Finite Group Databasebeta

← #26126590

Subgroups — #26126590

Order 12032.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#20249201C211yesmaximal, characteristic
2C2×#15688892C211yesmaximal, characteristic
3#20241419C212yesmaximal, complemented
4#20241416C212yesmaximal, complemented
5C2×C47×#324C211yesmaximal, characteristic
6C22×#10625329—21no—
7C2×#13094003C411yescharacteristic
8#15689062C411yescharacteristic
9#15689062C2211yescharacteristic
10#15689056—21no—
11#15689056—21no—
12#15689043C411yescharacteristic
13#15689043C2211yescharacteristic
14#15688996—21no—
15#15688986—21no—
16#15688983C411yescharacteristic
17#15688892—21no—
18#15688892C2212yes—
19C22×C4⋊C4×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C47×#324—21no—
22C47×#324C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C2×#10625329—24no—
25C2×#10625329—22no—
26C2×#10625329—41no—
27C2×#10625329—42no—
28C22×C47⋊C8—41no—
29#13094003—41no—
30#13094003—41no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47C2×C411yescharacteristic
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×Dic47⋊C4—21no—
36C2×C22⋊Dic47—21no—
37C2×C22⋊Dic47—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95C2×C411yescharacteristic
45C47⋊#95C2×C411yescharacteristic
46C23×C4×C47C2×C411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C22×C8×C47—41no—
50C2×C4⋊C4×C47—41no—
51C2×C4⋊C4×C47—41no—
52C2×C4⋊C4×C47C2×C411yescharacteristic
53C2×C4⋊C4×C47D411yescharacteristic
54C2×C4⋊C4×C47D411yescharacteristic
55C2×C4⋊C4×C47C2311yescharacteristic
56C22⋊C8×C47—22no—
57C22⋊C8×C47—41no—
58C24×C47C22⋊C411yescharacteristic
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—81no—
67C22×C4×C47—81no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47SD1612yes—
72C22×C4×C47Q1612yes—
73C22×C4×C47C22×C411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79C22×Dic47—21no—
80C2×Dic94—41no—
81C2×Dic94—81no—
82C2×C8×C47—42no—
83C2×C8×C47—82no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C22⋊Dic47—41no—
89C22⋊Dic47—41no—
90C22⋊Dic47—42no—
91C4⋊Dic47—41no—
92C4⋊Dic47—81no—
93Dic47⋊C4—42no—
94Dic47⋊C4—81no—
95Dic47⋊C4—81no—
96#10625329—24no—
97#10625329—44no—
98#10625329—42no—
99C2×C47⋊C8—44no—
100C2×C47⋊C8—42no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47#9911yescharacteristic
104C23×C47#9911yescharacteristic
105C23×C47#10312yes—
106C23×C47#10412yes—
107C23×C47C2×C22⋊C411yescharacteristic
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—42no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—81no—
126C2×C4×C47—81no—
127C2×C4×C47—81no—
128C2×C4×C47#10312yes—
129C2×C4×C47C2×SD1611yescharacteristic
130C2×C4×C47C2×Q1611yescharacteristic
131C2×Dic47—41no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—42no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142Dic94—82no—
143C8×C47—82no—
144C47⋊C8—44no—
145#11722—471nomaximal, Sylow
146C22×C47—21no—
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—42no—
153C22×C47—42no—
154C22×C47#32812yes—
155C22×C47#32812yes—
156C22×C47C2×#9911yescharacteristic
157C22×C47C2×#10311yescharacteristic
158C22×C47C2×#10411yescharacteristic
159C4×C47—21no—
160C4×C47—22no—
161C4×C47—21no—
162C4×C47—41no—
163C4×C47—41no—
164C4×C47—41no—
165C4×C47—41no—
166Dic47—81no—
167Dic47—81no—
168Dic47—81no—
169Dic47—81no—
170#2627—471no—
171C2×#343—471no—
172C2×#324D4711yescharacteristic
173#1354—472no—
174#1353—472no—
175C2×C47—21no—
176C2×C47—22no—
177C2×C47—22no—
