Finite Group Databasebeta

← #26248241

Subgroups — #26248241

Order 12032.

1158 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 207 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
1C2×#15689278C212yesmaximal
2#20245972C211yesmaximal, characteristic
3#20244536C212yesmaximal, complemented
4#20244488C212yesmaximal, complemented
5#20243729C214yesmaximal, complemented
6#20243701C212yesmaximal
7#20243459C211yesmaximal, characteristic
8C2×C47×#541C211yesmaximal, characteristic
9C23×Dic94C2211yescharacteristic
10C22×#10625337C2214yes—
11C2×#13093983C2211yescharacteristic
12#15689278—28no—
13C2×#13093948C2214yes—
14C2×#13093947C2214yes—
15C2×C22⋊(C47⋊C8)C2212yes—
16#15689029C2212yes—
17#15689029C2212yes—
18#15689028—22no—
19#15688984—22no—
20#15688885C2214yes—
21#15688878C2212yes—
22#15688878C2212yes—
23C47×#541—28no—
24C2×C4⋊Q8×C47C2211yescharacteristic
25C2×#126×C47—22no—
26C2×#125×C47—22no—
27C2×#125×C47C2211yescharacteristic
28C2×C22⋊Q8×C47C2212yes—
29C2×C4×Q8×C47C2212yes—
30C2×#117×C47C2211yescharacteristic
31C22×Dic94—21no—
32C22×Dic94—24no—
33C22×Dic94—24no—
34C22×Dic94—42no—
35C22×Dic94C2312yes—
36C23×Dic47—21no—
37C2×#10625337—28no—
38C2×#10625337—28no—
39C2×#10625337—44no—
40C22×C47⋊C8C2314yes—
41#13093983—42no—
42C2×C4⋊Dic47C2312yes—
43C2×C22⋊Dic47—21no—
44C2×C4×Dic47—21no—
45#13093948—44no—
46#13093947—28no—
47C22⋊(C47⋊C8)—24no—
48C47⋊#95—24no—
49C47⋊#95—24no—
50C22×Q8×C47C2312yes—
51C23×C4×C47C2311yescharacteristic
52C4⋊Q8×C47—24no—
53#126×C47—216no—
54#125×C47—216no—
55#125×C47—24no—
56C22⋊Q8×C47—24no—
57C22⋊Q8×C47D418yes—
58C4×Q8×C47—24no—
59C4×Q8×C47—24no—
60#117×C47—24no—
61C2×C4⋊C4×C47—22no—
62C2×C4⋊C4×C47—22no—
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47—22no—
65C2×C4⋊C4×C47D412yes—
66C2×C4⋊C4×C47C2312yes—
67C2×C22⋊C4×C47—22no—
68C2×C22⋊C4×C47D412yes—
69C2×C42×C47—22no—
70C2×C42×C47C2311yescharacteristic
71C2×C42×C47C2311yescharacteristic
72C24×C47C2×D411yescharacteristic
73C2×Q8×C47—22no—
74C2×Q8×C47—24no—
75C2×Q8×C47—24no—
76C2×Q8×C47C2×D414yes—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47C2×D412yes—
82C22×C4×C47C2×D412yes—
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D414yes—
85C22×C4×C47C2411yescharacteristic
86C22×Dic47—24no—
87C22×Dic47—22no—
88C22×Dic47—22no—
89C22×Dic47—42no—
90C22×Dic47—42no—
91C2×Dic94—24no—
92C2×Dic94—24no—
93C2×Dic94—24no—
94C2×Dic94—22no—
95C2×Dic94—416no—
96C2×Dic94—42no—
97C2×Dic94—42no—
98C2×Dic94#4114yes—
99C4⋊C4×C47—28no—
100C4⋊C4×C47—28no—
101C4⋊C4×C47—28no—
102C4⋊C4×C47—28no—
103C4⋊C4×C47—24no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47C2×D414yes—
106C22⋊C4×C47—28no—
107C22⋊C4×C47—24no—
108C42×C47—28no—
109C42×C47—22no—
110C42×C47—22no—
111C22⋊Dic47—42no—
112#10625337—416no—
113C4⋊Dic47—24no—
114C4×Dic47—42no—
115C2×C47⋊C8—24no—
116C2×C47⋊C8—24no—
117C2×C47⋊C8—24no—
118C23×C47—21no—
119C23×C47—21no—
120C23×C47—24no—
121C23×C47#12012yes—
122C23×C47C22×D411yescharacteristic
123Q8×C47—28no—
124Q8×C47—44no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—28no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47—24no—
131C2×C4×C47—24no—
132C2×C4×C47—24no—
133C2×C4×C47—24no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47—44no—
137C2×C4×C47#12012yes—
138C2×C4×C47C22×D412yes—
139C2×C4×C47C2×#4112yes—
140C2×C4×C47H(5,𝔽2)12yes—
141C2×Dic47—24no—
142C2×Dic47—48no—
143C2×Dic47—42no—
144C2×Dic47—42no—
145C2×Dic47—42no—
146C2×Dic47—44no—
147Dic94—28no—
148Dic94—48no—
149Dic94—44no—
150C47⋊C8—44no—
151#23943—471nomaximal, Sylow
152C22×C47—22no—
153C22×C47—24no—
154C22×C47—42no—
155C22×C47—42no—
156C22×C47C2×#12011yescharacteristic
157C22×C47#52511yescharacteristic
158C22×C47#53511yescharacteristic
159C22×C47#57814yes—
160C4×C47—22no—
161C4×C47—44no—
162C4×C47—42no—
