Finite Group Databasebeta

← #26113034

Subgroups — #26113034

Order 12032.

488 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 113 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#20253187C211yesmaximal, characteristic
2#20252411C211yesmaximal, characteristic
3#20243399C211yesmaximal, characteristic
4#20241953C211yesmaximal, characteristic, complemented
5#20241790C211yesmaximal, characteristic, complemented
6#20241248C211yesmaximal, characteristic, complemented
7C47×#1301C211yesmaximal, characteristic, complemented
8#15689680C2211yescharacteristic
9C2×C4×D188C2211yescharacteristic
10#15689424—22no—
11#15689414C412yescomplemented
12#15689404C412yescomplemented
13C8×D188—22no—
14#15689188—22no—
15#15689176C2211yescharacteristic
16#15689165—21no—
17#15689129—21no—
18C2×#13093946—21no—
19C22⋊C8×D47—22no—
20#15688989C2211yescharacteristic
21#15688886—21no—
22#15688861—22no—
23#15688853—22no—
24C47×#431C2211yescharacteristic
25C47×#423C2211yescharacteristic
26C2×C4⋊C8×C47C2211yescharacteristic
27C22×D188D411yescharacteristic
28C22×C4×D47—21no—
29C2×C8×D47—24no—
30#13094024C2×C411yescharacteristic
31#13094024C2×C411yescharacteristic
32#13094016—21no—
33#13094003—21no—
34#13094003—21no—
35C4×Dic94—21no—
36C4×Dic94—21no—
37C4×D188—21no—
38C4×D188—21no—
39C4×D188C2×C412yes—
40C4×D188D412yes—
41#13093983—21no—
42C2×C4⋊Dic47—21no—
43C2×C4⋊Dic47D411yescharacteristic
44C2×(C2×D47)⋊C4—21no—
45#13093958—22no—
46#13093956—22no—
47C4⋊(C47⋊C8)—21no—
48#13093946—22no—
49#13093946—22no—
50#13093946—42no—
51C22⋊(C47⋊C8)—21no—
52(C2×D47)⋊C8—24no—
53C4×C47⋊C8D412yes—
54C47⋊#95—21no—
55C47⋊#95—21no—
56C2×M16×C47—21no—
57C22×C8×C47—21no—
58C2×C42×C47C2311yescharacteristic
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47C2×C412yes—
64C22⋊C8×C47D412yes—
65C4×C8×C47—21no—
66C23×D47—21no—
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47#4111yescharacteristic
70C22×Dic47—21no—
71C22×Dic47—41no—
72C2×D188—21no—
73C2×D188—21no—
74C2×D188—21no—
75C2×D188C22⋊C412yes—
76C2×D188SD1612yes—
77C2×D188Q1612yes—
78C2×C4×D47—22no—
79C2×C4×D47—22no—
80C2×C4×D47—82no—
81C2×Dic94—21no—
82C2×Dic94—21no—
83M16×C47—41no—
84M16×C47—41no—
85C2×C8×C47—22no—
86C2×C8×C47—22no—
87C2×C8×C47—41no—
88C2×C8×C47—41no—
89C2×C8×C47—41no—
90C42×C47C22×C411yescharacteristic
91C42×C47C2×D411yescharacteristic
92C42×C47C2×D411yescharacteristic
93C42×C47#4111yescharacteristic
94C22⋊Dic47—41no—
95(C2×D47)⋊C4—22no—
96(C2×D47)⋊C4—22no—
97C4⋊Dic47—21no—
98C4⋊Dic47—21no—
99C4⋊Dic47—21no—
100C4⋊Dic47—21no—
101C4⋊Dic47—21no—
102C4⋊Dic47—41no—
103C4⋊Dic47—41no—
104C4⋊Dic47C22⋊C412yes—
105Dic47⋊C4—41no—
106Dic47⋊C4—41no—
107C4×Dic47—41no—
108C2×C47⋊C8—22no—
109C8×D47—84no—
110C23×C47#12111yescharacteristic
111C22×D47—22no—
112C22×D47—22no—
113C22×D47—22no—
114C22×D47—81no—
115C22×D47—81no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47C2×C22⋊C411yescharacteristic
119C2×C4×C47C4×D411yescharacteristic
120C2×C4×C47C4×D411yescharacteristic
121C2×C4×C47#12011yescharacteristic
122C2×C4×C47#12111yescharacteristic
123C2×C4×C47#12311yescharacteristic
124C2×C4×C47C2×SD1611yescharacteristic
125C2×C4×C47C2×Q1611yescharacteristic
126C2×C4×C47#13511yescharacteristic
127C2×C4×C47#13511yescharacteristic
128C2×Dic47—22no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—81no—
132C2×Dic47—81no—
133C2×Dic47—81no—
134C2×Dic47—81no—
135D188—22no—
136D188—22no—
137D188—22no—
138D188#10314yes—
139C4×D47—44no—
140C4×D47—82no—
141Dic94—42no—
142Dic94—81no—
143C8×C47—42no—
144C8×C47—42no—
145C8×C47—81no—
146C47⋊C8—42no—
147#9303—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47#38611yescharacteristic
151C22×C47#46511yescharacteristic
152C22×C47#46711yescharacteristic
153C2×D47—24no—
154C2×D47—42no—
155C2×D47—42no—
156C2×D47—82no—
157C2×D47—82no—
158C4×C47—21no—
159C4×C47—21no—
160C4×C47—21no—
161C4×C47—21no—
162C4×C47C2×#10311yescharacteristic
163C4×C47#41611yescharacteristic
164C4×C47#46511yescharacteristic
165C4×C47#46711yescharacteristic
166Dic47—42no—
167Dic47—81no—
168Dic47—81no—
169Dic47—161no—
170#3077—471no—
171#2972—471no—
172#1804—471no—
