41×C47—24no—
31C2×#41×C47—24no—
32C2×#41×C47—48no—
33C22×Q8×C47C2×C411yescharacteristic
34C22×D4×C47—22no—
35C22×D4×C47—22no—
36C22×D4×C47—21no—
37C22×D4×C47—21no—
38C22×D4×C47C2×C412yes—
39C22×D4×C47D411yescharacteristic
40C23×C4×C47—22no—
41C23×C4×C47C2×C412yes—
42#127×C47—28no—
43#127×C47—22no—
44#127×C47D414yes—
45#121×C47—216no—
46#121×C47—24no—
47#121×C47—24no—
48C4×Q8×C47—22no—
49C4×Q8×C47D414yes—
50C4×D4×C47—28no—
51C4×D4×C47—22no—
52C4×D4×C47—22no—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47C2×C411yescharacteristic
55C2×C4⋊C4×C47D411yescharacteristic
56C2×C22⋊C4×C47—22no—
57C2×C22⋊C4×C47D412yes—
58C2×C42×C47—21no—
59C2×C42×C47C2311yescharacteristic
60C24×C47—22no—
61C24×C47C22⋊C412yes—
62#41×C47—416no—
63#41×C47—48no—
64#41×C47—48no—
65C2×Q8×C47—24no—
66C2×Q8×C47—21no—
67C2×Q8×C47C22⋊C412yes—
68C2×D4×C47—28no—
69C2×D4×C47—28no—
70C2×D4×C47—22no—
71C2×D4×C47—22no—
72C2×D4×C47—24no—
73C2×D4×C47—24no—
74C2×D4×C47—22no—
75C2×D4×C47—21no—
76C2×D4×C47—21no—
77C2×D4×C47—48no—
78C2×D4×C47—44no—
79C2×D4×C47—42no—
80C2×D4×C47—41no—
81C2×D4×C47—41no—
82C2×D4×C47—82no—
83C2×D4×C47—84no—
84C2×D4×C47C22⋊C414yes—
85C22×C4×C47—21no—
86C22×C4×C47—28no—
87C22×C4×C47—21no—
88C22×C4×C47—22no—
89C22×C4×C47—22no—
90C22×C4×C47—48no—
91C22×C4×C47—44no—
92C22×C4×C47C22⋊C412yes—
93C22×C4×C47C22×C411yescharacteristic
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47C2×D411yescharacteristic
96C4⋊C4×C47—21no—
97C4⋊C4×C47—24no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47C22⋊C412yes—
100C22⋊C4×C47—28no—
101C22⋊C4×C47—24no—
102C42×C47—24no—
103C42×C47C2×D412yes—
104C42×C47C2×D412yes—
105#10625329—42no—
106#10625329—42no—
107C2×C47⋊C8—42no—
108C2×C47⋊C8—42no—
109C23×C47—22no—
110C23×C47—24no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47—44no—
114C23×C47—44no—
115C23×C47—42no—
116C23×C47—42no—
117C23×C47—42no—
118C23×C47—42no—
119C23×C47C2×C22⋊C411yescharacteristic
120Q8×C47—24no—
121Q8×C47—22no—
122Q8×C47#10014yescomplemented
123D4×C47—44no—
124D4×C47—416no—
125D4×C47—44no—
126D4×C47—42no—
127D4×C47—44no—
128D4×C47—42no—
129D4×C47—44no—
130D4×C47—84no—
131D4×C47—88no—
132C2×C4×C47—24no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—41no—
138C2×C4×C47—416no—
139C2×C4×C47—44no—
140C2×C4×C47—42no—
141C2×C4×C47—44no—
142C2×C4×C47—42no—
143C2×C4×C47C2×C22⋊C411yescharacteristic
144C2×C4×C47C2×C22⋊C411yescharacteristic
145C2×C4×C47#12012yes—
146C2×C4×C47#12012yes—
147C2×C4×C47#13512yes—
148C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C47#13711yescharacteristic
150C47⋊C8—82no—
151C47⋊C8—82no—
152#8385—471nomaximal, Sylow
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—42no—
156C22×C47—44no—
157C22×C47—42no—
158C22×C47—44no—
159C22×C47—42no—
160C22×C47—44no—
161C22×C47—82no—
162C22×C47—82no—
163C22×C47—84no—
164C22×C47—84no—
165C22×C47#37911yescharacteristic
166C22×C47#41611yescharacteristic
167C22×C47#41711yescharacteristic
168C4×C47—22no—
169C4×C47—24no—
170C4×C47—21no—
171C4×C47—42no—
172C4×C47—42no—
173C4×C47C2×#10012yes—
174C4×C47#44912yes—
175C4×C47#45212yes—
176C2×#550D4711yescharacteristic
177#1800—471no—
178#1545—472no—
179#1487—472no—
180#1460—471no—
181C2×C47—21no—
182C2×C47—21no—
183C2×C47—82no—
184C2×C47—84no—
185C2×C47#178311yescharacteristic
186C2×C47#178811yescharacteristic
187C2×C47#179511yescharacteristic
188C22×#41Dic4712yes—
189#550—24no—
190#550—24no—
191C2×#127—21no—
192C2×#127C2×D4711yescharacteristic
193C2×#121—22no—
194C2×#121Dic4712yes—
