Finite Group Databasebeta

← #26110249

Subgroups — #26110249

Order 12032.

532 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 123 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#20243681C211yesmaximal, characteristic
2#20243565C212yesmaximal, complemented
3C4×#13093939C211yesmaximal, characteristic
4#20241235C212yesmaximal, complemented
5C47×#2973C211yesmaximal, characteristic
6C23×C47⋊C8C2211yescharacteristic
7C2×Dic47⋊C8—22no—
8C2×#13093939C412yes—
9C2×C22⋊(C47⋊C8)—21no—
10C2×C22⋊(C47⋊C8)C2211yescharacteristic
11C4×C22⋊Dic47C2212yes—
12#15688890C411yescharacteristic
13#15688890C411yescharacteristic
14#15688878—21no—
15#15688877—22no—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#435—22no—
18C47×#435—22no—
19C47×#432—21no—
20C47×#407—21no—
21C2×C22⋊C8×C47C2212yes—
22C22×C47⋊C8—21no—
23C22×C47⋊C8—22no—
24C22×C47⋊C8—21no—
25C22×C47⋊C8—42no—
26C2×C22⋊Dic47C2×C412yes—
27C2×C22⋊Dic47C2×C412yes—
28C2×C4×Dic47—22no—
29Dic47⋊C8—44no—
30#13093939—24no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—42no—
33C47⋊#95—22no—
34C22×D4×C47C2×C411yescharacteristic
35C23×C4×C47D411yescharacteristic
36C23×C4×C47C2311yescharacteristic
37C2×M16×C47—21no—
38C2×M16×C47—21no—
39C22×C8×C47D412yes—
40C22×C8×C47D412yes—
41C4×D4×C47—22no—
42C4×D4×C47—22no—
43C2×C4⋊C4×C47Q811yescharacteristic
44C2×C22⋊C4×C47C2×C411yescharacteristic
45C2×C22⋊C4×C47Q811yescharacteristic
46C2×C42×C47D411yescharacteristic
47C4⋊C8×C47—21no—
48C4⋊C8×C47—21no—
49C22⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C22⋊C8×C47—22no—
52#97×C47—22no—
53C24×C47C4⋊C411yescharacteristic
54C24×C47C22×C411yescharacteristic
55C2×D4×C47—22no—
56C2×D4×C47—22no—
57C2×D4×C47—21no—
58C2×D4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47C4⋊C411yescharacteristic
66C22×C4×C47C4⋊C411yescharacteristic
67C22×C4×C47C4⋊C411yescharacteristic
68C22×C4×C47C2×D412yes—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×Q811yescharacteristic
71C22×C4×C47#4112yes—
72C22×Dic47—22no—
73C22×Dic47—22no—
74M16×C47—81no—
75M16×C47—81no—
76C2×C8×C47—22no—
77C2×C8×C47—22no—
78C2×C8×C47—42no—
79C2×C8×C47—42no—
80C4⋊C4×C47—22no—
81C22⋊C4×C47—22no—
82C22⋊C4×C47—22no—
83C42×C47—21no—
84C42×C47—21no—
85C22⋊Dic47—22no—
86C22⋊Dic47—22no—
87C22⋊Dic47—22no—
88C22⋊Dic47M1614yescomplemented
89C4×Dic47—42no—
90C4×Dic47—42no—
91C2×C47⋊C8—22no—
92C2×C47⋊C8—24no—
93C2×C47⋊C8—41no—
94C2×C47⋊C8—42no—
95C2×C47⋊C8—41no—
96C2×C47⋊C8—41no—
97C2×C47⋊C8—41no—
98C2×C47⋊C8—41no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47C2×C4⋊C411yescharacteristic
107C23×C47C4×D412yes—
108C23×C47C2×M1612yes—
109C23×C47#13111yescharacteristic
110C23×C47#13111yescharacteristic
111D4×C47—42no—
112D4×C47—42no—
113C2×C4×C47—21no—
114C2×C4×C47—22no—
115C2×C4×C47—21no—
116C2×C4×C47—22no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—81no—
129C2×C4×C47#12011yescharacteristic
130C2×C4×C47C22⋊Q812yes—
131C2×C4×C47#12311yescharacteristic
132C2×Dic47—24no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C8×C47—42no—
139C8×C47—42no—
140C47⋊C8—42no—
141C47⋊C8—81no—
142#8852—471nomaximal, Sylow
143C22×C47—22no—
144C22×C47—21no—
145C22×C47—22no—
146C22×C47—21no—
147C22×C47—42no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—42no—
151C22×C47—41no—
152C22×C47—41no—
153C22×C47—41no—
154C22×C47—81no—
155C22×C47#38511yescharacteristic
156C22×C47#42311yescharacteristic
157C22×C47#42411yescharacteristic
158C22×C47#43512yes—
159C22×C47#43612yes—
160C4×C47—21no—
161C4×C47—22no—
162C4×C47—41no—
163C4×C47—41no—
164C4×C47—41no—
165C4×C47B(4,𝔽2)11yescharacteristic
166C4×C47#45811yescharacteristic