178C2×C47—21no—
179C2×C47C2×#32811yescharacteristic
180C2×C47#212111yescharacteristic
181C2×C47#213311yescharacteristic
182C22×M16—941no—
183C2×C22⋊Q8—471no—
184C22×C4⋊C4C2×D4711yescharacteristic
185C2×C22⋊C8—21no—
186#400—471no—
187#400—471no—
188#398—941no—
189#398—941no—
190#395—471no—
191#395—471no—
192#385—941no—
193#381—941no—
194#380—471no—
195#343—472no—
196#343—941no—
197#324—21no—
198#324C2×D4712yes—
199C47#1172211yescharacteristic, complemented, Sylow
200C22×Q8—471no—
201C23×C4C4×D4711yescharacteristic
202C23×C4D18811yescharacteristic
203C23×C4C22⋊D4711yescharacteristic
204C2×M16—944no—
205C2×M16—942no—
206C2×M16—1881no—
207C2×M16—1882no—
208C22×C8—41no—
209C22×C8—1881no—
210C22⋊Q8—1881no—
211C22⋊Q8—1881no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—471no—
215C2×C4⋊C4—941no—
216C2×C4⋊C4—941no—
217C2×C4⋊C4—941no—
218C2×C4⋊C4—941no—
219C2×C4⋊C4C4×D4711yescharacteristic
220C2×C4⋊C4D18811yescharacteristic
221C2×C4⋊C4C22⋊D4711yescharacteristic
222C2×C4⋊C4C22×D4711yescharacteristic
223C2×C22⋊C4—941no—
224C2×C22⋊C4—941no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227#95—471no—
228#95—471no—
229#95—941no—
230#95—941no—
231#95—941no—
232#95—941no—
233#95—941no—
234#95—941no—
235C24(C2×D47)⋊C411yescharacteristic
236C2×Q8—3761no—
237C2×Q8—1881no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—941no—
248C22×C4—941no—
249C22×C4—941no—
250C22×C4—941no—
251C22×C4#1062532512yes—
252C22×C4Dic18812yes—
253C22×C4(C2×D47)⋊C411yescharacteristic
254C22×C4(C2×D47)⋊C411yescharacteristic
255C22×C4(C2×D47)⋊C411yescharacteristic
256C22×C4C2×C4×D4711yescharacteristic
257C22×C4C2×D18811yescharacteristic
258C22×C4C2×C22⋊D4711yescharacteristic
259M16—944no—
260M16—1884no—
261M16—1882no—
262C2×C8—42no—
263C2×C8—82no—
264C2×C8—1882no—
265C2×C8—1884no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—3761no—
271C4⋊C4—3761no—
272C4⋊C4—3761no—
273C4⋊C4—1881no—
274C4⋊C4—1882no—
275C22⋊C4—1881no—
276C22⋊C4—1881no—
277C22⋊C4—1882no—
278C23—41no—
279C23—41no—
280C23#1309393811yescharacteristic
281C23#1309393811yescharacteristic
282C23#1309394912yes—
283C23#1309395012yes—
284C23C2×(C2×D47)⋊C411yescharacteristic
285Q8—3762no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—1881no—
307C2×C4—1881no—
308C2×C4—1881no—
309C2×C4—1881no—
310C2×C4—1881no—
311C2×C4—1881no—
312C2×C4—1882no—
313C2×C4—1881no—
314C2×C4—1881no—
315C2×C4—1881no—
316C2×C4—1881no—
317C2×C4#1309394912yes—
318C2×C4C2×#1062532511yescharacteristic
319C2×C4C2×Dic18811yescharacteristic
320C8—82no—
321C8—1884no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22#1568886112yes—
331C22#1568886112yes—
332C22C2×#1309393811yescharacteristic
333C22C2×#1309394911yescharacteristic
334C22C2×#1309395011yescharacteristic
335C4—21no—
336C4—22no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—3761no—
343C4—3761no—
344C4—3761no—
345C4—3761no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—22no—
350C2C2×#1568886111yescharacteristic
351C2#2024503211yescharacteristic
352C2#2024507511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC2×C47×#324C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenterC2×#324D47
hypocenterC47#11722
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #20241419C2semidirect
4. #20241416C2semidirect
199. C47#11722semidirect

Maximal subgroups

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

subgroupindex
#202492012
C2×#156888922
#202414192
#202414162
C2×C47×#3242
#1172247