195C2×C4×Q8C2×D4711yescharacteristic
196C2×C4×D4—21no—
197C2×C4×D4C2×D4711yescharacteristic
198#431—472no—
199#423—472no—
200C2×#100—941no—
201#343—941no—
202#331—944no—
203Q8⋊C8—944no—
204#323—942no—
205C47#838511yescharacteristic, complemented, Sylow
206C2×#41—24no—
207C2×#41—24no—
208C2×#41—48no—
209C22×Q8C2×Dic4711yescharacteristic
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4C2×Dic4712yes—
215C22×D4C22⋊D4711yescharacteristic
216C23×C4—22no—
217C23×C4C2×Dic4712yes—
218C2×M16—942no—
219#127—28no—
220#127—22no—
221#127C22⋊D4714yes—
222#121—216no—
223#121—24no—
224#121—24no—
225C4×Q8—22no—
226C4×Q8C22⋊D4714yes—
227C4×D4—28no—
228C4×D4—22no—
229C4×D4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C2×Dic4711yescharacteristic
232C2×C4⋊C4C22⋊D4711yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4C22⋊D4712yes—
235C2×C42—21no—
236C2×C42C22×D4711yescharacteristic
237C4⋊C8—942no—
238C4⋊C8—942no—
239C4⋊C8—942no—
240#100—1882no—
241C22⋊C8—942no—
242C4×C8—942no—
243C24—22no—
244C24C22⋊Dic4712yes—
245#41—416no—
246#41—48no—
247#41—48no—
248C2×Q8—24no—
249C2×Q8—21no—
250C2×Q8C22⋊Dic4712yes—
251C2×D4—28no—
252C2×D4—28no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—82no—
266C2×D4—84no—
267C2×D4C22⋊Dic4714yes—
268C22×C4—21no—
269C22×C4—28no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4C22⋊Dic4712yes—
276C22×C4C22×Dic4711yescharacteristic
277C22×C4C2×C22⋊D4711yescharacteristic
278C22×C4C2×C22⋊D4711yescharacteristic
279M16—1882no—
280M16—1882no—
281C2×C8—1882no—
282C2×C8—1882no—
283C4⋊C4—21no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4C22⋊Dic4712yes—
287C22⋊C4—28no—
288C22⋊C4—24no—
289C42—24no—
290C42C2×C22⋊D4712yes—
291C42C2×C22⋊D4712yes—
292C23—22no—
293C23—24no—
294C23—22no—
295C23—22no—
296C23—44no—
297C23—44no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23C2×C22⋊Dic4711yescharacteristic
303Q8—24no—
304Q8—22no—
305Q8#1309394114yescomplemented
306D4—44no—
307D4—416no—
308D4—42no—
309D4—44no—
310D4—42no—
311D4—44no—
312D4—44no—
313D4—84no—
314D4—88no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—416no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4C2×C22⋊Dic4711yescharacteristic
327C2×C4C2×C22⋊Dic4711yescharacteristic
328C2×C4#1309399312yes—
329C2×C4#1309399312yes—
330C2×C4#1309404412yes—
331C2×C4#1309404911yescharacteristic
332C2×C4#1309405411yescharacteristic
333C8—3762no—
334C8—3762no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—44no—
339C22—42no—
340C22—44no—
341C22—44no—
342C22—42no—
343C22—82no—
344C22—82no—
345C22—84no—
346C22—84no—
347C22#1568897911yescharacteristic
348C22#1568911911yescharacteristic
349C22#1568912411yescharacteristic
350C4—24no—
351C4—21no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4C2×#1309394112yes—
356C4#1568926812yes—
357C4#1568928512yes—
358C2—21no—
359C2—21no—
360C2—84no—
361C2—82no—
362C2#2024330911yescharacteristic
363C2#2024332811yescharacteristic
364C2#2024335811yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC2×C47×#550C2
socleC22×C47#379
cosocleC2×C42×C47C23
hypercenterC2×#550D47
hypocenterC47#8385
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #20243380C2semidirect
2. #20242056C2semidirect
3. #20241817C2semidirect
4. #20241709C2semidirect
122. Q8×C47#100semidirect
205. C47#8385semidirect
305. Q8#13093941semidirect

Maximal subgroups

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

subgroupindex
#202433802
#202420562
#202418172
#202417092
C2×C47×#5502
#838547