Finite Group Databasebeta

← #26126485

Subgroups — #26126485

Order 12032.

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

normal subgroups only

This group has 514 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
1C2×#15688892C211yesmaximal, characteristic, complemented
2C2×#15688849C211yesmaximal, characteristic, complemented
3#20241394C212yesmaximal, complemented
4#20241391C212yesmaximal, complemented
5C47×#2618C211yesmaximal, characteristic
6C22×#10625329—21no—
7C2×C22⋊(C47⋊C8)—21no—
8#15688892—21no—
9#15688892C2212yes—
10#15688849—21no—
11#15688849C2212yes—
12C2×#123×C47—21no—
13C2×#121×C47C411yescharacteristic
14C22×C4⋊C4×C47C2211yescharacteristic
15C47×#398C411yescharacteristic
16C47×#398C2211yescharacteristic
17C47×#396—21no—
18C47×#396—21no—
19C47×#394C411yescharacteristic
20C47×#394C2211yescharacteristic
21C47×#386—21no—
22C47×#380C411yescharacteristic
23C2×#10625329—24no—
24C2×#10625329—22no—
25C2×#10625329—41no—
26C2×#10625329—42no—
27C22×C47⋊C8—41no—
28C22×C47⋊C8—41no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—41no—
31C22×D4×C47—21no—
32C22×D4×C47C2×C411yescharacteristic
33C23×C4×C47C2×C411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C23×C4×C47D411yescharacteristic
36#123×C47—42no—
37#123×C47—42no—
38#121×C47—41no—
39#121×C47—41no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—41no—
44C2×C4⋊C4×C47—41no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C4⋊C4×C47C2311yescharacteristic
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—21no—
55C2×C22⋊C4×C47—21no—
56#95×C47—21no—
57#95×C47—21no—
58#95×C47C2×C411yescharacteristic
59#95×C47C2×C411yescharacteristic
60C24×C47—21no—
61C24×C47C22⋊C411yescharacteristic
62C2×D4×C47—41no—
63C2×D4×C47—41no—
64C2×D4×C47—41no—
65C2×D4×C47—42no—
66C2×D4×C47—42no—
67C2×D4×C47—81no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—22no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—81no—
79C22×C4×C47—81no—
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22⋊C411yescharacteristic
83C22×C4×C47D812yes—
84C22×C4×C47SD1612yes—
85C22×C4×C47C22×C411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—42no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—81no—
97C22⋊C4×C47—42no—
98C22⋊C4×C47—42no—
99C22⋊C4×C47—42no—
100C22⋊C4×C47—41no—
101C22⋊C4×C47—41no—
102C22⋊C4×C47—81no—
103C22⋊C4×C47—81no—
104#10625329—24no—
105#10625329—44no—
106#10625329—42no—
107C2×C47⋊C8—44no—
108C2×C47⋊C8—42no—
109C2×C47⋊C8—42no—
110C2×C47⋊C8—82no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47#9911yescharacteristic
121C23×C47#9911yescharacteristic
122C23×C47#10212yes—
123C23×C47#10412yes—
124C23×C47C2×C22⋊C411yescharacteristic
125D4×C47—82no—
126D4×C47—82no—
127D4×C47—82no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—42no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—81no—
155C2×C4×C47—81no—
156C2×C4×C47—81no—
157C2×C4×C47#10212yes—
158C2×C4×C47C2×D811yescharacteristic
159C2×C4×C47C2×SD1611yescharacteristic
160C47⋊C8—44no—
161C47⋊C8—82no—
162#11698—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—42no—
170C22×C47—42no—
171C22×C47—81no—
172C22×C47—81no—
173C22×C47—81no—
174C22×C47—81no—
175C22×C47—81no—
176C22×C47—81no—
177C22×C47—81no—
178C22×C47#32712yes—
179C22×C47#32712yes—
180C22×C47C2×#9911yescharacteristic
181C22×C47C2×#10211yescharacteristic
182C22×C47C2×#10411yescharacteristic
183C4×C47—21no—
184C4×C47—22no—
185C4×C47—21no—
186C4×C47—41no—
187C4×C47—41no—
188C4×C47—41no—
189C4×C47—41no—
190C4×C47—81no—
