Finite Group Databasebeta

← #26107715

Subgroups — #26107715

Order 12032.

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

normal subgroups only

This group has 684 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
1C2×#15689735C211yesmaximal, characteristic
2#20241784C212yesmaximal, complemented
3#20241782C212yesmaximal, complemented
4#20241702C211yesmaximal, characteristic
5C47×#1458C211yesmaximal, characteristic
6#15689735—24no—
7#15689735—24no—
8C2×#13094024C411yescharacteristic
9C2×#13094017—21no—
10C2×#13094017—21no—
11C2×#13094016C411yescharacteristic
12C2×#13094014—21no—
13C2×#13094014—21no—
14C2×#13094003C411yescharacteristic
15C2×#13094003C411yescharacteristic
16C2×C4×Dic94C2211yescharacteristic
17C2×C4×Dic94C2211yescharacteristic
18#15689176C2212yes—
19#15688855—24no—
20#15688853—24no—
21#15688849—21no—
22#15688847—21no—
23C2×#117×C47C2211yescharacteristic
24C47×#431C2212yes—
25C47×#324—21no—
26C47×#323—21no—
27C22×Dic94C2×C411yescharacteristic
28C22×Dic94C2×C411yescharacteristic
29#13094024—22no—
30#13094024—22no—
31#13094017—28no—
32#13094017—28no—
33#13094016—22no—
34#13094016—22no—
35#13094014—28no—
36#13094014—28no—
37#13094003—22no—
38#13094003—22no—
39#13094003—22no—
40#13094003—22no—
41C4×Dic94—22no—
42C4×Dic94—22no—
43C4×Dic94D414yes—
44C4×Dic94D414yes—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47—21no—
47C2×C4⋊Dic47C2×C411yescharacteristic
48C2×C4⋊Dic47C2×C411yescharacteristic
49C2×Dic47⋊C4—21no—
50C2×Dic47⋊C4—21no—
51C2×Dic47⋊C4—21no—
52C2×Dic47⋊C4—21no—
53C2×Dic47⋊C4—21no—
54C2×Dic47⋊C4—21no—
55C2×C22⋊Dic47—21no—
56C2×C22⋊Dic47—21no—
57C2×C4×Dic47—21no—
58C2×C4×Dic47—21no—
59C4⋊(C47⋊C8)—22no—
60C22⋊(C47⋊C8)—22no—
61C4×C47⋊C8—22no—
62C23×C4×C47C2×C411yescharacteristic
63#117×C47—22no—
64#117×C47—22no—
65C2×C4⋊C4×C47D411yescharacteristic
66C2×C4⋊C4×C47D411yescharacteristic
67C2×C22⋊C4×C47D411yescharacteristic
68C2×C22⋊C4×C47D411yescharacteristic
69C2×C42×C47C2×C411yescharacteristic
70C2×C42×C47C2311yescharacteristic
71C4⋊C8×C47—22no—
72C22⋊C8×C47—22no—
73C4×C8×C47—22no—
74C24×C47C22⋊C411yescharacteristic
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×Dic47—21no—
86C22×Dic47—21no—
87C22×Dic47—21no—
88C22×Dic47—21no—
89C2×Dic94—22no—
90C2×Dic94—22no—
91C2×Dic94—22no—
92C2×Dic94—22no—
93C2×Dic94—21no—
94C2×Dic94—21no—
95C2×Dic94C22⋊C412yes—
96C2×Dic94C22⋊C412yes—
97C2×C8×C47—42no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—22no—
100C22⋊C4×C47—22no—
101C22⋊C4×C47—22no—
102C42×C47—21no—
103C42×C47—21no—
104C42×C47C2×D412yes—
105C42×C47C2×D412yes—
106C22⋊Dic47—24no—
107C22⋊Dic47—24no—
108C4⋊Dic47—24no—
109C4⋊Dic47—24no—
110C4⋊Dic47—21no—
111C4⋊Dic47—21no—
112C4⋊Dic47C22⋊C412yes—
113C4⋊Dic47C22⋊C412yes—
114Dic47⋊C4—24no—
115Dic47⋊C4—24no—
116Dic47⋊C4—24no—
117Dic47⋊C4—24no—
118Dic47⋊C4—24no—
119Dic47⋊C4—24no—
120C4×Dic47—24no—
121C4×Dic47—24no—
122C2×C47⋊C8—42no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47—41no—
127C23×C47—41no—
128C23×C47C2×C22⋊C411yescharacteristic
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47C2×C22⋊C411yescharacteristic
141C2×C4×C47C2×C22⋊C411yescharacteristic
142C2×C4×C47#12012yes—
143C2×C4×C47#12012yes—
144C2×C4×C47#13112yes—
145C2×C4×C47#13112yes—
146C2×Dic47—22no—
147C2×Dic47—22no—
148C2×Dic47—22no—
149C2×Dic47—22no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156Dic94—24no—
157Dic94—24no—
158Dic94—42no—
159Dic94—42no—
160C8×C47—82no—
161C47⋊C8—82no—
162#8327—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—81no—
169C22×C47—81no—
170C22×C47#37911yescharacteristic
171C22×C47#40811yescharacteristic
172C22×C47#40811yescharacteristic
173C4×C47—22no—
174C4×C47—22no—
175C4×C47—41no—
176C4×C47—41no—
177C4×C47—41no—
178C4×C47#40812yes—
179C4×C47#40812yes—
180Dic47—42no—
181Dic47—42no—
182Dic47—42no—
183Dic47—42no—
184C2×#541—471no—
185#1479—472no—
186#1479—472no—
187#1458—471no—
188#1458D4711yescharacteristic
189C2×C47—21no—
