Finite Group Databasebeta

← #26108807

Subgroups — #26108807

Order 12032.

910 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 139 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#20242114C212yesmaximal, complemented
2#20242012C212yesmaximal, complemented
3#20241722C212yesmaximal, complemented
4C2×C47×#536C211yesmaximal, characteristic
5#15689180C2214yes—
6#15688872—24no—
7#15688863—24no—
8#15688849—22no—
9#15688847—22no—
10C22×#41×C47C411yescharacteristic
11C23×Q8×C47C411yescharacteristic
12C47×#536—24no—
13C47×#536—24no—
14C2×C4⋊Q8×C47C411yescharacteristic
15C2×C4⋊Q8×C47C2211yescharacteristic
16C2×#124×C47C411yescharacteristic
17C2×#124×C47C2211yescharacteristic
18C2×#123×C47—22no—
19C2×C22⋊Q8×C47—22no—
20C2×#117×C47C2211yescharacteristic
21C4⋊(C47⋊C8)—24no—
22C22⋊(C47⋊C8)—24no—
23#13093935—24no—
24C2×#41×C47—22no—
25C2×#41×C47—22no—
26C2×#41×C47—42no—
27C2×#41×C47—42no—
28C22×Q8×C47—22no—
29C22×Q8×C47—22no—
30C22×Q8×C47—21no—
31C22×Q8×C47—42no—
32C22×Q8×C47—42no—
33C22×Q8×C47C2×C411yescharacteristic
34C22×Q8×C47C2×C411yescharacteristic
35C22×Q8×C47C2×C411yescharacteristic
36C22×D4×C47—21no—
37C22×D4×C47C2×C411yescharacteristic
38C23×C4×C47—21no—
39C23×C4×C47—21no—
40C23×C4×C47C2×C411yescharacteristic
41C4⋊Q8×C47—22no—
42C4⋊Q8×C47—22no—
43C4⋊Q8×C47—22no—
44C4⋊Q8×C47D414yes—
45#124×C47—22no—
46#124×C47—22no—
47#124×C47—22no—
48#124×C47D414yes—
49#123×C47—216no—
50C22⋊Q8×C47—216no—
51#117×C47—22no—
52#117×C47—22no—
53C2×C4⋊C4×C47—22no—
54C2×C4⋊C4×C47—22no—
55C2×C4⋊C4×C47D412yes—
56C2×C22⋊C4×C47—22no—
57C2×C22⋊C4×C47—22no—
58C2×C22⋊C4×C47D412yes—
59C2×C42×C47C2×C411yescharacteristic
60C2×C42×C47C2311yescharacteristic
61C24×C47—21no—
62C24×C47C22⋊C411yescharacteristic
63#41×C47—48no—
64#41×C47—48no—
65C2×Q8×C47—24no—
66C2×Q8×C47—21no—
67C2×Q8×C47—21no—
68C2×Q8×C47—21no—
69C2×Q8×C47—48no—
70C2×Q8×C47—48no—
71C2×Q8×C47—42no—
72C2×Q8×C47—41no—
73C2×Q8×C47—41no—
74C2×Q8×C47—41no—
75C2×Q8×C47—41no—
76C2×Q8×C47—41no—
77C2×Q8×C47—41no—
78C2×Q8×C47—42no—
79C2×Q8×C47C22⋊C412yes—
80C2×Q8×C47C22⋊C412yes—
81C2×Q8×C47C22⋊C412yes—
82C2×D4×C47—24no—
83C2×D4×C47—21no—
84C2×D4×C47—41no—
85C2×D4×C47—41no—
86C2×D4×C47—42no—
87C2×D4×C47—42no—
88C2×D4×C47C22⋊C412yes—
89C22×C4×C47—22no—
90C22×C4×C47—22no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—24no—
94C22×C4×C47—24no—
95C22×C4×C47—21no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—42no—
99C22×C4×C47—42no—
100C22×C4×C47—42no—
101C22×C4×C47—42no—
102C22×C4×C47C22⋊C411yescharacteristic
103C22×C4×C47C22⋊C412yes—
104C22×C4×C47C22×C411yescharacteristic
105C22×C4×C47C2×D412yes—
106C4⋊C4×C47—28no—
107C4⋊C4×C47—28no—
108C4⋊C4×C47—24no—
109C22⋊C4×C47—28no—
110C22⋊C4×C47—28no—
111C22⋊C4×C47—24no—
112C42×C47—21no—
113C42×C47—21no—
114C42×C47C2×D412yes—
115C42×C47C2×D412yes—
116C2×C47⋊C8—44no—
117C23×C47—21no—
118C23×C47—21no—
119C23×C47—22no—
120C23×C47—42no—
121C23×C47—44no—
122C23×C47—41no—
123C23×C47—41no—
124C23×C47#10012yes—
125C23×C47C2×C22⋊C411yescharacteristic
126Q8×C47—42no—
127Q8×C47—42no—
128Q8×C47—42no—
129Q8×C47—48no—
130Q8×C47—42no—
131Q8×C47—42no—
132Q8×C47—42no—
133D4×C47—42no—
134D4×C47—42no—
135D4×C47—48no—
136C2×C4×C47—24no—
137C2×C4×C47—24no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47—22no—
145C2×C4×C47—22no—
146C2×C4×C47—42no—
147C2×C4×C47—48no—
148C2×C4×C47—48no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—42no—
156C2×C4×C47#10012yes—
157C2×C4×C47C2×C22⋊C411yescharacteristic
158C2×C4×C47C2×C22⋊C411yescharacteristic
159C2×C4×C47#12014yes—
160C47⋊C8—84no—
161#8524—471nomaximal, Sylow
162C22×C47—22no—
163C22×C47—22no—
164C22×C47—42no—
165C22×C47—42no—
166C22×C47—42no—
167C22×C47—44no—
168C22×C47—42no—
169C22×C47—82no—
170C22×C47—82no—
171C22×C47#37911yescharacteristic
