Finite Group Databasebeta

← #26106794

Subgroups — #26106794

Order 12032.

698 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×Dic47C211yesmaximal, characteristic
2#20243152C212yesmaximal
3C2×#15688849C211yesmaximal, characteristic
4#20241230C212yesmaximal
5C2×C47×#324C211yesmaximal, characteristic
6C22×C4⋊Dic47C411yescharacteristic
7C22×C4×Dic47C411yescharacteristic
8C22×C4×Dic47C411yescharacteristic
9C22×C4×Dic47C2211yescharacteristic
10C2×C22⋊(C47⋊C8)C2212yes—
11C2×#13093934—22no—
12C4⋊C4×Dic47—22no—
13C4⋊C4×Dic47—22no—
14C2×C47⋊#95C411yescharacteristic
15C2×C47⋊#95C2211yescharacteristic
16#15688877—22no—
17#15688849C414yescomplemented
18C22×C4⋊C4×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2212yes—
20C47×#324C414yescomplemented
21C23×Dic47C2×C411yescharacteristic
22C23×Dic47C2×C411yescharacteristic
23C23×Dic47C2×C411yescharacteristic
24C23×Dic47C2×C411yescharacteristic
25C22×C47⋊C8—22no—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47—22no—
28C2×C4⋊Dic47—22no—
29C2×C4×Dic47—22no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47—22no—
33C2×C4×Dic47—21no—
34C2×C4×Dic47—21no—
35C2×C4×Dic47—42no—
36C2×C4×Dic47—42no—
37C2×C4×Dic47D412yes—
38C2×C4×Dic47D412yes—
39C22⋊(C47⋊C8)—22no—
40C22⋊(C47⋊C8)C2×C414yes—
41#13093934—44no—
42C47⋊#95—22no—
43C47⋊#95—22no—
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47C2×C411yescharacteristic
46C23×C4×C47C2311yescharacteristic
47C22×C8×C47—22no—
48C2×C4⋊C4×C47—22no—
49C2×C4⋊C4×C47—22no—
50C2×C4⋊C4×C47C2×C412yes—
51C2×C4⋊C4×C47C2×C412yes—
52C22⋊C8×C47—22no—
53C22⋊C8×C47C2×C414yes—
54C24×C47C22×C411yescharacteristic
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47C4212yes—
66C22×C4×C47C4212yes—
67C22×C4×C47M1612yes—
68C22×C4×C47M1612yes—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D412yes—
72C22×C4×C47#4112yes—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—22no—
76C22×Dic47—22no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—41no—
84C22×Dic47—41no—
85C22×Dic47—41no—
86C22×Dic47—41no—
87C22×Dic47—41no—
88C22×Dic47—41no—
89C22×Dic47C22⋊C412yes—
90C22×Dic47C22⋊C411yescharacteristic
91C22×Dic47C22⋊C411yescharacteristic
92C22×Dic47M1612yes—
93C22×Dic47M1612yes—
94C2×C8×C47—24no—
95C2×C8×C47—42no—
96C2×C8×C47—42no—
97C4⋊C4×C47—24no—
98C4⋊C4×C47—24no—
99C4⋊Dic47—28no—
100C4×Dic47—24no—
101C4×Dic47—24no—
102C4×Dic47—24no—
103C4×Dic47—42no—
104C4×Dic47—42no—
105C4×Dic47—42no—
106C2×C47⋊C8—24no—
107C2×C47⋊C8—42no—
108C2×C47⋊C8—42no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47C2×C4211yescharacteristic
114C23×C47C2×C22⋊C411yescharacteristic
115C23×C47#11711yescharacteristic
116C23×C47C2×M1611yescharacteristic
117C23×C47C2×M1611yescharacteristic
118C23×C47C2×M1611yescharacteristic
119C23×C47C2×M1611yescharacteristic
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—22no—
124C2×C4×C47—22no—
125C2×C4×C47—22no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—42no—
136C2×C4×C47#9714yescomplemented
137C2×C4×C47C4×D412yes—
138C2×C4×C47C4×D412yes—
139C2×Dic47—24no—
140C2×Dic47—24no—
141C2×Dic47—21no—
142C2×Dic47—21no—
143C2×Dic47—21no—
144C2×Dic47—21no—
145C2×Dic47—22no—
146C2×Dic47—22no—
147C2×Dic47—22no—
148C2×Dic47—22no—
149C2×Dic47—22no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158C2×Dic47—42no—
159C2×Dic47—42no—
160C2×Dic47#9912yes—
161C2×Dic47#10112yes—
162C8×C47—44no—
163C47⋊C8—44no—
164#8153—471nomaximal, Sylow
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—21no—
176C22×C47—21no—
177C22×C47C4×C22⋊C411yescharacteristic
178C22×C47C2×#9711yescharacteristic
179C22×C47C4×M1611yescharacteristic
180C22×C47#40711yescharacteristic
181C22×C47#40711yescharacteristic
182C22×C47C2×#9911yescharacteristic
183C22×C47#41011yescharacteristic
184C22×C47C2×#10111yescharacteristic
185C22×C47#41311yescharacteristic
186C22×C47#43211yescharacteristic
187C22×C47#43211yescharacteristic
188C4×C47—24no—
189C4×C47—22no—
190C4×C47—22no—
191C4×C47—42no—
192Dic47—22no—
193Dic47—22no—
194Dic47—42no—
195Dic47—42no—
196Dic47—42no—
197C2×C4×C4⋊C4—471no—
198#1749—472no—
199C2×#324—471no—
200C2×#324D4711yescharacteristic
201#1280—472no—
202C2×C47—21no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47—21no—
206C2×C47#146311yescharacteristic
207C2×C47#146411yescharacteristic