173#1518—471no—
174#1481—471no—
175#1301D4711yescharacteristic, complemented
176#1286—471no—
177C2×C47—21no—
178C2×C47—21no—
179C2×C47#188111yescharacteristic
180C2×C47#189211yescharacteristic
181C2×C47#190411yescharacteristic
182D47—44no—
183#533—471no—
184C2×C4×D4—471no—
185#478—942no—
186#476—472no—
187#474—472no—
188C8×D4—942no—
189C8×D4—942no—
190#431—471no—
191#431C2×D4711yescharacteristic
192#428—941no—
193#423C2×D4711yescharacteristic
194C2×C4⋊C8C2×D4711yescharacteristic
195#418—941no—
196C2×#102—941no—
197C2×C22⋊C8—942no—
198#382—471no—
199#340—941no—
200#328—942no—
201Q8⋊C8—942no—
202C47#930311yescharacteristic, complemented, Sylow
203C22×D4—471no—
204C23×C4—941no—
205C2×M16—21no—
206C22×C8—21no—
207C22×C8—944no—
208C4⋊Q8—471no—
209C4⋊Q8—471no—
210#125—941no—
211C22⋊Q8—941no—
212C22⋊Q8—941no—
213C4×Q8—941no—
214C4×Q8—941no—
215C4×D4—472no—
216C4×D4—472no—
217C4×D4—941no—
218C4×D4—941no—
219#117—941no—
220C2×C4⋊C4—471no—
221C2×C4⋊C4—941no—
222C2×C22⋊C4—941no—
223C2×C42C22×D4711yescharacteristic
224#107—942no—
225#106—942no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—941no—
231C4⋊C8C4×D4712yes—
232#102—942no—
233#102—942no—
234#102—1882no—
235C22⋊C8—944no—
236C22⋊C8—941no—
237C22⋊C8D18812yes—
238C4×C8—21no—
239C4×C8—472no—
240#95—941no—
241#95—941no—
242C24—941no—
243C2×Q8—941no—
244C2×Q8—941no—
245C2×D4—472no—
246C2×D4—472no—
247C2×D4—472no—
248C2×D4—941no—
249C2×D4—941no—
250C2×D4—941no—
251C22×C4—941no—
252C22×C4—942no—
253C22×C4—942no—
254C22×C4—3762no—
255C22×C4—1881no—
256C22×C4C2×D18811yescharacteristic
257C22×C4D4×D4711yescharacteristic
258C22×C4#1062535411yescharacteristic
259M16—41no—
260M16—41no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—942no—
267C2×C8—3764no—
268C4⋊C4—472no—
269C4⋊C4—941no—
270C4⋊C4—941no—
271C4⋊C4—941no—
272C4⋊C4—941no—
273C4⋊C4—941no—
274C4⋊C4—1881no—
275C4⋊C4—1881no—
276C4⋊C4—1881no—
277C4⋊C4—1881no—
278C22⋊C4—942no—
279C22⋊C4—942no—
280C22⋊C4—1881no—
281C42—1881no—
282C42C2×C4×D4711yescharacteristic
283C42#1062535011yescharacteristic
284C42D4×D4711yescharacteristic
285C42D4×D4711yescharacteristic
286C23—942no—
287C23—942no—
288C23—942no—
289C23—3761no—
290C23—3761no—
291C23#1309399911yescharacteristic
292Q8—3761no—
293Q8—1882no—
294D4—474no—
295D4—942no—
296D4—942no—
297D4—942no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—942no—
301C2×C4—3762no—
302C2×C4—3761no—
303C2×C4—3761no—
304C2×C4—3761no—
305C2×C4—3761no—
306C2×C4—1884no—
307C2×C4—1881no—
308C2×C4—1881no—
309C2×C4C22⋊C4×D4711yescharacteristic
310C2×C4#1309398611yescharacteristic
311C2×C4C4×D18811yescharacteristic
312C2×C4#1309399411yescharacteristic
313C2×C4#1309399711yescharacteristic
314C2×C4#1309400911yescharacteristic
315C2×C4SD16×D4711yescharacteristic
316C2×C4Q16×D4711yescharacteristic
317C2×C4#1309404311yescharacteristic
318C2×C4#1309404511yescharacteristic
319C8—42no—
320C8—42no—
321C8—81no—
322C8—1882no—
323C22—21no—
324C22—21no—
325C22—944no—
326C22—3762no—
327C22—3762no—
328C22—1882no—
329C22—1882no—
330C22#1568900111yescharacteristic
331C22#1568935311yescharacteristic
332C22#1568936511yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—3761no—
338C4—3761no—
339C4—1882no—
340C4—7521no—
341C4#103×D4711yescharacteristic
342C4#1568911611yescharacteristic
343C4#1568935111yescharacteristic
344C4#1568936311yescharacteristic
345C2—21no—
346C2—21no—
347C2—1884no—
348C2#2024375111yescharacteristic
349C2#2024381011yescharacteristic
350C2#2024387411yescharacteristic
subgroupG/H
centerC22#15689001
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1301C2
socleC22×C47#386
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1301D47
hypocenterC47#9303
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
4. #20241953C2semidirect
5. #20241790C2semidirect
6. #20241248C2semidirect
7. C47×#1301C2semidirect
11. #15689414C4semidirect
12. #15689404C4semidirect
175. #1301D47semidirect
202. C47#9303semidirect

Maximal subgroups

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

subgroupindex
#202531872
#202524112
#202433992
#202419532
#202417902
#202412482
C47×#13012
#930347