Finite Group Databasebeta

← #26182353

Subgroups — #26182353

Order 12032.

970 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13094014C211yesmaximal, characteristic
2#20252017C212yesmaximal
3#20241742C218yesmaximal, complemented
4#20241718C211yesmaximal, characteristic
5#20241717C212yesmaximal
6C47×#1463C211yesmaximal, characteristic
7C2×#13094014—22no—
8C2×#13094014—22no—
9C2×#13094014—22no—
10C2×#13094014—22no—
11C2×#13094014C414yes—
12C2×#13094014C2214yes—
13C22×C4⋊Dic47C411yescharacteristic
14C22×Dic47⋊C4C411yescharacteristic
15C22×Dic47⋊C4C411yescharacteristic
16C22×Dic47⋊C4C411yescharacteristic
17C22×Dic47⋊C4C2211yescharacteristic
18C22×C4×Dic47C2211yescharacteristic
19#15689177—22no—
20#15689177C2214yes—
21#15689173—22no—
22#15689173C2214yes—
23#15689146—22no—
24#15689087C2212yes—
25M16×Dic47—22no—
26#15688849C2214yes—
27#15688848C2218yes—
28C22×C4⋊C4×C47C2211yescharacteristic
29C47×#407C2212yes—
30C47×#324C2214yes—
31C23×Dic47C2×C411yescharacteristic
32C23×Dic47C2×C411yescharacteristic
33C23×Dic47C2×C411yescharacteristic
34C23×Dic47C2×C411yescharacteristic
35C2×#10625329—22no—
36#13094014—24no—
37#13094014—24no—
38#13094014—24no—
39#13094014—24no—
40#13094014—24no—
41#13094014—24no—
42#13094014—24no—
43#13094014—24no—
44C2×C4⋊Dic47—21no—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47C2×C414yes—
48C2×Dic47⋊C4—22no—
49C2×Dic47⋊C4—22no—
50C2×Dic47⋊C4—21no—
51C2×Dic47⋊C4—22no—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4—21no—
54C2×Dic47⋊C4—21no—
55C2×Dic47⋊C4—21no—
56C2×Dic47⋊C4—21no—
57C2×Dic47⋊C4—22no—
58C2×Dic47⋊C4—21no—
59C2×Dic47⋊C4C2×C414yes—
60C2×Dic47⋊C4C2×C414yes—
61C2×Dic47⋊C4C2×C414yes—
62C2×Dic47⋊C4C2×C412yes—
63C2×Dic47⋊C4C2312yes—
64C2×C4×Dic47C2×C412yes—
65C2×C4×Dic47D412yes—
66C2×C4×Dic47D412yes—
67C2×C4×Dic47D412yes—
68C2×C4×Dic47D412yes—
69C2×C4×Dic47C2312yes—
70Dic47⋊C8—24no—
71Dic47⋊C8—24no—
72C22⋊(C47⋊C8)—22no—
73C22⋊(C47⋊C8)C2314yes—
74#13093934—24no—
75C8×Dic47—24no—
76C23×C4×C47C2×C411yescharacteristic
77C23×C4×C47C2×C411yescharacteristic
78C23×C4×C47C2311yescharacteristic
79C2×M16×C47—22no—
80C2×C4⋊C4×C47—22no—
81C2×C4⋊C4×C47—22no—
82C2×C4⋊C4×C47C2×C412yes—
83C2×C4⋊C4×C47C2312yes—
84C22⋊C8×C47—22no—
85C22⋊C8×C47C2314yes—
86C24×C47C22×C411yescharacteristic
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—42no—
92C22×C4×C47—42no—
93C22×C4×C47—42no—
94C22×C4×C47C22×C412yes—
95C22×C4×C47C22×C412yes—
96C22×C4×C47C22×C411yescharacteristic
97C22×C4×C47C2×D412yes—
98C22×C4×C47C2×D412yes—
99C22×C4×C47C2411yescharacteristic
100C22×Dic47—24no—
101C22×Dic47—21no—
102C22×Dic47—21no—
103C22×Dic47—21no—
104C22×Dic47—21no—
105C22×Dic47—42no—
106C22×Dic47—42no—
107C22×Dic47—42no—
108C22×Dic47C22⋊C411yescharacteristic
109C22×Dic47C22⋊C412yes—
110C22×Dic47C22⋊C411yescharacteristic
111C22×Dic47C22⋊C411yescharacteristic
112C22×Dic47C22⋊C412yes—
113C22×Dic47C22⋊C411yescharacteristic
114C22×Dic47C22×C412yes—
115C22×Dic47C22×C412yes—
116C22×Dic47C22×C412yes—
