Finite Group Databasebeta

← #26108929

Subgroups — #26108929

Order 12032.

842 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×#13094017C211yesmaximal, characteristic
2#20243466C212yesmaximal, complemented
3#20241717C212yesmaximal, complemented
4#20241715C211yesmaximal, characteristic
5C47×#1462C211yesmaximal, characteristic
6C2×#13094017—24no—
7C2×#13094017—24no—
8C2×#13094017—22no—
9C2×#13094017—22no—
10C22×C4⋊Dic47C2211yescharacteristic
11C22×Dic47⋊C4C411yescharacteristic
12C22×Dic47⋊C4C2211yescharacteristic
13C22×C22⋊Dic47C411yescharacteristic
14C22×C22⋊Dic47C411yescharacteristic
15C22×C4×Dic47C411yescharacteristic
16#15689087C2212yes—
17#15688891—24no—
18#15688848—24no—
19#15688847—22no—
20C22×C22⋊C4×C47C2211yescharacteristic
21C47×#407C2212yes—
22C47×#323—22no—
23C23×Dic47C2×C411yescharacteristic
24C23×Dic47C2×C411yescharacteristic
25C23×Dic47C2×C411yescharacteristic
26C23×Dic47C2×C411yescharacteristic
27C2×#10625329—22no—
28#13094017—216no—
29#13094017—216no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—22no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47D414yes—
34C2×Dic47⋊C4—22no—
35C2×Dic47⋊C4—22no—
36C2×Dic47⋊C4—21no—
37C2×Dic47⋊C4—21no—
38C2×Dic47⋊C4—22no—
39C2×Dic47⋊C4—22no—
40C2×Dic47⋊C4D414yes—
41C2×C22⋊Dic47—22no—
42C2×C22⋊Dic47—22no—
43C2×C22⋊Dic47—22no—
44C2×C22⋊Dic47—22no—
45C2×C22⋊Dic47—22no—
46C2×C22⋊Dic47—22no—
47C2×C4×Dic47—22no—
48C2×C4×Dic47—22no—
49C2×C4×Dic47—22no—
50C22⋊(C47⋊C8)—22no—
51C22⋊(C47⋊C8)—22no—
52C25×C47C2×C411yescharacteristic
53C23×C4×C47C2×C411yescharacteristic
54C23×C4×C47C2311yescharacteristic
55C2×M16×C47—22no—
56C2×C22⋊C4×C47—22no—
57C2×C22⋊C4×C47—22no—
58C2×C22⋊C4×C47D412yes—
59C2×C22⋊C4×C47D412yes—
60C22⋊C8×C47—22no—
61C22⋊C8×C47—22no—
62C24×C47—21no—
63C24×C47—21no—
64C24×C47—42no—
65C24×C47—42no—
66C24×C47—42no—
67C24×C47C22⋊C412yes—
68C24×C47C22×C411yescharacteristic
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—42no—
72C22×C4×C47—42no—
73C22×C4×C47C22⋊C412yes—
74C22×C4×C47C2×D412yes—
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D412yes—
77C22×C4×C47C2×D411yescharacteristic
78C22×Dic47—21no—
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C22×Dic47—21no—
85C22×Dic47—21no—
86C22×Dic47—42no—
87C22×Dic47—42no—
88C22×Dic47—42no—
89C22×Dic47—42no—
90C22×Dic47C22⋊C412yes—
91C22×Dic47C22⋊C412yes—
92C22×Dic47C22⋊C412yes—
93C22×Dic47C22⋊C412yes—
94M16×C47—44no—
95C2×C8×C47—42no—
96C22⋊C4×C47—24no—
97C22⋊C4×C47—24no—
98C22⋊Dic47—28no—
99C22⋊Dic47—28no—
100C4⋊Dic47—24no—
101C4⋊Dic47—24no—
102Dic47⋊C4—28no—
103Dic47⋊C4—24no—
104Dic47⋊C4—24no—
105C4×Dic47—28no—
106#10625329—44no—
107C2×C47⋊C8—42no—
108C23×C47—22no—
109C23×C47—22no—
110C23×C47—22no—
111C23×C47—22no—
112C23×C47—24no—
113C23×C47—42no—
114C23×C47—42no—
115C23×C47—42no—
116C23×C47—42no—
117C23×C47—42no—
118C23×C47—42no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—88no—
122C23×C47C2×C22⋊C411yescharacteristic
123C23×C47C2×C22⋊C411yescharacteristic
124C23×C47C2×C22⋊C411yescharacteristic
125C2×C4×C47—24no—
126C2×C4×C47—22no—
127C2×C4×C47—41no—
128C2×C4×C47—42no—
129C2×C4×C47—41no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47#12012yes—
134C2×C4×C47#12012yes—
135C2×Dic47—22no—
136C2×Dic47—22no—
137C2×Dic47—22no—
138C2×Dic47—22no—
139C2×Dic47—24no—
140C2×Dic47—42no—
141C2×Dic47—42no—
142C2×Dic47—42no—
143C2×Dic47—42no—
144C2×Dic47—42no—
145C2×Dic47—42no—
146C2×Dic47—42no—
147C2×Dic47—42no—
148C2×Dic47—42no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C8×C47—82no—
152C47⋊C8—82no—
153#8556—471nomaximal, Sylow
154C22×C47—28no—
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—42no—
162C22×C47—42no—
163C22×C47—42no—
164C22×C47—84no—
165C22×C47—84no—
166C22×C47—84no—
167C22×C47#37911yescharacteristic
168C22×C47#41012yes—
169C22×C47#41312yes—
170C22×C47#41312yes—
171C4×C47—22no—
172C4×C47—22no—