163C4×C47—42no—
164C4×C47—44no—
165C4×C47#54614yes—
166Dic47—44no—
167Dic47—82no—
168C2×#541D4711yescharacteristic
169C2×#451—472no—
170#2310—471no—
171#2026—472no—
172#2016—472no—
173#1878—474no—
174#1873—472no—
175#1818—471no—
176C2×C47—42no—
177C2×C47#240411yescharacteristic
178C2×C47#301112yes—
179C2×C47#312712yes—
180C2×C47#322312yes—
181C23×Q8—471no—
182C22×Q16—474no—
183#541—28no—
184C2×C4⋊Q8C2×D4711yescharacteristic
185C2×#126—22no—
186C2×#125—22no—
187C2×#125C2×D4711yescharacteristic
188C2×C22⋊Q8C2×D4712yes—
189C2×C4×Q8C2×D4712yes—
190C2×#117—471no—
191C2×#117C2×D4711yescharacteristic
192#451—948no—
193C2×#103—474no—
194C2×#103—474no—
195C2×C22⋊C8—472no—
196#391—472no—
197#391—472no—
198#391—942no—
199#381—942no—
200#340—474no—
201#336—472no—
202#336—472no—
203C47#2394311yescharacteristic, complemented, Sylow
204C22×Q8—472no—
205C22×Q8—944no—
206C22×Q8—941no—
207C22×Q8—944no—
208C22×Q8—1882no—
209C22×Q8C22×D4712yes—
210C23×C4—941no—
211C23×C4C22×D4711yescharacteristic
212C2×Q16—948no—
213C2×Q16—948no—
214C2×Q16—1884no—
215C22×C8—474no—
216C4⋊Q8—24no—
217#126—216no—
218#125—216no—
219#125—24no—
220C22⋊Q8—24no—
221C22⋊Q8C22⋊D4718yes—
222C4×Q8—24no—
223C4×Q8—24no—
224#117—24no—
225#117—1882no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—472no—
231C2×C4⋊C4C22⋊D4712yes—
232C2×C4⋊C4C22×D4712yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—941no—
235C2×C22⋊C4C22⋊D4712yes—
236C2×C42—22no—
237C2×C42—941no—
238C2×C42C22×D4711yescharacteristic
239C2×C42C22×D4711yescharacteristic
240#103—948no—
241#103—1884no—
242C22⋊C8—944no—
243#95—944no—
244#95—944no—
245C24C2×C22⋊D4711yescharacteristic
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—474no—
250C2×Q8—944no—
251C2×Q8—942no—
252C2×Q8—944no—
253C2×Q8—944no—
254C2×Q8—18816no—
255C2×Q8—1882no—
256C2×Q8—1882no—
257C2×Q8C2×C22⋊D4714yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—944no—
263C22×C4—942no—
264C22×C4—942no—
265C22×C4—1882no—
266C22×C4—1882no—
267C22×C4C2×C22⋊D4712yes—
268C22×C4C2×C22⋊D4712yes—
269C22×C4C2×C22⋊D4711yescharacteristic
270C22×C4C2×C22⋊D4714yes—
271C22×C4C23×D4711yescharacteristic
272Q16—18816no—
273C2×C8—944no—
274C2×C8—944no—
275C2×C8—944no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—944no—
283C4⋊C4C2×C22⋊D4714yes—
284C22⋊C4—28no—
285C22⋊C4—24no—
286C22⋊C4—1882no—
287C42—28no—
288C42—22no—
289C42—22no—
290C42—1882no—
291C23—21no—
292C23—24no—
293C23—21no—
294C23#1309399312yes—
295C23C22×C22⋊D4711yescharacteristic
296Q8—28no—
297Q8—44no—
298Q8—948no—
299Q8—1888no—
300Q8—1884no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—28no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—944no—
314C2×C4—1884no—
315C2×C4—1882no—
316C2×C4—1888no—
317C2×C4—1882no—
318C2×C4—1882no—
319C2×C4#1309399312yes—
320C2×C4C22×C22⋊D4712yes—
321C2×C4#41×D4712yes—
322C2×C4#1309407012yes—
323C8—1884no—
324C22—24no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22C2×#1309399311yescharacteristic
329C22#1568963011yescharacteristic
330C22#1568968711yescharacteristic
331C22#1568993814yes—
332C4—22no—
333C4—42no—
334C4—42no—
335C4—44no—
336C4—44no—
337C4—3762no—
338C4—1884no—
339C4#1568976114yes—
340C2—42no—
341C2#2024670311yescharacteristic
342C2#2025268312yes—
343C2#2025368312yes—
344C2#2025446512yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#541C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenterC2×#541D47
hypocenterC47#23943
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
3. #20244536C2semidirect
4. #20244488C2semidirect
5. #20243729C2semidirect
203. C47#23943semidirect

Maximal subgroups

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

subgroupindex
C2×#156892782
#202459722
#202445362
#202444882
#202437292
#202437012
#202434592
C2×C47×#5412
#2394347