172C22×C47C2×#10011yescharacteristic
173C22×C47C2×#10011yescharacteristic
174C4×C47—24no—
175C4×C47—22no—
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—42no—
179C4×C47—42no—
180C4×C47—42no—
181C4×C47—42no—
182C2×#536D4711yescharacteristic
183#1558—472no—
184#1533—472no—
185#1464—472no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—42no—
189C2×C47—82no—
190C2×C47#178311yescharacteristic
191C2×C47#179612yes—
192C22×#41Dic4711yescharacteristic
193C23×Q8Dic4711yescharacteristic
194#536—24no—
195#536—24no—
196C2×C4⋊Q8Dic4711yescharacteristic
197C2×C4⋊Q8C2×D4711yescharacteristic
198C2×#124Dic4711yescharacteristic
199C2×#124C2×D4711yescharacteristic
200C2×#123—22no—
201C2×C22⋊Q8—22no—
202C2×#117C2×D4711yescharacteristic
203#432—474no—
204#333—944no—
205#329—944no—
206#324—942no—
207#323—942no—
208C47#852411yescharacteristic, complemented, Sylow
209C2×#41—22no—
210C2×#41—22no—
211C2×#41—42no—
212C2×#41—42no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—21no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8C2×Dic4711yescharacteristic
219C22×Q8C2×Dic4711yescharacteristic
220C22×Q8C2×Dic4711yescharacteristic
221C22×D4—21no—
222C22×D4C2×Dic4711yescharacteristic
223C23×C4—21no—
224C23×C4—21no—
225C23×C4C2×Dic4711yescharacteristic
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8C22⋊D4714yes—
230#124—22no—
231#124—22no—
232#124—22no—
233#124C22⋊D4714yes—
234#123—216no—
235C22⋊Q8—216no—
236#117—22no—
237#117—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C22⋊D4712yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4C22⋊D4712yes—
244C2×C42C2×Dic4711yescharacteristic
245C2×C42C22×D4711yescharacteristic
246C4⋊C8—944no—
247C22⋊C8—944no—
248#97—944no—
249C24—21no—
250C24C22⋊Dic4711yescharacteristic
251#41—48no—
252#41—48no—
253C2×Q8—24no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—42no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8C22⋊Dic4712yes—
268C2×Q8C22⋊Dic4712yes—
269C2×Q8C22⋊Dic4712yes—
270C2×D4—24no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4C22⋊Dic4712yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4C22⋊Dic4711yescharacteristic
291C22×C4C22⋊Dic4712yes—
292C22×C4C22×Dic4711yescharacteristic
293C22×C4C2×C22⋊D4712yes—
294C2×C8—1884no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C22⋊C4—28no—
299C22⋊C4—28no—
300C22⋊C4—24no—
301C42—21no—
302C42—21no—
303C42C2×C22⋊D4712yes—
304C42C2×C22⋊D4712yes—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—42no—
309C23—44no—
310C23—41no—
311C23—41no—
312C23#1309394112yes—
313C23C2×C22⋊Dic4711yescharacteristic
314Q8—42no—
315Q8—42no—
316Q8—42no—
317Q8—48no—
318Q8—42no—
319Q8—42no—
320Q8—42no—
321D4—42no—
322D4—42no—
323D4—48no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4#1309394112yes—
345C2×C4C2×C22⋊Dic4711yescharacteristic
346C2×C4C2×C22⋊Dic4711yescharacteristic
347C2×C4#1309399314yes—
348C8—3764no—
349C22—22no—
350C22—22no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—44no—
356C22—82no—
357C22—82no—
358C22#1568897911yescharacteristic
359C22C2×#1309394111yescharacteristic
360C22C2×#1309394111yescharacteristic
361C4—22no—
362C4—21no—
363C4—21no—
364C4—24no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4—42no—
369C2—21no—
370C2—21no—
371C2—42no—
372C2—82no—
373C2#2024330911yescharacteristic
374C2#2024336312yes—
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#536C2
socleC22×C47#379
prime radicalC22×#41×C47C4
cosocleC2×C42×C47C23
hypercenterC2×#536D47
hypocenterC47#8524
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. #20242114C2semidirect
2. #20242012C2semidirect
3. #20241722C2semidirect
208. C47#8524semidirect

Maximal subgroups

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

subgroupindex
#202421142
#202420122
#202417222
C2×C47×#5362
#852447