173C4×C47—42no—
174Dic47—42no—
175Dic47—42no—
176Dic47—42no—
177Dic47—42no—
178C22×#126—471no—
179#1821—472no—
180#1463—472no—
181#1462—471no—
182#1462D4711yescharacteristic
183C2×C47—22no—
184C2×C47—22no—
185C2×C47—22no—
186C2×C47—82no—
187C2×C47#178211yescharacteristic
188C2×C47#178511yescharacteristic
189C2×C47#178511yescharacteristic
190C2×#126—944no—
191C2×#126—944no—
192C2×#126—942no—
193C2×#126—942no—
194C22×C4⋊C4—471no—
195C22×C4⋊C4—471no—
196C22×C4⋊C4—471no—
197C22×C22⋊C4—471no—
198C22×C22⋊C4—471no—
199C22×C22⋊C4C2×D4711yescharacteristic
200C22×C42—471no—
201#407—472no—
202#407C2×D4712yes—
203#343—944no—
204#324—944no—
205#323—22no—
206#323—942no—
207C47#855611yescharacteristic, complemented, Sylow
208C25C4×D4711yescharacteristic
209C23×C4—471no—
210C23×C4—471no—
211C23×C4—471no—
212C23×C4—471no—
213C23×C4C4×D4711yescharacteristic
214C23×C4C22×D4711yescharacteristic
215C2×M16—22no—
216C2×M16—942no—
217#126—9416no—
218#126—9416no—
219C2×C4⋊C4—474no—
220C2×C4⋊C4—474no—
221C2×C4⋊C4—942no—
222C2×C4⋊C4—942no—
223C2×C4⋊C4—942no—
224C2×C4⋊C4—941no—
225C2×C4⋊C4—941no—
226C2×C4⋊C4—942no—
227C2×C4⋊C4—942no—
228C2×C4⋊C4—941no—
229C2×C4⋊C4—941no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—942no—
233C2×C22⋊C4—942no—
234C2×C22⋊C4—942no—
235C2×C22⋊C4—942no—
236C2×C22⋊C4—942no—
237C2×C22⋊C4—942no—
238C2×C22⋊C4D18812yes—
239C2×C22⋊C4C22⋊D4712yes—
240C2×C42—942no—
241C2×C42—942no—
242C2×C42—942no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—942no—
246C22⋊C8—942no—
247C24—21no—
248C24—21no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24(C2×D47)⋊C412yes—
253C24C2×C4×D4711yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—472no—
259C22×C4—472no—
260C22×C4—472no—
261C22×C4—472no—
262C22×C4—941no—
263C22×C4—941no—
264C22×C4—941no—
265C22×C4—941no—
266C22×C4—941no—
267C22×C4—941no—
268C22×C4—941no—
269C22×C4—941no—
270C22×C4—1882no—
271C22×C4—1882no—
272C22×C4—1882no—
273C22×C4—1882no—
274C22×C4(C2×D47)⋊C412yes—
275C22×C4C2×D18811yescharacteristic
276C22×C4D4×D4712yes—
277C22×C4D4×D4712yes—
278C22×C4C2×C22⋊D4711yescharacteristic
279M16—44no—
280M16—1884no—
281C2×C8—42no—
282C2×C8—1882no—
283C4⋊C4—948no—
284C4⋊C4—944no—
285C4⋊C4—944no—
286C4⋊C4—944no—
287C4⋊C4—944no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—948no—
291C22⋊C4—948no—
292C42—948no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—24no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—88no—
307C23C22⋊C4×D4711yescharacteristic
308C23C22⋊C4×D4711yescharacteristic
309C23C2×(C2×D47)⋊C411yescharacteristic
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—942no—
319C2×C4—942no—
320C2×C4—942no—
321C2×C4—942no—
322C2×C4—944no—
323C2×C4—1882no—
324C2×C4—1882no—
325C2×C4—1882no—
326C2×C4—1882no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1882no—
330C2×C4—1882no—
331C2×C4—1882no—
332C2×C4—1882no—
333C2×C4—1882no—
334C2×C4#1309399212yes—
335C2×C4#1309399412yes—
336C8—82no—
337C8—3762no—
338C22—28no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—84no—
349C22—84no—
350C22—84no—
351C22#1568897811yescharacteristic
352C22#1568909512yes—
353C22#1568910412yes—
354C22#1568910512yes—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—1882no—
359C4—1882no—
360C4—1882no—
361C4—1882no—
362C2—22no—
363C2—22no—
364C2—22no—
365C2—82no—
366C2#2024330411yescharacteristic
367C2#2024331311yescharacteristic
368C2#2024331411yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1462C2
socleC22×C47#379
prime radicalC25×C47C2×C4
cosocleC23×C4×C47C23
hypercenter#1462D47
hypocenterC47#8556
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #20243466C2semidirect
3. #20241717C2semidirect
207. C47#8556semidirect

Maximal subgroups

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

subgroupindex
C22×#130940172
#202434662
#202417172
#202417152
C47×#14622
#855647