Finite Group Databasebeta

← #26108828

Subgroups — #26108828

Order 12032.

648 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 131 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#20244354C211yesmaximal, characteristic, complemented
2#20242015C211yesmaximal, characteristic, complemented
3#20241926C212yesmaximal, complemented
4#20241848C212yesmaximal, complemented
5C47×#3553C211yesmaximal, characteristic
6#15689180C2212yes—
7C2×C4×C47⋊C8C2211yescharacteristic
8#15688989C2211yescharacteristic
9#15688885—22no—
10#15688865—22no—
11#15688860—22no—
12#15688857—22no—
13Q8⋊(C47⋊C8)—22no—
14D4⋊(C47⋊C8)—22no—
15C47×#564C411yescharacteristic, complemented
16C47×#563C412yescomplemented
17C47×#562C411yescharacteristic, complemented
18C47×#554—22no—
19C47×#553—22no—
20C47×#552—22no—
21C47×#541—22no—
22C47×#539—22no—
23C47×#539—22no—
24C47×#529C2212yes—
25C2×#125×C47C2211yescharacteristic
26C22×C47⋊C8—21no—
27C2×C4⋊Dic47—21no—
28C4⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30#13093935—22no—
31C4×C47⋊C8—22no—
32C47⋊#95—22no—
33C4⋊Q8×C47—22no—
34#126×C47—22no—
35#126×C47—22no—
36#126×C47—22no—
37#126×C47—22no—
38#126×C47—22no—
39#125×C47—22no—
40#125×C47—22no—
41#125×C47—22no—
42#125×C47—22no—
43#125×C47—22no—
44#125×C47—22no—
45#125×C47C2×C411yescharacteristic
46#125×C47C2×C411yescharacteristic
47#125×C47C2×C412yes—
48#125×C47D412yes—
49#124×C47C2×C412yes—
50#123×C47—22no—
51#123×C47—22no—
52#123×C47—22no—
53#123×C47—22no—
54#123×C47—22no—
55C22⋊Q8×C47—22no—
56C22⋊Q8×C47—22no—
57C22⋊Q8×C47—22no—
58C22⋊Q8×C47—22no—
59C22⋊Q8×C47—22no—
60C22⋊Q8×C47—22no—
61C22⋊Q8×C47D412yes—
62#121×C47—22no—
63#121×C47—22no—
64#121×C47D412yes—
65C4×Q8×C47—22no—
66C4×Q8×C47D412yes—
67C4×D4×C47—22no—
68C4×D4×C47—22no—
69C4×D4×C47—22no—
70C4×D4×C47—22no—
71C4×D4×C47D412yes—
72#117×C47—22no—
73#117×C47—22no—
74C2×C4⋊C4×C47—22no—
75C2×C4⋊C4×C47—22no—
76C2×C4⋊C4×C47D412yes—
77C2×C42×C47C2311yescharacteristic
78C2×Q8×C47—22no—
79C2×Q8×C47C22⋊C412yes—
80C2×D4×C47—22no—
81C2×D4×C47—22no—
82C2×D4×C47C22⋊C412yes—
83C22×C4×C47—22no—
84C22×C4×C47—22no—
85C22×C4×C47—22no—
86C22×C4×C47—22no—
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D412yes—
89C22×Dic47—41no—
90C4⋊C4×C47—22no—
91C4⋊C4×C47—22no—
92C4⋊C4×C47—22no—
93C4⋊C4×C47—22no—
94C4⋊C4×C47—22no—
95C4⋊C4×C47—22no—
96C4⋊C4×C47—22no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—22no—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47C22⋊C412yes—
106C4⋊C4×C47C22⋊C412yes—
107C4⋊C4×C47C22⋊C412yes—
108C4⋊C4×C47C22⋊C412yes—
109C22⋊C4×C47—22no—
110C22⋊C4×C47—22no—
111C22⋊C4×C47—22no—
112C22⋊C4×C47—22no—
113C22⋊C4×C47—22no—
114C22⋊C4×C47—22no—
115C22⋊C4×C47—22no—
116C22⋊C4×C47—22no—
117C22⋊C4×C47—22no—
118C42×C47—22no—
119C42×C47—22no—
120C42×C47—22no—
121C42×C47C22×C411yescharacteristic
122C42×C47C2×D411yescharacteristic
123C42×C47C2×D412yes—
124C4⋊Dic47—42no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—41no—
127C2×C47⋊C8—41no—
128C2×C47⋊C8—42no—
129C23×C47—22no—
130C23×C47#12011yescharacteristic
131Q8×C47—42no—
132Q8×C47—82no—
133D4×C47—42no—
134D4×C47—42no—
135D4×C47—42no—
136D4×C47—82no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—22no—
140C2×C4×C47—22no—
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47—42no—
145C2×C4×C47—82no—
146C2×C4×C47—82no—
147C2×C4×C47—82no—
148C2×C4×C47—82no—
149C2×C4×C47C2×C22⋊C411yescharacteristic
150C2×C4×C47C2×C22⋊C412yes—
151C2×C4×C47#12012yes—
152C2×C4×C47#12011yescharacteristic
153C2×Dic47—82no—
154C2×Dic47—81no—
155C47⋊C8—42no—
156C47⋊C8—82no—
157#8529—471nomaximal, Sylow
158C22×C47—22no—
159C22×C47—42no—
160C22×C47—42no—
161C22×C47—42no—
162C22×C47#37911yescharacteristic
163C22×C47#45411yescharacteristic
