Finite Group Databasebeta

← #26134958

Subgroups — #26134958

Order 12032.

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

normal subgroups only

This group has 492 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#20243376C211yesmaximal, characteristic
2C4×#13093939C211yesmaximal, characteristic
3#20241513C212yesmaximal
4#20241513C212yesmaximal
5C47×#2566C211yesmaximal, characteristic
6C2×#13093939C2212yes—
7C2×#13093939C2212yes—
8#15688983—21no—
9C4×C22⋊Dic47—21no—
10#15688890—21no—
11#15688890—21no—
12C2×#125×C47C411yescharacteristic, complemented
13C2×C4×D4×C47C2211yescharacteristic
14C47×#396—21no—
15C47×#396—21no—
16C47×#390C2211yescharacteristic
17C47×#390C2211yescharacteristic
18C47×#385—21no—
19C47×#385—21no—
20C47×#384C411yescharacteristic, complemented
21C47×#382C411yescharacteristic, complemented
22C4×C4⋊C4×C47C411yescharacteristic, complemented
23C2×C4⋊Dic47—41no—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—22no—
26C2×C4×Dic47—41no—
27#13093939—22no—
28#13093939—24no—
29#13093939Q814yes—
30C47⋊#95—41no—
31C47⋊#95—41no—
32C22×D4×C47C2311yescharacteristic
33C23×C4×C47D411yescharacteristic
34C23×C4×C47D411yescharacteristic
35#125×C47—41no—
36#125×C47—41no—
37C4×D4×C47—41no—
38C4×D4×C47—41no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47D411yescharacteristic
53C2×C42×C47C2×C411yescharacteristic
54C2×C42×C47C2×C411yescharacteristic
55C2×C42×C47C2×C411yescharacteristic
56#95×C47—21no—
57#95×C47—21no—
58#95×C47—21no—
59C24×C47C2×D411yescharacteristic
60C24×C47C2×D411yescharacteristic
61C2×D4×C47—41no—
62C2×D4×C47—41no—
63C2×D4×C47C2×Q812yes—
64C2×D4×C47#4112yes—
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×C47C22×C411yescharacteristic
80C22×Dic47—42no—
81C22×Dic47—42no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—42no—
84C4⋊C4×C47—42no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—81no—
90C4⋊C4×C47—81no—
91C22⋊C4×C47—41no—
92C22⋊C4×C47—41no—
93C22⋊C4×C47—42no—
94C22⋊C4×C47—42no—
95C22⋊C4×C47—81no—
96C22⋊C4×C47—81no—
97C42×C47—21no—
98C42×C47—21no—
99C42×C47—41no—
100C42×C47—41no—
101C22⋊Dic47—24no—
102C22⋊Dic47—42no—
103C22⋊Dic47—42no—
104C22⋊Dic47—42no—
105C4⋊Dic47—82no—
106C4×Dic47—82no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47C2×C22⋊C411yescharacteristic
116C23×C47C22⋊Q812yes—
117C23×C47#12411yescharacteristic
118C23×C47C4⋊Q811yescharacteristic
119D4×C47—42no—
120D4×C47—42no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—22no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—41no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
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—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—81no—
145C2×C4×C47—81no—
146C2×C4×C47#9911yescharacteristic
147C2×C4×C47#9911yescharacteristic
148C2×C4×C47C4×Q812yes—
149C2×Dic47—44no—
150C2×Dic47—82no—
151C2×Dic47—82no—
152C2×Dic47—82no—
153C2×Dic47—82no—
154C2×Dic47—82no—
155#13300—471nomaximal, Sylow
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—22no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—42no—
162C22×C47—41no—
163C22×C47—41no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—81no—
169C22×C47—81no—
170C22×C47#39111yescharacteristic
171C22×C47C2×#9911yescharacteristic
172C22×C47#41011yescharacteristic
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—22no—
176C4×C47—42no—
177C4×C47—41no—
178C4×C47—41no—
179C4×C47—41no—
180C4×C47—81no—
181C4×C47—81no—
182C4×C47—81no—
183Dic47—84no—
184Dic47—162no—
185#2566D4711yescharacteristic
186#1799—471no—
187C4×#99—471no—
188#1393—472no—
189#1393—472no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47—42no—