190C2×C47—21no—
191C2×C47—81no—
192C2×C47#178011yescharacteristic
193C2×C47#179611yescharacteristic
194C2×C47#179611yescharacteristic
195#541—944no—
196#541—944no—
197C2×C4⋊Q8—471no—
198C2×#126—941no—
199C2×#126—941no—
200C2×#125—471no—
201C2×#125—941no—
202C2×#125—941no—
203C2×C22⋊Q8—471no—
204C2×C22⋊Q8—471no—
205C2×C4×Q8—471no—
206C2×C4×Q8—471no—
207C2×#117C2×D4711yescharacteristic
208#431—472no—
209#431C2×D4712yes—
210Q8⋊C8—944no—
211Q8⋊C8—944no—
212#324—21no—
213#324—941no—
214#323—21no—
215#323—941no—
216C47#832711yescharacteristic, complemented, Sylow
217C22×Q8—471no—
218C22×Q8—471no—
219C23×C4C4×D4711yescharacteristic
220C4⋊Q8—942no—
221C4⋊Q8—942no—
222#126—948no—
223#126—948no—
224#125—942no—
225#125—942no—
226#125—948no—
227#125—948no—
228C22⋊Q8—942no—
229C22⋊Q8—942no—
230C22⋊Q8—942no—
231C22⋊Q8—942no—
232C4×Q8—474no—
233C4×Q8—474no—
234C4×Q8—942no—
235C4×Q8—942no—
236#117—22no—
237#117—22no—
238C2×C4⋊C4—471no—
239C2×C4⋊C4—471no—
240C2×C4⋊C4—941no—
241C2×C4⋊C4—941no—
242C2×C4⋊C4—941no—
243C2×C4⋊C4—941no—
244C2×C4⋊C4—941no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4—941no—
247C2×C4⋊C4—941no—
248C2×C4⋊C4D18811yescharacteristic
249C2×C4⋊C4C22⋊D4711yescharacteristic
250C2×C22⋊C4—941no—
251C2×C22⋊C4—941no—
252C2×C22⋊C4D18811yescharacteristic
253C2×C22⋊C4C22⋊D4711yescharacteristic
254C2×C42—941no—
255C2×C42—941no—
256C2×C42C4×D4711yescharacteristic
257C2×C42C22×D4711yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—942no—
260C22⋊C8—22no—
261C22⋊C8—942no—
262C4×C8—22no—
263C4×C8—942no—
264C24(C2×D47)⋊C411yescharacteristic
265C2×Q8—472no—
266C2×Q8—472no—
267C2×Q8—941no—
268C2×Q8—941no—
269C2×Q8—942no—
270C2×Q8—942no—
271C2×Q8—942no—
272C2×Q8—942no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—941no—
278C22×C4—941no—
279C22×C4—941no—
280C22×C4—941no—
281C22×C4(C2×D47)⋊C411yescharacteristic
282C22×C4(C2×D47)⋊C411yescharacteristic
283C22×C4(C2×D47)⋊C411yescharacteristic
284C22×C4C2×C4×D4711yescharacteristic
285C22×C4C2×D18811yescharacteristic
286C22×C4C2×C22⋊D4711yescharacteristic
287C2×C8—42no—
288C2×C8—1882no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—472no—
292C4⋊C4—472no—
293C4⋊C4—944no—
294C4⋊C4—944no—
295C4⋊C4—944no—
296C4⋊C4—944no—
297C4⋊C4—944no—
298C4⋊C4—944no—
299C4⋊C4—941no—
300C4⋊C4—941no—
301C4⋊C4—944no—
302C4⋊C4—944no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—944no—
306C22⋊C4—944no—
307C42—21no—
308C42—21no—
309C42—944no—
310C42—944no—
311C42D4×D4712yes—
312C42D4×D4712yes—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23C2×(C2×D47)⋊C411yescharacteristic
319Q8—944no—
320Q8—944no—
321Q8—1882no—
322Q8—1882no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—942no—
335C2×C4—942no—
336C2×C4—942no—
337C2×C4—942no—
338C2×C4—1882no—
339C2×C4—1882no—
340C2×C4—1881no—
341C2×C4—1881no—
342C2×C4—1881no—
343C2×C4—1881no—
344C2×C4C22⋊C4×D4711yescharacteristic
345C2×C4C22⋊C4×D4711yescharacteristic
346C2×C4#1309399212yes—
347C2×C4#1309399412yes—
348C2×C4#1309403012yes—
349C2×C4#1309403212yes—
350C8—82no—
351C8—3762no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—81no—
359C22#1568897811yescharacteristic
360C22#1568908911yescharacteristic
361C22#1568909111yescharacteristic
362C4—22no—
363C4—22no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—1882no—
368C4—1882no—
369C4—1882no—
370C4—1882no—
371C4#1568908812yes—
372C4#1568908812yes—
373C2—21no—
374C2—21no—
375C2—81no—
376C2#2024329911yescharacteristic
377C2#2024336111yescharacteristic
378C2#2024336411yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1458C2
socleC22×C47#379
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1458D47
hypocenterC47#8327
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #20241784C2semidirect
3. #20241782C2semidirect
216. C47#8327semidirect

Maximal subgroups

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

subgroupindex
C2×#156897352
#202417842
#202417822
#202417022
C47×#14582
#832747