167Dic47—44no—
168Dic47—82no—
169#2973D4711yescharacteristic
170#1869—471no—
171#1850—472no—
172C4×#99—471no—
173#1282—472no—
174C2×C47—22no—
175C2×C47—22no—
176C2×C47—21no—
177C2×C47—21no—
178C2×C47—81no—
179C2×C47#185111yescharacteristic
180C2×C47#186711yescharacteristic
181C2×C47#186811yescharacteristic
182C23×C8—471no—
183C2×C4×D4C2×D4711yescharacteristic
184#435—22no—
185#435—22no—
186#432—21no—
187C2×C4⋊C8—942no—
188C2×#99—472no—
189#407—21no—
190C2×C22⋊C8—471no—
191C2×C22⋊C8—941no—
192C2×C22⋊C8C2×D4712yes—
193C4×C22⋊C4—472no—
194#342—471no—
195#342—471no—
196#336—942no—
197#336—941no—
198C47#885211yescharacteristic, complemented, Sylow
199C22×D4C4×D4711yescharacteristic
200C23×C4C22⋊D4711yescharacteristic
201C23×C4C22×D4711yescharacteristic
202C2×M16—21no—
203C2×M16—21no—
204C22×C8—941no—
205C22×C8—942no—
206C22×C8—941no—
207C22×C8—1882no—
208C22×C8C22⋊D4712yes—
209C22×C8C22⋊D4712yes—
210C4×D4—22no—
211C4×D4—22no—
212C2×C4⋊C4Dic9411yescharacteristic
213C2×C22⋊C4—472no—
214C2×C22⋊C4—472no—
215C2×C22⋊C4Dic9411yescharacteristic
216C2×C22⋊C4C4×D4711yescharacteristic
217C2×C42—942no—
218C2×C42C22⋊D4711yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—1884no—
222#99—944no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—942no—
227C22⋊C8—1882no—
228#97—22no—
229#95—942no—
230C24Dic47⋊C411yescharacteristic
231C24C2×C4×D4711yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—21no—
235C2×D4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—942no—
243C22×C4—942no—
244C22×C4Dic47⋊C411yescharacteristic
245C22×C4Dic47⋊C411yescharacteristic
246C22×C4Dic47⋊C411yescharacteristic
247C22×C4C2×Dic9411yescharacteristic
248C22×C4#1062535012yes—
249C22×C4C2×C22⋊D4712yes—
250C22×C4C2×C22⋊D4711yescharacteristic
251M16—81no—
252M16—81no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—942no—
258C2×C8—944no—
259C2×C8—1881no—
260C2×C8—1881no—
261C2×C8—1881no—
262C2×C8—1882no—
263C2×C8—1881no—
264C2×C8—1881no—
265C4⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—474no—
269C22⋊C4—942no—
270C22⋊C4—942no—
271C22⋊C4—942no—
272C42—21no—
273C42—21no—
274C42—1882no—
275C42—1882no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×Dic47⋊C411yescharacteristic
284C23C4×C22⋊D4712yes—
285C23M16×D4712yes—
286C23#1309403211yescharacteristic
287C23#1309403211yescharacteristic
288D4—42no—
289D4—42no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—944no—
307C2×C4—1882no—
308C2×C4—1882no—
309C2×C4—1882no—
310C2×C4—1882no—
311C2×C4—1882no—
312C2×C4#1309399311yescharacteristic
313C2×C4#1309400312yes—
314C2×C4#1309400711yescharacteristic
315C8—42no—
316C8—42no—
317C8—3761no—
318C8—1882no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1568899611yescharacteristic
332C22#1568914611yescharacteristic
333C22#1568915111yescharacteristic
334C22#1568919212yes—
335C22#1568919912yes—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—3762no—
342C4—1884no—
343C4#1568931011yescharacteristic
344C4#1568931311yescharacteristic
345C2—22no—
346C2—22no—
347C2—21no—
348C2—21no—
349C2—81no—
350C2#2024357111yescharacteristic
351C2#2024366911yescharacteristic
352C2#2024367411yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2973C2
socleC22×C47#385
prime radicalC22×D4×C47C2×C4
cosocleC23×C4×C47C23
hypercenter#2973D47
hypocenterC47#8852
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20243565C2semidirect
4. #20241235C2semidirect
88. C22⋊Dic47M16semidirect
198. C47#8852semidirect

Maximal subgroups

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

subgroupindex
#202436812
#202435652
C4×#130939392
#202412352
C47×#29732
#885247