Finite Group Databasebeta

← #26218367

Subgroups — #26218367

Order 12032.

914 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689078C211yesmaximal, characteristic
2#20245871C211yesmaximal, characteristic
3#20243949C214yesmaximal, complemented
4#20243942C214yesmaximal, complemented
5#20243084C211yesmaximal, characteristic
6#20243084C211yesmaximal, characteristic
7#20243084C211yesmaximal, characteristic
8#20243082C211yesmaximal, characteristic
9C22×C4⋊C8×C47C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C2×#13093983C2211yescharacteristic
13C2×#13093983C2211yescharacteristic
14#15689078—24no—
15C2×#13093934C2212yes—
16C2×C8×Dic47C2212yes—
17#15689021C412yes—
18#15689021C412yes—
19#15688996C412yes—
20#15688996C412yes—
21C4×Dic47⋊C4C2214yes—
22#15688878C2214yes—
23#15688877C2212yes—
24#15688877C2214yes—
25#15688877C2214yes—
26#15688877C2212yes—
27C23×C8×C47C2211yescharacteristic
28C23×C8×C47C2211yescharacteristic
29C22×C42×C47C2211yescharacteristic
30C2×C4⋊C8×C47—24no—
31C2×C4⋊C8×C47C2212yes—
32C2×C4⋊C8×C47C2212yes—
33C2×#10625329—22no—
34C2×#10625329—22no—
35C2×#10625329—22no—
36C2×#10625329—22no—
37C2×#10625329—22no—
38C2×#10625329Q814yes—
39C22×C47⋊C8C2312yes—
40C22×C47⋊C8C2312yes—
41#13093983—22no—
42#13093983—22no—
43#13093983—22no—
44#13093983—22no—
45C2×C4⋊Dic47C2×C412yes—
46C2×C4⋊Dic47C2×C411yescharacteristic
47C2×C4⋊Dic47C2×C411yescharacteristic
48C2×Dic47⋊C4C2×C414yes—
49C2×Dic47⋊C4C2×C414yes—
50C2×C22⋊Dic47C2×C412yes—
51C2×C22⋊Dic47C2×C412yes—
52C2×C4×Dic47C2312yes—
53C2×C4×Dic47C2312yes—
54#13093934—24no—
55C8×Dic47—24no—
56C47⋊#95C2×C414yes—
57C47⋊#95C2×C414yes—
58C23×C4×C47C2×C411yescharacteristic
59C23×C4×C47C2×C411yescharacteristic
60C23×C4×C47C2311yescharacteristic
61C22×C8×C47—22no—
62C22×C8×C47—22no—
63C22×C8×C47—22no—
64C22×C8×C47—22no—
65C22×C8×C47—22no—
66C22×C8×C47D414yes—
67C22×C8×C47C2312yes—
68C22×C8×C47C2311yescharacteristic
69C22×C8×C47C2311yescharacteristic
70C2×C42×C47—22no—
71C2×C42×C47—22no—
72C2×C42×C47C2×C412yes—
73C2×C42×C47C2312yes—
74C4⋊C8×C47—24no—
75C4⋊C8×C47—24no—
76C24×C47C22×C411yescharacteristic
77C22×C4×C47—22no—
78C22×C4×C47—24no—
79C22×C4×C47—22no—
80C22×C4×C47—42no—
81C22×C4×C47C22×C412yes—
82C22×C4×C47C22×C412yes—
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C2×D412yes—
85C22×C4×C47C2×Q812yes—
86C22×C4×C47#4112yes—
87C22×C4×C47#4112yes—
88C22×C4×C47#4112yes—
89C22×C4×C47#4112yes—
90C22×C4×C47C2411yescharacteristic
91C22×Dic47C22×C414yes—
92C22×Dic47C22×C412yes—
93C22×Dic47C22×C412yes—
94C2×C8×C47—28no—
95C2×C8×C47—21no—
96C2×C8×C47—21no—
97C2×C8×C47—21no—
98C2×C8×C47—21no—
99C2×C8×C47—22no—
100C2×C8×C47—22no—
101C2×C8×C47—22no—
102C2×C8×C47—44no—
103C2×C8×C47C2×D412yes—
104C2×C8×C47C2×D412yes—
105C42×C47—24no—
106C42×C47—22no—
107C42×C47—22no—
108C22⋊Dic47—24no—
109C22⋊Dic47—24no—
110C4⋊Dic47—24no—
111C4⋊Dic47—22no—
112C4⋊Dic47—22no—
113Dic47⋊C4—28no—
114Dic47⋊C4—44no—
115C4×Dic47—22no—
116C4×Dic47—22no—
117C4×Dic47—22no—