208C2×C47#146511yescharacteristic
209C2×C47#146711yescharacteristic
210C2×C47#174911yescharacteristic
211C2×C47C4×#9911yescharacteristic
212C2×C47C4×#10111yescharacteristic
213C22×C4⋊C4—471no—
214C22×C4⋊C4C2×D4711yescharacteristic
215C22×C42—471no—
216C22×C42—471no—
217C22×C42—471no—
218C2×C22⋊C8—472no—
219C2×C22⋊C8C2×D4712yes—
220C2×#97—942no—
221C4×C4⋊C4—942no—
222C4×C4⋊C4—942no—
223C2×#95—471no—
224C2×#95—471no—
225#336—942no—
226#324—474no—
227#324Dic4714yescomplemented
228C47#815311yescharacteristic, complemented, Sylow
229C23×C4—471no—
230C23×C4—471no—
231C23×C4—471no—
232C23×C4—471no—
233C23×C4C4×D4711yescharacteristic
234C23×C4C4×D4711yescharacteristic
235C23×C4C22×D4711yescharacteristic
236C22×C8—22no—
237C22×C8—942no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—942no—
241C2×C4⋊C4—942no—
242C2×C4⋊C4—942no—
243C2×C4⋊C4C4×D4712yes—
244C2×C4⋊C4C2×Dic4712yes—
245C2×C42—472no—
246C2×C42—472no—
247C2×C42—941no—
248C2×C42—941no—
249C2×C42—942no—
250C2×C42—942no—
251C2×C42—941no—
252C2×C42—941no—
253C2×C42—1882no—
254C2×C42—1882no—
255C22⋊C8—22no—
256C22⋊C8—474no—
257C22⋊C8—942no—
258C22⋊C8C2×Dic4714yes—
259#97—1884no—
260#95—942no—
261#95—942no—
262C24C2×C4×D4711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—471no—
274C22×C4—471no—
275C22×C4—472no—
276C22×C4—472no—
277C22×C4—472no—
278C22×C4—942no—
279C22×C4—942no—
280C22×C4—941no—
281C22×C4—941no—
282C22×C4—941no—
283C22×C4—941no—
284C22×C4—941no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4—1881no—
289C22×C4—1881no—
290C22×C4—1881no—
291C22×C4—1881no—
292C22×C4—1881no—
293C22×C4—1881no—
294C22×C4#1062532412yes—
295C22×C4#1062532412yes—
296C22×C4C4×Dic4712yes—
297C22×C4C4×Dic4712yes—
298C22×C4C2×C4×D4711yescharacteristic
299C22×C4D4×D4712yes—
300C22×C4#1062535212yes—
301C22×C4C22×Dic4711yescharacteristic
302C2×C8—24no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—944no—
306C2×C8—1882no—
307C2×C8—1882no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—948no—
311C42—944no—
312C42—944no—
313C42—944no—
314C42—1882no—
315C42—1882no—
316C42—1882no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C4×Dic4711yescharacteristic
322C23C22⋊C4×D4711yescharacteristic
323C23#1309398011yescharacteristic
324C23M16×D4711yescharacteristic
325C23M16×D4711yescharacteristic
326C23C2×#1062532411yescharacteristic
327C23C2×#1062532411yescharacteristic
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—472no—
345C2×C4—472no—
346C2×C4—944no—
347C2×C4—944no—
348C2×C4—941no—
349C2×C4—941no—
350C2×C4—941no—
351C2×C4—941no—
352C2×C4—942no—
353C2×C4—942no—
354C2×C4—942no—
355C2×C4—942no—
356C2×C4—942no—
357C2×C4—1882no—
358C2×C4—1882no—
359C2×C4—1882no—
360C2×C4—1882no—
361C2×C4—1881no—
362C2×C4—1881no—
363C2×C4—1881no—
364C2×C4—1881no—
365C2×C4—1882no—
366C2×C4—1882no—
367C2×C4#1309393414yescomplemented
368C2×C4(C22⋊D47)⋊C412yes—
369C2×C4D4×Dic4712yes—
370C8—44no—
371C8—1884no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22C22⋊C4×Dic4711yescharacteristic
385C22C2×#1309393411yescharacteristic
386C22M16×Dic4711yescharacteristic
387C22#1568908511yescharacteristic
388C22#1568908511yescharacteristic
389C22#99×D4711yescharacteristic
390C22#1568909511yescharacteristic
391C22#101×D4711yescharacteristic
392C22#1568910611yescharacteristic
393C22#1568917711yescharacteristic
394C22#1568917711yescharacteristic
395C4—24no—
396C4—22no—
397C4—22no—
398C4—42no—
399C4—942no—
400C4—942no—
401C4—1882no—
402C4—1882no—
403C4—1882no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2#2024171611yescharacteristic
409C2#2024172111yescharacteristic
410C2#2024172511yescharacteristic
411C2#2024173011yescharacteristic
412C2#2024315211yescharacteristic
413C2#99×Dic4711yescharacteristic
414C2#101×Dic4711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC23×C47C2×C42
FrattiniC23×C4C22×D47
FittingC2×C47×#324C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22×C4
cosocleC23×C4×C47C23
hypercenterC2×#324D47
hypocenterC47#8153
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
17. #15688849C4semidirect
20. C47×#324C4semidirect
136. C2×C4×C47#97semidirect
227. #324Dic47semidirect
228. C47#8153semidirect
367. C2×C4#13093934semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×Dic472
#202431522
C2×#156888492
#202412302
C2×C47×#3242
#815347