Finite Group Databasebeta

← #26125854

Subgroups — #26125854

Order 12032.

440 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 91 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#20241718C211yesmaximal, characteristic, complemented
2C2×#15688849C211yesmaximal, characteristic, complemented
3#20241332C212yesmaximal, complemented
4#20241320C212yesmaximal, complemented
5C47×#2504C211yesmaximal, characteristic
6#15689087—21no—
7C2×C22⋊(C47⋊C8)—21no—
8#15688849—21no—
9#15688849—21no—
10#15688849C2212yes—
11#15688849C2212yes—
12C2×#121×C47C411yescharacteristic
13C22×C4⋊C4×C47C2211yescharacteristic
14C47×#397—21no—
15C47×#397—21no—
16C47×#390C411yescharacteristic
17C47×#390C2211yescharacteristic
18C47×#386—21no—
19C47×#386—21no—
20C47×#382C411yescharacteristic
21C47×#382C2211yescharacteristic
22C4×C22⋊C4×C47C411yescharacteristic
23C2×#10625329—41no—
24C22×C47⋊C8—41no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
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#121×C47—41no—
35#121×C47—41no—
36C2×C4⋊C4×C47—41no—
37C2×C4⋊C4×C47—41no—
38C2×C4⋊C4×C47C2×C411yescharacteristic
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C4⋊C4×C47C2311yescharacteristic
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C2×C42×C47C2×C411yescharacteristic
50C2×C42×C47C2×C411yescharacteristic
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56C24×C47—21no—
57C24×C47C22⋊C411yescharacteristic
58C2×D4×C47—22no—
59C2×D4×C47—41no—
60C2×D4×C47—41no—
61C2×D4×C47—42no—
62C2×D4×C47—42no—
63C2×D4×C47—41no—
64C2×D4×C47—41no—
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—41no—
89C22⋊C4×C47—42no—
90C22⋊C4×C47—42no—
91C22⋊C4×C47—81no—
92C22⋊C4×C47—81no—
93C22⋊C4×C47—81no—
94C22⋊C4×C47—81no—
95C42×C47—22no—
96C42×C47—41no—
97#10625329—82no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C2×C47⋊C8—82no—
101C23×C47—22no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47#9911yescharacteristic
113C23×C47#9911yescharacteristic
114C23×C47C2×C22⋊C411yescharacteristic
115D4×C47—82no—
116D4×C47—82no—
117D4×C47—82no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
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—42no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—41no—
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—81no—
145C2×C4×C47—81no—
146C2×C4×C47#13112yes—
147C47⋊C8—82no—
148C47⋊C8—82no—
149#11553—471nomaximal, Sylow
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—42no—
158C22×C47—42no—
159C22×C47—81no—
160C22×C47—81no—
161C22×C47—81no—
162C22×C47—81no—
163C22×C47—81no—
164C22×C47—81no—
165C22×C47—81no—
166C22×C47#35512yes—
167C22×C47#40811yescharacteristic
168C22×C47C2×#9911yescharacteristic
169C22×C47#41311yescharacteristic
170C4×C47—22no—
171C4×C47—41no—
172C4×C47—41no—
173C4×C47—41no—
174C4×C47—41no—
175C4×C47—41no—
176C4×C47—81no—
177C4×C47—81no—
178C4×C47—81no—
179#2504D4711yescharacteristic
180#1463—471no—
181C2×#324—471no—
182#1324—472no—
183#1318—472no—
184C2×C47—21no—
185C2×C47—21no—
186C2×C47—22no—
187C2×C47—41no—
188C2×C47—161no—
189C2×C47#146811yescharacteristic
190C2×C47#212711yescharacteristic
191C2×C47C2×#35511yescharacteristic
192C2×#121Dic4711yescharacteristic
193C22×C4⋊C4C2×D4711yescharacteristic
194#407—941no—
195C2×C22⋊C8—941no—
196#397—21no—
197#397—21no—
198#390Dic4711yescharacteristic
199#390C2×D4711yescharacteristic
200#386—21no—
201#386—21no—
202#382Dic4711yescharacteristic
203#382C2×D4711yescharacteristic
204C4×C22⋊C4Dic4711yescharacteristic
205#324—472no—
206#324—472no—
207#324—941no—
208#324—941no—
209C47#1155311yescharacteristic, complemented, Sylow
210C22×D4—21no—
211C22×D4C2×Dic4711yescharacteristic
212C23×C4C2×Dic4711yescharacteristic
213C23×C4C22⋊D4711yescharacteristic
214C23×C4C22⋊D4711yescharacteristic
215C2×M16—1881no—
216C22×C8—1881no—
217#121—41no—
218#121—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4C2×Dic4711yescharacteristic
222C2×C4⋊C4C2×Dic4711yescharacteristic
223C2×C4⋊C4C22⋊D4711yescharacteristic
224C2×C4⋊C4C22⋊D4711yescharacteristic
225C2×C4⋊C4C22×D4711yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42C2×Dic4711yescharacteristic
233C2×C42C2×Dic4711yescharacteristic
234C22⋊C8—942no—
235C22⋊C8—942no—
236C22⋊C8—1881no—
237C22⋊C8—1881no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243C24—21no—
244C24C22⋊Dic4711yescharacteristic
245C2×D4—22no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4C22⋊Dic4711yescharacteristic
266C22×C4C22⋊Dic4711yescharacteristic
267C22×C4C22⋊Dic4711yescharacteristic
268C22×C4C22×Dic4711yescharacteristic
269C22×C4C2×C22⋊D4711yescharacteristic
270C22×C4C2×C22⋊D4711yescharacteristic
271M16—3762no—
272C2×C8—3762no—
273C2×C8—1882no—
274C2×C8—1882no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C42—22no—
287C42—41no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23#1309393911yescharacteristic
300C23#1309393911yescharacteristic
301C23C2×C22⋊Dic4711yescharacteristic
302D4—82no—
303D4—82no—
304D4—82no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
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—81no—
332C2×C4—81no—
333C2×C4#1309403112yes—
334C8—3762no—
335C8—3762no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22#1568892012yes—
353C22#1568909011yescharacteristic
354C22C2×#1309393911yescharacteristic
355C22#1568910711yescharacteristic
356C4—22no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—81no—
363C4—81no—
364C4—81no—
365C2—21no—
366C2—21no—
367C2—22no—
368C2—41no—
369C2—161no—
370C2#2024173611yescharacteristic
371C2#2024505711yescharacteristic
372C2C2×#1568892011yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2504C2
socleC22×C47C2×#99
prime radicalC2×#121×C47C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2504D47
hypocenterC47#11553
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
1. #20241718C2semidirect
2. C2×#15688849C2semidirect
3. #20241332C2semidirect
4. #20241320C2semidirect
209. C47#11553semidirect

Maximal subgroups

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

subgroupindex
#202417182
C2×#156888492
#202413322
#202413202
C47×#25042
#1155347