164C22×C47#45611yescharacteristic
165C4×C47—21no—
166C4×C47—21no—
167C4×C47—22no—
168C4×C47—22no—
169C4×C47—42no—
170C4×C47—42no—
171C4×C47—42no—
172C4×C47—42no—
173C4×C47—42no—
174Dic47—161no—
175#3553D4711yescharacteristic
176#1991—471no—
177#1534—471no—
178#1512—472no—
179#1494—472no—
180C2×C47—21no—
181C2×C47—21no—
182C2×C47—82no—
183C2×C47#179311yescharacteristic
184C2×C47#179412yes—
185#564Dic4711yescharacteristic, complemented
186#563Dic4712yescomplemented
187#562Dic4711yescharacteristic, complemented
188#554—22no—
189#553—22no—
190#552—22no—
191#541—22no—
192#539—22no—
193#539—22no—
194#529C2×D4712yes—
195C2×#125C2×D4711yescharacteristic
196#432—472no—
197C2×C4×C8—471no—
198#382—471no—
199#340—942no—
200#330—942no—
201#328—942no—
202#327—942no—
203Q8⋊C8—942no—
204D4⋊C8—942no—
205C47#852911yescharacteristic, complemented, Sylow
206C22×C8—941no—
207C4⋊Q8—22no—
208#126—22no—
209#126—22no—
210#126—22no—
211#126—22no—
212#126—22no—
213#125—22no—
214#125—22no—
215#125—22no—
216#125—22no—
217#125—22no—
218#125—22no—
219#125C2×Dic4711yescharacteristic
220#125C2×Dic4711yescharacteristic
221#125C2×Dic4712yes—
222#125C22⋊D4712yes—
223#124C2×Dic4712yes—
224#123—22no—
225#123—22no—
226#123—22no—
227#123—22no—
228#123—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8C22⋊D4712yes—
236#121—22no—
237#121—22no—
238#121C22⋊D4712yes—
239C4×Q8—22no—
240C4×Q8C22⋊D4712yes—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4C22⋊D4712yes—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—941no—
251C2×C4⋊C4C22⋊D4712yes—
252C2×C42C22×D4711yescharacteristic
253C4⋊C8—942no—
254C22⋊C8—942no—
255#97—942no—
256C4×C8—942no—
257#95—942no—
258C2×Q8—22no—
259C2×Q8C22⋊Dic4712yes—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4C22⋊Dic4712yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—1881no—
268C22×C4C2×C22⋊D4711yescharacteristic
269C22×C4C2×C22⋊D4712yes—
270C2×C8—942no—
271C2×C8—1882no—
272C2×C8—1881no—
273C2×C8—1881no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—1882no—
290C4⋊C4C22⋊Dic4712yes—
291C4⋊C4C22⋊Dic4712yes—
292C4⋊C4C22⋊Dic4712yes—
293C4⋊C4C22⋊Dic4712yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42C22×Dic4711yescharacteristic
307C42C2×C22⋊D4712yes—
308C42C2×C22⋊D4711yescharacteristic
309C23—22no—
310C23#1309399311yescharacteristic
311Q8—42no—
312Q8—82no—
313D4—42no—
314D4—42no—
315D4—42no—
316D4—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—3762no—
330C2×C4—3761no—
331C2×C4C2×C22⋊Dic4711yescharacteristic
332C2×C4C2×C22⋊Dic4712yes—
333C2×C4#1309399312yes—
334C2×C4#1309399311yescharacteristic
335C8—3762no—
336C8—1882no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22#1568897911yescharacteristic
342C22#1568929611yescharacteristic
343C22#1568930611yescharacteristic
344C4—21no—
345C4—21no—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—7521no—
354C2—21no—
355C2—21no—
356C2—82no—
357C2#2024334711yescharacteristic
358C2#2024335212yes—
subgroupG/H
centerC22#15688979
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3553C2
socleC22×C47#379
cosocleC2×C42×C47C23
hypercenter#3553D47
hypocenterC47#8529
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3553

Complements

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

NH ≅ G/Ntype
1. #20244354C2semidirect
2. #20242015C2semidirect
3. #20241926C2semidirect
4. #20241848C2semidirect
15. C47×#564C4semidirect
16. C47×#563C4semidirect
17. C47×#562C4semidirect
185. #564Dic47semidirect
186. #563Dic47semidirect
187. #562Dic47semidirect
205. C47#8529semidirect

Maximal subgroups

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

subgroupindex
#202443542
#202420152
#202419262
#202418482
C47×#35532
#852947