Finite Group Databasebeta

← #26126974

Subgroups — #26126974

Order 12032.

448 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 87 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#20243468C211yesmaximal, characteristic, complemented
2#20241718C211yesmaximal, characteristic, complemented
3#20241394C212yesmaximal, complemented
4#20241391C212yesmaximal, complemented
5C47×#2618C211yesmaximal, characteristic
6#15689087—21no—
7#15689087—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—22no—
24C2×#10625329—41no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—41no—
27C22⋊(C47⋊C8)—41no—
28C22⋊(C47⋊C8)—41no—
29C22×D4×C47—21no—
30C22×D4×C47C2×C411yescharacteristic
31C23×C4×C47C2×C411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C23×C4×C47D411yescharacteristic
34#123×C47—44no—
35#121×C47—41no—
36#121×C47—41no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—41no—
41C2×C4⋊C4×C47—41no—
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C4⋊C4×C47C2311yescharacteristic
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47C2×C411yescharacteristic
56#95×C47C2×C411yescharacteristic
57C24×C47—21no—
58C24×C47C22⋊C411yescharacteristic
59C2×D4×C47—41no—
60C2×D4×C47—42no—
61C2×D4×C47—42no—
62C2×D4×C47—41no—
63C2×D4×C47—41no—
64C2×D4×C47—81no—
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—22no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—41no—
76C22×C4×C47—81no—
77C22×C4×C47—81no—
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22×C411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C4⋊C4×C47—41no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—42no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—42no—
91C4⋊C4×C47—81no—
92C22⋊C4×C47—42no—
93C22⋊C4×C47—42no—
94C22⋊C4×C47—42no—
95C22⋊C4×C47—42no—
96C22⋊C4×C47—81no—
97C22⋊C4×C47—81no—
98#10625329—42no—
99#10625329—42no—
100#10625329—82no—
101C2×C47⋊C8—42no—
102C2×C47⋊C8—42no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47#9911yescharacteristic
115C23×C47#9911yescharacteristic
116C23×C47C2×C22⋊C411yescharacteristic
117D4×C47—82no—
118D4×C47—82no—
119D4×C47—82no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
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—41no—
145C2×C4×C47—81no—
146C2×C4×C47—81no—
147C2×C4×C47—81no—
148C2×C4×C47AGL(1,ℤ/8)12yes—
149C47⋊C8—82no—
150C47⋊C8—82no—
151#11805—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—42no—
161C22×C47—42no—
162C22×C47—81no—
163C22×C47—81no—
164C22×C47—81no—
165C22×C47—81no—
166C22×C47—81no—
167C22×C47—81no—
168C22×C47—81no—
169C22×C47C2×#9911yescharacteristic
170C22×C47#41811yescharacteristic
171C22×C47#42111yescharacteristic
172C4×C47—22no—
173C4×C47—41no—
174C4×C47—41no—
175C4×C47—41no—
176C4×C47—41no—
177C4×C47—41no—
178C4×C47—81no—
179C4×C47—81no—
180C4×C47—81no—
181#2618D4711yescharacteristic
182#1821—471no—
183#1463—471no—
184#1346—472no—
185#1345—472no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—41no—
189C2×C47—41no—
190C2×C47—161no—
191C2×C47#151011yescharacteristic
192C2×C47#212111yescharacteristic
193C2×C47#213311yescharacteristic
194C2×#123—21no—
195C2×#121Dic4711yescharacteristic
196C22×C4⋊C4C2×D4711yescharacteristic
197#407—941no—
198#407—941no—
199#398Dic4711yescharacteristic
200#398C2×D4711yescharacteristic
201#396—21no—
202#396—21no—
203#394Dic4711yescharacteristic
204#394C2×D4711yescharacteristic
205#386—21no—
206#380Dic4711yescharacteristic
207#343—472no—
208#343—941no—
209#324—472no—
210#324—941no—
211C47#1180511yescharacteristic, complemented, Sylow
212C22×D4—21no—
213C22×D4C2×Dic4711yescharacteristic
214C23×C4C2×Dic4711yescharacteristic
215C23×C4C22⋊D4711yescharacteristic
216C23×C4C22⋊D4711yescharacteristic
217C2×M16—942no—
218C2×M16—1881no—
219#123—44no—
220#121—41no—
221#121—41no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C2×Dic4711yescharacteristic
228C2×C4⋊C4C2×Dic4711yescharacteristic
229C2×C4⋊C4C22⋊D4711yescharacteristic
230C2×C4⋊C4C22⋊D4711yescharacteristic
231C2×C4⋊C4C22×D4711yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C22⋊C8—942no—
239C22⋊C8—1881no—
240C22⋊C8—1881no—
241C22⋊C8—1881no—
242#95—21no—
243#95—21no—
244#95C2×Dic4711yescharacteristic
245#95C2×Dic4711yescharacteristic
246C24—21no—
247C24C22⋊Dic4711yescharacteristic
248C2×D4—41no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—81no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4C22⋊Dic4711yescharacteristic
268C22×C4C22⋊Dic4711yescharacteristic
269C22×C4C22⋊Dic4711yescharacteristic
270C22×C4C22×Dic4711yescharacteristic
271C22×C4C2×C22⋊D4711yescharacteristic
272C22×C4C2×C22⋊D4711yescharacteristic
273M16—3762no—
274M16—1882no—
275M16—1882no—
276C2×C8—1882no—
277C2×C8—1882no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—81no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23#1309393911yescharacteristic
304C23#1309393911yescharacteristic
305C23C2×C22⋊Dic4711yescharacteristic
306D4—82no—
307D4—82no—
308D4—82no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
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—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4#1309405012yes—
338C8—3762no—
339C8—3762no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22C2×#1309393911yescharacteristic
358C22#1568912811yescharacteristic
359C22#1568914011yescharacteristic
360C4—22no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—81no—
367C4—81no—
368C4—81no—
369C2—21no—
370C2—21no—
371C2—41no—
372C2—41no—
373C2—161no—
374C2#2024191711yescharacteristic
375C2#2024503411yescharacteristic
376C2#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#11805
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. #20243468C2semidirect
2. #20241718C2semidirect
3. #20241394C2semidirect
4. #20241391C2semidirect
211. C47#11805semidirect

Maximal subgroups

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

subgroupindex
#202434682
#202417182
#202413942
#202413912
C47×#26182
#1180547