118C4×Dic47—22no—
119#10625329—28no—
120#10625329—44no—
121C2×C47⋊C8—22no—
122C2×C47⋊C8—22no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8C2×Q814yes—
127C23×C47—22no—
128C23×C47—22no—
129C23×C47#11712yes—
130C23×C47C4×D412yes—
131C23×C47C4×Q812yes—
132C23×C47C23×C411yescharacteristic
133C2×C4×C47—28no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—44no—
140C2×C4×C47#11712yes—
141C2×C4×C47C4×D412yes—
142C2×C4×C47C4×Q812yes—
143C2×C4×C47C22⋊Q814yes—
144C2×C4×C47#12314yes—
145C2×C4×C47#12514yes—
146C2×C4×C47#12614yes—
147C2×C4×C47#13114yes—
148C2×C4×C47C22×D411yescharacteristic
149C2×C4×C47C22×Q811yescharacteristic
150C2×C4×C47C2×#4111yescharacteristic
151C2×C4×C47C2×#4111yescharacteristic
152C2×C4×C47C2×#4111yescharacteristic
153C2×C4×C47C2×#4111yescharacteristic
154C2×Dic47—24no—
155C2×Dic47—22no—
156C2×Dic47—22no—
157C2×Dic47—24no—
158C2×Dic47—24no—
159C2×Dic47—24no—
160C2×Dic47—24no—
161C8×C47—22no—
162C8×C47—22no—
163C8×C47—42no—
164C47⋊C8—24no—
165C47⋊C8—42no—
166#21024—471nomaximal, Sylow
167C22×C47—24no—
168C22×C47—22no—
169C22×C47—22no—
170C22×C47—24no—
171C22×C47#38714yes—
172C22×C47C2×#11711yescharacteristic
173C22×C47C2×C4×D411yescharacteristic
174C22×C47C2×C4×Q811yescharacteristic
175C22×C47C2×#13112yes—
176C22×C47#56812yes—
177C4×C47—24no—
178C4×C47—42no—
179C4×C47#38714yes—
180C4×C47C2×C22⋊Q811yescharacteristic
181C4×C47C2×#12311yescharacteristic
182C4×C47C2×#12511yescharacteristic
183C4×C47C2×#12611yescharacteristic
184Dic47—44no—
185Dic47—42no—
186Dic47—42no—
187C22×C4⋊C8D4711yescharacteristic
188C2×#405—471no—
189#2296—471no—
190#1917—474no—
191#1916—474no—
192#1729—471no—
193#1729—471no—
194#1729—471no—
195#1728—471no—
196C2×C47—24no—
197C2×C47C2×#38711yescharacteristic
198C2×C47#292512yes—
199C2×C47#292911yescharacteristic
200C2×C47#293011yescharacteristic
201C2×C47#296212yes—
202C22×M16—471no—
203C22×M16—471no—
204C23×C8C2×D4711yescharacteristic
205C23×C8C2×D4711yescharacteristic
206C2×#117—471no—
207C2×#117—471no—
208C22×C42C2×D4711yescharacteristic
209C2×C4⋊C8—24no—
210C2×C4⋊C8C2×D4712yes—
211C2×C4⋊C8C2×D4712yes—
212#405—944no—
213C2×#97—472no—
214C2×C4×C8—472no—
215#389—472no—
216#389—472no—
217#385—472no—
218#385—472no—
219C4×C4⋊C4—474no—
220#336—472no—
221#336—474no—
222#336—474no—
223#336—472no—
224#336—474no—
225C47#2102411yescharacteristic, complemented, Sylow
226C23×C4C4×D4711yescharacteristic
227C23×C4C4×D4711yescharacteristic
228C23×C4C22×D4711yescharacteristic
229C2×M16—474no—
230C2×M16—942no—
231C2×M16—942no—
232C2×M16—942no—
233C2×M16—942no—
234C2×M16—942no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—472no—
241C22×C8—472no—
242C22×C8C22⋊D4714yes—
243C22×C8C22×D4712yes—
244C22×C8C22×D4711yescharacteristic
245C22×C8C22×D4711yescharacteristic
246#117—942no—
247#117—942no—
248#117—942no—
249#117—942no—
250C2×C4⋊C4—472no—
251C2×C4⋊C4—471no—
252C2×C4⋊C4—471no—
253C2×C4⋊C4—474no—
254C2×C4⋊C4—474no—
255C2×C22⋊C4—472no—
