Finite Group Databasebeta

← #26125834

Subgroups — #26125834

Order 12032.

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

normal subgroups only

This group has 496 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#20241334C212yesmaximal, complemented
4#20241320C212yesmaximal, complemented
5C47×#2501C211yesmaximal, characteristic
6#15689087—21no—
7C2×C22⋊(C47⋊C8)—21no—
8#15688849—21no—
9#15688849—21no—
10#15688849C2212yes—
11#15688849C2212yes—
12C2×#123×C47—21no—
13C2×#123×C47—21no—
14C2×#121×C47C411yescharacteristic
15C22×C4⋊C4×C47C2211yescharacteristic
16C47×#399—21no—
17C47×#399—21no—
18C47×#390C411yescharacteristic
19C47×#390C2211yescharacteristic
20C47×#384C411yescharacteristic
21C47×#384C2211yescharacteristic
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#123×C47—44no—
35#123×C47—44no—
36#121×C47—41no—
37#121×C47—41no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—41no—
43C2×C4⋊C4×C47—41no—
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47C2311yescharacteristic
49C2×C22⋊C4×C47—21no—
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×C42×C47C2×C411yescharacteristic
56C2×C42×C47C2×C411yescharacteristic
57#95×C47—21no—
58C24×C47—21no—
59C24×C47C22⋊C411yescharacteristic
60C2×D4×C47—22no—
61C2×D4×C47—41no—
62C2×D4×C47—41no—
63C2×D4×C47—42no—
64C2×D4×C47—42no—
65C2×D4×C47—41no—
66C2×D4×C47—41no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—22no—
76C22×C4×C47—21no—
77C22×C4×C47—41no—
78C22×C4×C47—42no—
79C22×C4×C47—42no—
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22⋊C411yescharacteristic
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—42no—
91C4⋊C4×C47—42no—
92C4⋊C4×C47—42no—
93C4⋊C4×C47—42no—
94C4⋊C4×C47—41no—
95C22⋊C4×C47—24no—
96C22⋊C4×C47—42no—
97C22⋊C4×C47—42no—
98C22⋊C4×C47—42no—
99C22⋊C4×C47—42no—
100C22⋊C4×C47—42no—
101C42×C47—22no—
102C42×C47—41no—
103#10625329—82no—
104C2×C47⋊C8—42no—
105C2×C47⋊C8—42no—
106C2×C47⋊C8—82no—
107C23×C47—22no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47#9911yescharacteristic
119C23×C47#9911yescharacteristic
120C23×C47C2×C22⋊C411yescharacteristic
121D4×C47—82no—
122D4×C47—82no—
123D4×C47—82no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—42no—
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—81no—
151C2×C4×C47—81no—
152C2×C4×C47#13112yes—
153C47⋊C8—82no—
154C47⋊C8—82no—
155#11549—471nomaximal, Sylow
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—42no—
164C22×C47—42no—
165C22×C47—81no—
166C22×C47—81no—
167C22×C47—81no—
168C22×C47—81no—
169C22×C47—81no—
170C22×C47—81no—
171C22×C47—81no—
172C22×C47#35312yes—
173C22×C47#40811yescharacteristic
174C22×C47C2×#9911yescharacteristic
175C22×C47#41311yescharacteristic
176C4×C47—22no—
177C4×C47—42no—
178C4×C47—42no—
179C4×C47—41no—
180C4×C47—41no—
181C4×C47—41no—
182C4×C47—41no—
183C4×C47—41no—
184C4×C47—81no—
185#2501D4711yescharacteristic
186#1463—471no—
187C2×#324—471no—
188#1325—472no—
189#1318—472no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47—22no—
193C2×C47—41no—
194C2×C47—161no—
195C2×C47#146811yescharacteristic
196C2×C47C2×#35311yescharacteristic
197C2×C47#213311yescharacteristic
198C2×#123—21no—
199C2×#123—21no—
200C2×#121Dic4711yescharacteristic
201C22×C4⋊C4C2×D4711yescharacteristic
202#407—941no—
203C2×C22⋊C8—941no—
204#399—21no—
205#399—21no—
206#390Dic4711yescharacteristic
207#390C2×D4711yescharacteristic
208#384Dic4711yescharacteristic
209#384C2×D4711yescharacteristic
210C4×C22⋊C4Dic4711yescharacteristic
211#324—472no—
212#324—472no—
213#324—941no—
214#324—941no—
215C47#1154911yescharacteristic, complemented, Sylow
216C22×D4—21no—
217C22×D4C2×Dic4711yescharacteristic
218C23×C4C2×Dic4711yescharacteristic
219C23×C4C22⋊D4711yescharacteristic
220C23×C4C22⋊D4711yescharacteristic
221C2×M16—1881no—
222C22×C8—1881no—
223#123—44no—
224#123—44no—
225#121—41no—
226#121—41no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4C2×Dic4711yescharacteristic
234C2×C4⋊C4C2×Dic4711yescharacteristic
235C2×C4⋊C4C22⋊D4711yescharacteristic
236C2×C4⋊C4C22⋊D4711yescharacteristic
237C2×C4⋊C4C22×D4711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C42C2×Dic4711yescharacteristic
245C2×C42C2×Dic4711yescharacteristic
246C22⋊C8—942no—
247C22⋊C8—942no—
248C22⋊C8—1881no—
249C22⋊C8—1881no—
250#95—21no—
251C24—21no—
252C24C22⋊Dic4711yescharacteristic
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4C22⋊Dic4711yescharacteristic
274C22×C4C22⋊Dic4711yescharacteristic
275C22×C4C22⋊Dic4711yescharacteristic
276C22×C4C22×Dic4711yescharacteristic
277C22×C4C2×C22⋊D4711yescharacteristic
278C22×C4C2×C22⋊D4711yescharacteristic
279M16—3762no—
280C2×C8—3762no—
281C2×C8—1882no—
282C2×C8—1882no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C22⋊C4—24no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C42—22no—
299C42—41no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23#1309393911yescharacteristic
312C23#1309393911yescharacteristic
313C23C2×C22⋊Dic4711yescharacteristic
314D4—82no—
315D4—82no—
316D4—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4#1309403112yes—
346C8—3762no—
347C8—3762no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—42no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#1568891612yes—
365C22#1568909011yescharacteristic
366C22C2×#1309393911yescharacteristic
367C22#1568910711yescharacteristic
368C4—22no—
369C4—42no—
370C4—42no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—81no—
377C2—21no—
378C2—21no—
379C2—22no—
380C2—41no—
381C2—161no—
382C2#2024173611yescharacteristic
383C2C2×#1568891611yescharacteristic
384C2#2024507711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2501C2
socleC22×C47C2×#99
prime radicalC2×#121×C47C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2501D47
hypocenterC47#11549
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
1. #20241718C2semidirect
2. C2×#15688849C2semidirect
3. #20241334C2semidirect
4. #20241320C2semidirect
215. C47#11549semidirect

Maximal subgroups

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

subgroupindex
#202417182
C2×#156888492
#202413342
#202413202
C47×#25012
#1154947