117C22×Dic47C22×C412yes—
118M16×C47—44no—
119C2×C8×C47—24no—
120C4⋊C4×C47—24no—
121C4⋊C4×C47—24no—
122C4⋊Dic47—24no—
123C4⋊Dic47—24no—
124Dic47⋊C4—24no—
125Dic47⋊C4—24no—
126Dic47⋊C4—24no—
127Dic47⋊C4—24no—
128Dic47⋊C4—24no—
129Dic47⋊C4—24no—
130Dic47⋊C4—24no—
131Dic47⋊C4—24no—
132C4×Dic47C22⋊C412yes—
133C4×Dic47C22⋊C412yes—
134C4×Dic47C22⋊C412yes—
135C4×Dic47C22⋊C412yes—
136C4×Dic47C2×D412yes—
137C4×Dic47C2×D412yes—
138C4×Dic47C2×D412yes—
139C4×Dic47C2×D412yes—
140#10625329—44no—
141C2×C47⋊C8—24no—
142C23×C47—22no—
143C23×C47—22no—
144C23×C47—22no—
145C23×C47C2×C22⋊C411yescharacteristic
146C23×C47C2×C22⋊C411yescharacteristic
147C23×C47C23×C411yescharacteristic
148C2×C4×C47—24no—
149C2×C4×C47—22no—
150C2×C4×C47—22no—
151C2×C4×C47—24no—
152C2×C4×C47—41no—
153C2×C4×C47—42no—
154C2×C4×C47—41no—
155C2×C4×C47—42no—
156C2×C4×C47—42no—
157C2×C4×C47—42no—
158C2×C4×C47—42no—
159C2×C4×C47—42no—
160C2×C4×C47C2×C22⋊C412yes—
161C2×C4×C47C22×D412yes—
162C2×Dic47—24no—
163C2×Dic47—22no—
164C2×Dic47—24no—
165C2×Dic47—24no—
166C2×Dic47—22no—
167C2×Dic47—22no—
168C2×Dic47—24no—
169C2×Dic47—41no—
170C2×Dic47—41no—
171C2×Dic47—42no—
172C2×Dic47—42no—
173C2×Dic47—42no—
174C2×Dic47—42no—
175C2×Dic47—42no—
176C2×Dic47—42no—
177C2×Dic47#10114yes—
178C2×Dic47C2×C22⋊C412yes—
179C2×Dic47C2×C22⋊C412yes—
180C2×Dic47C2×C22⋊C412yes—
181C2×Dic47C2×C22⋊C412yes—
182C8×C47—44no—
183C47⋊C8—44no—
184#17394—471nomaximal, Sylow
185C22×C47—28no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47#41012yes—
193C22×C47C2×#10112yes—
194C22×C47C22×C22⋊C411yescharacteristic
195C22×C47#56812yes—
196C4×C47—22no—
197C4×C47—22no—
198C4×C47—42no—
199C4×C47—42no—
200Dic47—22no—
201Dic47—22no—
202Dic47—22no—
203Dic47—42no—
204Dic47—42no—
205Dic47C2×#10114yes—
206C22×#125—471no—
207#2920—472no—
208#1470—478no—
209#1463—472no—
210#1463—471no—
211#1463D4711yescharacteristic
212C2×C47—22no—
213C2×C47—22no—
214C2×C47—22no—
215C2×C47#287911yescharacteristic
216C2×C47C2×#41011yescharacteristic
217C2×C47C22×#10111yescharacteristic
218C2×#125—474no—
219C2×#125—474no—
220C2×#125—942no—
221C2×#125—942no—
222C2×#125—942no—
223C2×#125—942no—
224C22×C4⋊C4—471no—
225C22×C4⋊C4—471no—
226C22×C4⋊C4—471no—
227C22×C4⋊C4—471no—
228C22×C4⋊C4—471no—
229C22×C4⋊C4C2×D4711yescharacteristic
230C22×C42—471no—
231#432—474no—
232#432—942no—
233#431—474no—
234#431—942no—
235#423—942no—
236#407—472no—
237#407C2×D4712yes—
238C4×M16—942no—
239#324—474no—
240#324—478no—
241#324C2×D4714yes—
242C47#1739411yescharacteristic, complemented, Sylow
243C23×C4—471no—
244C23×C4—471no—
245C23×C4—471no—
246C23×C4—471no—
247C23×C4C4×D4711yescharacteristic
248C23×C4C4×D4711yescharacteristic
249C23×C4C22×D4711yescharacteristic
250C2×M16—22no—
251C2×M16—942no—
252#125—944no—
253#125—944no—
254#125—944no—
255#125—944no—
256#125—944no—
257#125—944no—
258#125—944no—
259#125—944no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—474no—
263C2×C4⋊C4—474no—
264C2×C4⋊C4—474no—
265C2×C4⋊C4—472no—
266C2×C4⋊C4—474no—