193C2×C47—41no—
194C2×C47—41no—
195C2×C47#199611yescharacteristic
196C2×C47#212611yescharacteristic
197C2×C47#212711yescharacteristic
198C2×#125Dic4711yescharacteristic, complemented
199C2×C4×D4C2×D4711yescharacteristic
200C2×#99—472no—
201C2×#99—472no—
202#396—21no—
203#396—21no—
204#390C2×D4711yescharacteristic
205#390C2×D4711yescharacteristic
206#385—21no—
207#385—21no—
208#384Dic4711yescharacteristic, complemented
209#382Dic4711yescharacteristic, complemented
210#380—941no—
211C4×C4⋊C4Dic4711yescharacteristic, complemented
212C4×C22⋊C4—941no—
213#342—941no—
214#342—941no—
215C47#1330011yescharacteristic, complemented, Sylow
216C22×D4C22×D4711yescharacteristic
217C23×C4C22⋊D4711yescharacteristic
218C23×C4C22⋊D4711yescharacteristic
219#125—41no—
220#125—41no—
221C4×D4—41no—
222C4×D4—41no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—1881no—
229C2×C4⋊C4C2×Dic4711yescharacteristic
230C2×C4⋊C4C2×Dic4711yescharacteristic
231C2×C4⋊C4C2×Dic4711yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—942no—
237C2×C22⋊C4—942no—
238C2×C22⋊C4C22⋊D4711yescharacteristic
239C2×C22⋊C4C22⋊D4711yescharacteristic
240C2×C42—1881no—
241C2×C42C2×Dic4711yescharacteristic
242C2×C42C2×Dic4711yescharacteristic
243C2×C42C2×Dic4711yescharacteristic
244#99—474no—
245#99—942no—
246#99—944no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—1881no—
251#95—1881no—
252C24C2×C22⋊D4711yescharacteristic
253C24C2×C22⋊D4711yescharacteristic
254C2×D4—41no—
255C2×D4—41no—
256C2×D4Q8×D4712yes—
257C2×D4#1062535412yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—1882no—
269C22×C4—1882no—
270C22×C4C22⋊Dic4711yescharacteristic
271C22×C4C22⋊Dic4711yescharacteristic
272C22×C4C22⋊Dic4711yescharacteristic
273C22×C4C22⋊Dic4711yescharacteristic
274C22×C4C22×Dic4711yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C4⋊C4—81no—
284C4⋊C4—3762no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—944no—
292C22⋊C4—1882no—
293C22⋊C4—1882no—
294C22⋊C4—1882no—
295C42—21no—
296C42—21no—
297C42—41no—
298C42—41no—
299C42—3762no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23C2×C22⋊Dic4711yescharacteristic
309C23#1309400012yes—
310C23#1309401311yescharacteristic
311C23#1309402211yescharacteristic
312D4—42no—
313D4—42no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—3762no—
340C2×C4—3762no—
341C2×C4—3762no—
342C2×C4—3762no—
343C2×C4—3762no—
344C2×C4—1884no—
345C2×C4#1309393911yescharacteristic
346C2×C4#1309393911yescharacteristic
347C2×C4Q8×Dic4712yes—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22#1568903011yescharacteristic
363C22C2×#1309393911yescharacteristic
364C22#1568909711yescharacteristic
365C4—21no—
366C4—21no—
367C4—22no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—3764no—
376C4—7522no—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—41no—
381C2—41no—
382C2#2024437411yescharacteristic
383C2#2024505311yescharacteristic
384C2#2024505711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC22×D4C22×D47
FittingC47×#2566C2
socleC22×C47C2×#99
prime radical, cosocleC22×D4×C47C23
hypercenter#2566D47
hypocenterC47#13300
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2566

Complements

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

NH ≅ G/Ntype
12. C2×#125×C47C4semidirect
20. C47×#384C4semidirect
21. C47×#382C4semidirect
22. C4×C4⋊C4×C47C4semidirect
198. C2×#125Dic47semidirect
208. #384Dic47semidirect
209. #382Dic47semidirect
211. C4×C4⋊C4Dic47semidirect
215. C47#13300semidirect

Maximal subgroups

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

subgroupindex
#202433762
C4×#130939392
#202415132
C47×#25662
#1330047