256C2×C22⋊C4—472no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—472no—
260C2×C42—472no—
261C2×C42C4×D4712yes—
262C2×C42C22×D4712yes—
263C4⋊C8—24no—
264C4⋊C8—24no—
265#97—944no—
266C4×C8—944no—
267#95—474no—
268#95—474no—
269C24C2×C4×D4711yescharacteristic
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—474no—
275C22×C4—472no—
276C22×C4—472no—
277C22×C4C2×C4×D4712yes—
278C22×C4C2×C4×D4712yes—
279C22×C4C2×C4×D4711yescharacteristic
280C22×C4#1062535012yes—
281C22×C4#1062535212yes—
282C22×C4#1062535212yes—
283C22×C4Q8×D4712yes—
284C22×C4#1062535412yes—
285C22×C4C2×C22⋊D4712yes—
286C22×C4C23×D4711yescharacteristic
287M16—948no—
288M16—1884no—
289C2×C8—28no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—44no—
298C2×C8—474no—
299C2×C8—942no—
300C2×C8—942no—
301C2×C8—942no—
302C2×C8—942no—
303C2×C8—942no—
304C2×C8C2×C22⋊D4712yes—
305C2×C8C2×C22⋊D4712yes—
306C4⋊C4—944no—
307C4⋊C4—942no—
308C4⋊C4—942no—
309C4⋊C4—948no—
310C4⋊C4—1884no—
311C22⋊C4—944no—
312C22⋊C4—944no—
313C42—24no—
314C42—22no—
315C42—22no—
316C42—942no—
317C42—942no—
318C42—942no—
319C42—942no—
320C23—22no—
321C23—22no—
322C23#1309398212yes—
323C23C4×C22⋊D4712yes—
324C23#1309398912yes—
325C23C22×C4×D4711yescharacteristic
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—44no—
333C2×C4—944no—
334C2×C4—942no—
335C2×C4—942no—
336C2×C4—944no—
337C2×C4—944no—
338C2×C4—944no—
339C2×C4—944no—
340C2×C4#1309398212yes—
341C2×C4C4×C22⋊D4712yes—
342C2×C4#1309398912yes—
343C2×C4#1309400014yes—
344C2×C4#1309400514yes—
345C2×C4#1309401414yes—
346C2×C4#1309401914yes—
347C2×C4#1309403014yes—
348C2×C4C22×C22⋊D4711yescharacteristic
349C2×C4C2×Q8×D4711yescharacteristic
350C2×C4C2×#1062535211yescharacteristic
351C2×C4C2×#1062535211yescharacteristic
352C2×C4C2×#1062535011yescharacteristic
353C2×C4C2×#1062535411yescharacteristic
354C8—22no—
355C8—22no—
356C8—42no—
357C8—944no—
358C8—1882no—
359C22—24no—
360C22—24no—
361C22—22no—
362C22—22no—
363C22#1568900414yes—
364C22C2×#1309398211yescharacteristic
365C22C2×C4×C22⋊D4711yescharacteristic
366C22C2×#1309398911yescharacteristic
367C22C2×#1309403012yes—
368C22#1568988912yes—
369C4—24no—
370C4—42no—
371C4—1884no—
372C4—1882no—
373C4—1882no—
374C4#1568900414yes—
375C4C2×#1309400011yescharacteristic
376C4C2×#1309400511yescharacteristic
377C4C2×#1309401411yescharacteristic
378C4C2×#1309401911yescharacteristic
379C2—24no—
380C2C2×#1568900411yescharacteristic
381C2#2025205312yes—
382C2#2025208211yescharacteristic
383C2#2025209111yescharacteristic
384C2#2025233112yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC22×C4⋊C8×C47C2
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenterC22×C4⋊C8D47
hypocenterC47#21024
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
3. #20243949C2semidirect
4. #20243942C2semidirect
225. C47#21024semidirect

Maximal subgroups

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

subgroupindex
C2×#156890782
#202458712
#202439492
#202439422
#202430842
#202430822
C22×C4⋊C8×C472
#2102447