191C4×C47—81no—
192C4×C47—81no—
193#2618D4711yescharacteristic
194C2×#343—471no—
195C2×#324—471no—
196#1346—472no—
197#1345—472no—
198C2×C47—21no—
199C2×C47—21no—
200C2×C47—22no—
201C2×C47—22no—
202C2×C47—161no—
203C2×C47C2×#32711yescharacteristic
204C2×C47#212111yescharacteristic
205C2×C47#213311yescharacteristic
206C22×M16—941no—
207C2×#123—21no—
208C2×#121Dic4711yescharacteristic
209C22×C4⋊C4C2×D4711yescharacteristic
210C2×C22⋊C8—941no—
211#398Dic4711yescharacteristic
212#398C2×D4711yescharacteristic
213#396—21no—
214#396—21no—
215#394Dic4711yescharacteristic
216#394C2×D4711yescharacteristic
217#386—21no—
218#380Dic4711yescharacteristic
219#343—472no—
220#343—941no—
221#324—472no—
222#324—941no—
223C47#1169811yescharacteristic, complemented, Sylow
224C22×D4—21no—
225C22×D4C2×Dic4711yescharacteristic
226C23×C4C2×Dic4711yescharacteristic
227C23×C4C22⋊D4711yescharacteristic
228C23×C4C22⋊D4711yescharacteristic
229C2×M16—944no—
230C2×M16—942no—
231C2×M16—1881no—
232C2×M16—1882no—
233C22×C8—1881no—
234C22×C8—1881no—
235#123—42no—
236#123—42no—
237#121—41no—
238#121—41no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4C2×Dic4711yescharacteristic
245C2×C4⋊C4C2×Dic4711yescharacteristic
246C2×C4⋊C4C22⋊D4711yescharacteristic
247C2×C4⋊C4C22⋊D4711yescharacteristic
248C2×C4⋊C4C22×D4711yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C22⋊C8—942no—
256C22⋊C8—1881no—
257#95—21no—
258#95—21no—
259#95C2×Dic4711yescharacteristic
260#95C2×Dic4711yescharacteristic
261C24—21no—
262C24C22⋊Dic4711yescharacteristic
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—81no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4#1062533412yes—
282C22×C4#1062533512yes—
283C22×C4C22⋊Dic4711yescharacteristic
284C22×C4C22⋊Dic4711yescharacteristic
285C22×C4C22⋊Dic4711yescharacteristic
286C22×C4C22×Dic4711yescharacteristic
287C22×C4C2×C22⋊D4711yescharacteristic
288C22×C4C2×C22⋊D4711yescharacteristic
289M16—944no—
290M16—1884no—
291M16—1882no—
292C2×C8—3762no—
293C2×C8—1882no—
294C2×C8—1882no—
295C2×C8—1884no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—81no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23#1309393911yescharacteristic
322C23#1309393911yescharacteristic
323C23#1309394512yes—
324C23#1309395112yes—
325C23C2×C22⋊Dic4711yescharacteristic
326D4—82no—
327D4—82no—
328D4—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4#1309394512yes—
359C2×C4C2×#1062533411yescharacteristic
360C2×C4C2×#1062533511yescharacteristic
361C8—3762no—
362C8—1884no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#1568885712yes—
379C22#1568885712yes—
380C22C2×#1309393911yescharacteristic
381C22C2×#1309394511yescharacteristic
382C22C2×#1309395111yescharacteristic
383C4—21no—
384C4—22no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—81no—
391C4—81no—
392C4—81no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2—22no—
397C2—161no—
398C2C2×#1568885711yescharacteristic
399C2#2024503411yescharacteristic
400C2#2024507711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2618C2
socleC22×C47C2×#99
prime radicalC2×#121×C47C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2618D47
hypocenterC47#11698
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
1. C2×#15688892C2semidirect
2. C2×#15688849C2semidirect
3. #20241394C2semidirect
4. #20241391C2semidirect
223. C47#11698semidirect

Maximal subgroups

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

subgroupindex
C2×#156888922
C2×#156888492
#202413942
#202413912
C47×#26182
#1169847