Finite Group Databasebeta

← #26112206

Subgroups — #26112206

Order 12032.

446 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253193C211yesmaximal, characteristic
2#20241951C211yesmaximal, characteristic
3#20241886C211yesmaximal, characteristic
4#20241783C211yesmaximal, characteristic
5#20241291C211yesmaximal, characteristic
6#20241246C211yesmaximal, characteristic
7C47×#2356C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689430—22no—
10#15689418C412yescomplemented
11#15689408C412yescomplemented
12#15689176C2211yescharacteristic
13#15689176C2211yescharacteristic
14#15689163—21no—
15#15689147C2211yescharacteristic
16#15689132—21no—
17#15688859—22no—
18#15688859—21no—
19#15688859—21no—
20Q8⋊(C47⋊C8)—22no—
21C2×C4×Q8×C47C2211yescharacteristic
22C47×#391—21no—
23C47×#387—22no—
24C47×#383—21no—
25C47×#382C2211yescharacteristic
26C4×C4⋊C4×C47—21no—
27C4×C4⋊C4×C47C2211yescharacteristic
28C47×#376—21no—
29C2×#10625329—21no—
30#13094024C2×C411yescharacteristic
31#13094024C2×C411yescharacteristic
32#13094016—21no—
33#13094003—21no—
34#13094003—21no—
35C4×Dic94—21no—
36C4×Dic94—21no—
37#13093983—21no—
38#13093957—22no—
39#13093955—22no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)C2×C412yes—
44#13093948—22no—
45#13093948—22no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48C4×C47⋊C8—21no—
49C4×C47⋊C8—21no—
50C22×Q8×C47D411yescharacteristic
51C4×Q8×C47—21no—
52C4×Q8×C47—21no—
53C4×Q8×C47C2×C412yes—
54C4×Q8×C47D412yes—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47D411yescharacteristic
59C2×C42×C47—21no—
60C2×C42×C47—22no—
61C2×C42×C47—21no—
62C2×C42×C47C2311yescharacteristic
63#95×C47—22no—
64#95×C47—22no—
65#95×C47—22no—
66#95×C47—21no—
67#95×C47D411yescharacteristic
68#95×C47D411yescharacteristic
69#95×C47D412yes—
70C2×Q8×C47—21no—
71C2×Q8×C47—21no—
72C2×Q8×C47—21no—
73C2×Q8×C47—21no—
74C2×Q8×C47—21no—
75C2×Q8×C47C22⋊C412yes—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—22no—
80C22×C4×C47—21no—
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C2×Dic94—21no—
85C2×Dic94—21no—
86C4⋊C4×C47—22no—
87C4⋊C4×C47—22no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—42no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47C22⋊C412yes—
93C42×C47—22no—
94C42×C47—22no—
95C42×C47—44no—
96C42×C47—41no—
97C42×C47—41no—
98C42×C47C22×C411yescharacteristic
99C42×C47C2×D411yescharacteristic
100C42×C47#4111yescharacteristic
101C42×C47#4111yescharacteristic
102C22⋊Dic47—41no—
103C4⋊Dic47—21no—
104C4⋊Dic47—21no—
105C4⋊Dic47—21no—
106C4⋊Dic47—21no—
107Dic47⋊C4—41no—
108Dic47⋊C4—41no—
109C4×Dic47—41no—
110#10625329—41no—
111#10625329—41no—
112C2×C47⋊C8—22no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8—41no—
115C23×C47#12011yescharacteristic
116Q8×C47—22no—
117Q8×C47—22no—
118Q8×C47—41no—
119Q8×C47—41no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—42no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—42no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47—42no—
136C2×C4×C47C2×C22⋊C411yescharacteristic
137C2×C4×C47C4×D411yescharacteristic
138C2×C4×C47C4×D411yescharacteristic
139C2×C4×C47#12111yescharacteristic
140C2×C4×C47#12111yescharacteristic
141C2×C4×C47#12311yescharacteristic
142C2×C4×C47#13511yescharacteristic
143C2×C4×C47#13511yescharacteristic
144C2×C4×C47#13711yescharacteristic
145C2×C4×C47#13711yescharacteristic
146C2×Dic47—41no—
147C2×Dic47—41no—
148Dic94—81no—
149Dic94—81no—
150C47⋊C8—42no—
151C47⋊C8—81no—
152C47⋊C8—81no—
153#9178—471nomaximal, Sylow
154C22×C47—21no—
155C22×C47—21no—
156C22×C47#38611yescharacteristic
157C22×C47#45211yescharacteristic
158C22×C47#45211yescharacteristic
159C4×C47—21no—
160C4×C47—21no—
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—42no—
164C4×C47—42no—
165C4×C47—42no—
166C4×C47—81no—
167C4×C47—82no—
168C4×C47#41611yescharacteristic
169C4×C47#41911yescharacteristic
