Finite Group Databasebeta

← #26127069

Subgroups — #26127069

Order 12032.

414 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 87 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#20243376C211yesmaximal, characteristic
2#20241729C211yesmaximal, characteristic
3#20241409C212yesmaximal, complemented
4#20241388C212yesmaximal, complemented
5C47×#2638C211yesmaximal, characteristic
6#15689180—21no—
7C2×#13093939—21no—
8#15688983—21no—
9#15688890C2212yes—
10#15688847—21no—
11#15688847C2212yes—
12C2×C22⋊Q8×C47C411yescharacteristic, complemented
13C2×C4×D4×C47C2211yescharacteristic
14C47×#398C411yescharacteristic, complemented
15C47×#397—21no—
16C47×#397C2211yescharacteristic
17C47×#395C411yescharacteristic, complemented
18C47×#394—21no—
19C47×#393C2211yescharacteristic
20C47×#387—21no—
21C47×#385—21no—
22C47×#381C411yescharacteristic, complemented
23C2×C4⋊Dic47—41no—
24C2×C22⋊Dic47—22no—
25C4⋊(C47⋊C8)—41no—
26#13093939—42no—
27C22⋊(C47⋊C8)—22no—
28#13093935—41no—
29C47⋊#95—41no—
30C22×Q8×C47C2×C411yescharacteristic
31C22×D4×C47D411yescharacteristic
32C23×C4×C47C2×C411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C22⋊Q8×C47—41no—
35C22⋊Q8×C47—41no—
36C4×D4×C47—41no—
37C4×D4×C47—41no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47D411yescharacteristic
48C2×C22⋊C4×C47C2311yescharacteristic
49C2×C42×C47D411yescharacteristic
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47C2×C411yescharacteristic
56#95×C47C2×C411yescharacteristic
57C24×C47C22⋊C411yescharacteristic
58C24×C47C2×D411yescharacteristic
59C2×Q8×C47—41no—
60C2×Q8×C47—81no—
61C2×D4×C47—21no—
62C2×D4×C47—21no—
63C2×D4×C47—41no—
64C2×D4×C47—41no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—81no—
74C22×C4×C47—81no—
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×Dic47—42no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—42no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—81no—
87C22⋊C4×C47—41no—
88C22⋊C4×C47—41no—
89C22⋊C4×C47—42no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C22⋊C4×C47—81no—
93C22⋊C4×C47—81no—
94C42×C47—21no—
95C42×C47—21no—
96C22⋊Dic47—42no—
97C22⋊Dic47—42no—
98C4⋊Dic47—82no—
99C2×C47⋊C8—42no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47C2×C22⋊C411yescharacteristic
110C23×C47AGL(1,ℤ/8)11yescharacteristic
111C23×C47#13711yescharacteristic
112Q8×C47—82no—
113D4×C47—42no—
114D4×C47—42no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—41no—
122C2×C4×C47—42no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—81no—
139C2×C4×C47—81no—
140C2×C4×C47—81no—
141C2×C4×C47#9911yescharacteristic
142C2×C4×C47#9911yescharacteristic
143C2×Dic47—82no—
144C2×Dic47—82no—
145C2×Dic47—82no—
146C47⋊C8—82no—
147#11826—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—41no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47—81no—
160C22×C47—81no—
161C22×C47—81no—
162C22×C47C2×#9911yescharacteristic
163C22×C47#41711yescharacteristic
164C22×C47#42111yescharacteristic
165C4×C47—21no—
166C4×C47—21no—
167C4×C47—41no—
168C4×C47—41no—
169C4×C47—41no—
170C4×C47—41no—
171C4×C47—81no—
172C4×C47—81no—
173C4×C47—81no—
174C4×C47—81no—
175Dic47—162no—
176#2638D4711yescharacteristic
177#1799—471no—
178#1466—471no—
179#1351—472no—
180#1344—472no—
181C2×C47—21no—
182C2×C47—21no—
183C2×C47—41no—
184C2×C47—41no—
185C2×C47—81no—
186C2×C47#151311yescharacteristic
187C2×C47#212111yescharacteristic
188C2×C47#212911yescharacteristic
189C2×C22⋊Q8Dic4711yescharacteristic, complemented
190C2×C4×D4C2×D4711yescharacteristic
191#432—941no—
192C2×#99—941no—
193#398Dic4711yescharacteristic, complemented
194#397—21no—
195#397C2×D4711yescharacteristic
196#395Dic4711yescharacteristic, complemented
197#394—21no—
198#393C2×D4711yescharacteristic
199#387—21no—
200#385—21no—
201#381Dic4711yescharacteristic, complemented
202#380—941no—
203#342—472no—
204#323—472no—
205#323—941no—
206C47#1182611yescharacteristic, complemented, Sylow
207C22×Q8C2×Dic4711yescharacteristic
208C22×D4C22⋊D4711yescharacteristic
209C23×C4C2×Dic4711yescharacteristic
210C23×C4C22⋊D4711yescharacteristic
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—1881no—
219C2×C4⋊C4C2×Dic4711yescharacteristic
220C2×C4⋊C4C2×Dic4711yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—942no—
226C2×C22⋊C4C22⋊D4711yescharacteristic
227C2×C22⋊C4C22×D4711yescharacteristic
228C2×C42C22⋊D4711yescharacteristic
229C4⋊C8—1881no—
230#99—1882no—
231C22⋊C8—942no—
232#97—1881no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—1881no—
239#95C2×Dic4711yescharacteristic
240#95C2×Dic4711yescharacteristic
241C24C22⋊Dic4711yescharacteristic
242C24C2×C22⋊D4711yescharacteristic
243C2×Q8—41no—
244C2×Q8—81no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—1882no—
260C22×C4C22⋊Dic4711yescharacteristic
261C22×C4C22⋊Dic4711yescharacteristic
262C22×C4C22⋊Dic4711yescharacteristic
263C22×C4C22×Dic4711yescharacteristic
264C22×C4C2×C22⋊D4711yescharacteristic
265C2×C8—1882no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—81no—
272C4⋊C4—3762no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C22⋊C4—1882no—
281C22⋊C4—1882no—
282C42—21no—
283C42—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23C2×C22⋊Dic4711yescharacteristic
294C23#1309404911yescharacteristic
295C23#1309405411yescharacteristic
296Q8—82no—
297D4—42no—
298D4—42no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—3762no—
326C2×C4—3762no—
327C2×C4—3762no—
328C2×C4#1309393911yescharacteristic
329C2×C4#1309393911yescharacteristic
330C8—3762no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#1309393911yescharacteristic
346C22#1568912411yescharacteristic
347C22#1568914011yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—7522no—
359C2—21no—
360C2—21no—
361C2—41no—
362C2—41no—
363C2—81no—
364C2#2024193111yescharacteristic
365C2#2024503411yescharacteristic
366C2#2024506311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2638C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2638D47
hypocenterC47#11826
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2638

Complements

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

NH ≅ G/Ntype
3. #20241409C2semidirect
4. #20241388C2semidirect
12. C2×C22⋊Q8×C47C4semidirect
14. C47×#398C4semidirect
17. C47×#395C4semidirect
22. C47×#381C4semidirect
189. C2×C22⋊Q8Dic47semidirect
193. #398Dic47semidirect
196. #395Dic47semidirect
201. #381Dic47semidirect
206. C47#11826semidirect

Maximal subgroups

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

subgroupindex
#202433762
#202417292
#202414092
#202413882
C47×#26382
#1182647