267C2×C4⋊C4—472no—
268C2×C4⋊C4—941no—
269C2×C4⋊C4—941no—
270C2×C4⋊C4—942no—
271C2×C4⋊C4—942no—
272C2×C4⋊C4—941no—
273C2×C4⋊C4—941no—
274C2×C4⋊C4—941no—
275C2×C4⋊C4—941no—
276C2×C4⋊C4—942no—
277C2×C4⋊C4—942no—
278C2×C4⋊C4—941no—
279C2×C4⋊C4—941no—
280C2×C4⋊C4—942no—
281C2×C4⋊C4—942no—
282C2×C4⋊C4C4×D4712yes—
283C2×C4⋊C4C22×D4712yes—
284C2×C42—472no—
285C2×C42—472no—
286C2×C42—472no—
287C2×C42—472no—
288C2×C42—472no—
289C2×C42—472no—
290C4⋊C8—944no—
291C4⋊C8—944no—
292C22⋊C8—22no—
293C22⋊C8—474no—
294C22⋊C8—942no—
295C22⋊C8C22×D4714yes—
296#97—944no—
297C4×C8—944no—
298C24C2×C4×D4711yescharacteristic
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—471no—
307C22×C4—472no—
308C22×C4—471no—
309C22×C4—472no—
310C22×C4—472no—
311C22×C4—471no—
312C22×C4—471no—
313C22×C4—472no—
314C22×C4—472no—
315C22×C4—472no—
316C22×C4—944no—
317C22×C4—941no—
318C22×C4—941no—
319C22×C4—941no—
320C22×C4—941no—
321C22×C4—1882no—
322C22×C4—1882no—
323C22×C4—1882no—
324C22×C4C2×C4×D4712yes—
325C22×C4C2×C4×D4712yes—
326C22×C4C2×C4×D4711yescharacteristic
327C22×C4D4×D4712yes—
328C22×C4D4×D4712yes—
329C22×C4C23×D4711yescharacteristic
330M16—44no—
331M16—1884no—
332C2×C8—24no—
333C2×C8—944no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—944no—
337C4⋊C4—944no—
338C4⋊C4—944no—
339C4⋊C4—944no—
340C4⋊C4—944no—
341C4⋊C4—944no—
342C4⋊C4—944no—
343C4⋊C4—944no—
344C4⋊C4—944no—
345C4⋊C4—944no—
346C42—472no—
347C42—472no—
348C42—472no—
349C42—472no—
350C42—472no—
351C42—472no—
352C42—472no—
353C42—472no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23C22⋊C4×D4711yescharacteristic
358C23C22⋊C4×D4711yescharacteristic
359C23C22×C4×D4711yescharacteristic
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—472no—
373C2×C4—474no—
374C2×C4—472no—
375C2×C4—472no—
376C2×C4—472no—
377C2×C4—944no—
378C2×C4—944no—
379C2×C4—942no—
380C2×C4—944no—
381C2×C4—942no—
382C2×C4—942no—
383C2×C4—944no—
384C2×C4—1881no—
385C2×C4—1881no—
386C2×C4—1882no—
387C2×C4—1882no—
388C2×C4—1882no—
389C2×C4—1882no—
390C2×C4—1882no—
391C2×C4—1882no—
392C2×C4C22⋊C4×D4712yes—
393C2×C4C2×D4×D4712yes—
394C8—44no—
395C8—1884no—
396C22—28no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22#1568909512yes—
404C22#101×D4712yes—
405C22C2×C22⋊C4×D4711yescharacteristic
406C22#1568988912yes—
407C4—22no—
408C4—22no—
409C4—42no—
410C4—42no—
411C4—474no—
412C4—942no—
413C4—942no—
414C4—942no—
415C4—1882no—
416C4—1882no—
417C2—22no—
418C2—22no—
419C2—22no—
420C2#2025183111yescharacteristic
421C2C2×#1568909511yescharacteristic
422C2C2×#101×D4711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1463C2
socleC22×C47C22×C22⋊C4
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenter#1463D47
hypocenterC47#17394
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #20241742C2semidirect
242. C47#17394semidirect

Maximal subgroups

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

subgroupindex
C22×#130940142
#202520172
#202417422
#202417182
#202417172
C47×#14632
#1739447