170C4×C47#48511yescharacteristic
171C4×C47#48511yescharacteristic
172Dic47—81no—
173Dic47—81no—
174#3078—471no—
175#2356D4711yescharacteristic
176#1518—471no—
177#1503—471no—
178#1479—471no—
179#1303—471no—
180#1286—471no—
181C2×C47—21no—
182C2×C47—21no—
183C2×C47#187311yescharacteristic
184C2×C47#190511yescharacteristic
185C2×C47#190611yescharacteristic
186#533—471no—
187C2×C4×Q8C2×D4711yescharacteristic
188#479—942no—
189#477—472no—
190#475—472no—
191#431—471no—
192#431—471no—
193#428—941no—
194#423—471no—
195#419—941no—
196#391—21no—
197#387—22no—
198#383—21no—
199#382C2×D4711yescharacteristic
200C4×C4⋊C4—21no—
201C4×C4⋊C4C2×D4711yescharacteristic
202#376—21no—
203#328—941no—
204#328—941no—
205#328—942no—
206Q8⋊C8—942no—
207C47#917811yescharacteristic, complemented, Sylow
208C22×Q8C22⋊D4711yescharacteristic
209C2×M16—941no—
210C4⋊Q8—471no—
211C4⋊Q8—471no—
212#125—941no—
213C22⋊Q8—941no—
214C22⋊Q8—941no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—941no—
218C4×Q8—941no—
219C4×Q8C4×D4712yes—
220C4×Q8D18812yes—
221#117—941no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C22⋊D4711yescharacteristic
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42C22×D4711yescharacteristic
230#107—942no—
231#106—942no—
232C4⋊C8—472no—
233C4⋊C8—941no—
234C4⋊C8—941no—
235C4⋊C8—941no—
236#103—942no—
237#103—942no—
238C22⋊C8—941no—
239C22⋊C8—941no—
240C4×C8—941no—
241C4×C8—941no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95C22⋊D4711yescharacteristic
247#95C22⋊D4711yescharacteristic
248#95C22⋊D4712yes—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—941no—
255C2×Q8—941no—
256C2×Q8(C2×D47)⋊C412yes—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4C2×C22⋊D4711yescharacteristic
263C22×C4C2×C22⋊D4711yescharacteristic
264C22×C4C2×C22⋊D4711yescharacteristic
265M16—1881no—
266M16—1881no—
267C2×C8—942no—
268C2×C8—1881no—
269C2×C8—1881no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—941no—
277C4⋊C4—941no—
278C4⋊C4—941no—
279C4⋊C4—941no—
280C4⋊C4—1881no—
281C4⋊C4—1881no—
282C4⋊C4(C2×D47)⋊C412yes—
283C22⋊C4—1881no—
284C42—22no—
285C42—22no—
286C42—44no—
287C42—41no—
288C42—41no—
289C42—1881no—
290C42C2×C4×D4711yescharacteristic
291C42C2×D18811yescharacteristic
292C42#1062535011yescharacteristic
293C42#1062535011yescharacteristic
294C23#1309399311yescharacteristic
295Q8—22no—
296Q8—22no—
297Q8—41no—
298Q8—41no—
299Q8—3761no—
300Q8—3761no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—1881no—
318C2×C4—1881no—
319C2×C4C2×(C2×D47)⋊C411yescharacteristic
320C2×C4C4×C22⋊D4711yescharacteristic
321C2×C4C4×C22⋊D4711yescharacteristic
322C2×C4#1309399811yescharacteristic
323C2×C4#1309399811yescharacteristic
324C2×C4#1309400711yescharacteristic
325C2×C4#1309404411yescharacteristic
326C2×C4#1309404411yescharacteristic
327C2×C4#1309405411yescharacteristic
328C2×C4#1309405411yescharacteristic
329C8—3761no—
330C8—3761no—
331C8—1882no—
332C22—21no—
333C22—21no—
334C22#1568900211yescharacteristic
335C22#1568928511yescharacteristic
336C22#1568928511yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—42no—
343C4—42no—
344C4—81no—
345C4—82no—
346C4—3761no—
347C4—3761no—
348C4#1568911711yescharacteristic
349C4#1568913111yescharacteristic
350C4#1568946411yescharacteristic
351C4#1568946411yescharacteristic
352C2—21no—
353C2—21no—
354C2#2024370011yescharacteristic
355C2#2024388211yescharacteristic
356C2#2024388911yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2356C2
socleC22×C47#386
prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#2356D47
hypocenterC47#9178
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
10. #15689418C4semidirect
11. #15689408C4semidirect
207. C47#9178semidirect

Maximal subgroups

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

subgroupindex
#202531932
#202419512
#202418862
#202417832
#202412912
#202412462
C47×#23562
#917847