Finite Group Databasebeta

← #26110244

Subgroups — #26110244

Order 12032.

558 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243681C211yesmaximal, characteristic
2#20243565C212yesmaximal, complemented
3C4×#13093939C211yesmaximal, characteristic
4#20241235C212yesmaximal, complemented
5C47×#2972C211yesmaximal, characteristic
6C23×C47⋊C8C2211yescharacteristic
7C2×Dic47⋊C8—22no—
8C2×#13093939C411yescharacteristic
9C2×#13093939C411yescharacteristic
10C2×C22⋊(C47⋊C8)—21no—
11C2×C22⋊(C47⋊C8)C2211yescharacteristic
12C4×C22⋊Dic47C2212yes—
13#15688890C412yes—
14#15688878—21no—
15#15688877—22no—
16C2×C4×D4×C47C2211yescharacteristic
17C8×D4×C47—22no—
18C8×D4×C47—22no—
19C47×#431—21no—
20C2×C22⋊C8×C47—21no—
21C2×C22⋊C8×C47C2212yes—
22C22×C47⋊C8—21no—
23C22×C47⋊C8—22no—
24C22×C47⋊C8—21no—
25C22×C47⋊C8—42no—
26C2×C22⋊Dic47C2×C412yes—
27C2×C22⋊Dic47C2×C412yes—
28C2×C4×Dic47—22no—
29Dic47⋊C8—44no—
30#13093939—22no—
31#13093939C814yescomplemented
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—42no—
34C47⋊#95—22no—
35C22×D4×C47C2×C411yescharacteristic
36C23×C4×C47D411yescharacteristic
37C23×C4×C47C2311yescharacteristic
38C22×C8×C47—21no—
39C22×C8×C47—21no—
40C22×C8×C47D412yes—
41C22×C8×C47D412yes—
42C4×D4×C47—22no—
43C4×D4×C47—22no—
44C2×C4⋊C4×C47Q811yescharacteristic
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47Q811yescharacteristic
47C2×C42×C47D411yescharacteristic
48C4⋊C8×C47—21no—
49C4⋊C8×C47—21no—
50C22⋊C8×C47—22no—
51C22⋊C8×C47—22no—
52C22⋊C8×C47—22no—
53C4×C8×C47—22no—
54C24×C47C4⋊C411yescharacteristic
55C24×C47C22×C411yescharacteristic
56C2×D4×C47—22no—
57C2×D4×C47—22no—
58C2×D4×C47C2×C812yes—
59C2×D4×C47M1612yes—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47—41no—
66C22×C4×C47C4⋊C411yescharacteristic
67C22×C4×C47C4⋊C411yescharacteristic
68C22×C4×C47C4⋊C411yescharacteristic
69C22×C4×C47C2×D412yes—
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×Q811yescharacteristic
72C22×C4×C47#4112yes—
73C22×Dic47—22no—
74C22×Dic47—22no—
75C2×C8×C47—22no—
76C2×C8×C47—22no—
77C2×C8×C47—42no—
78C2×C8×C47—42no—
79C2×C8×C47—81no—
80C2×C8×C47—81no—
81C4⋊C4×C47—22no—
82C22⋊C4×C47—22no—
83C22⋊C4×C47—22no—
84C42×C47—21no—
85C42×C47—21no—
86C22⋊Dic47—22no—
87C22⋊Dic47—22no—
88C22⋊Dic47—22no—
89C22⋊Dic47C2×C814yescomplemented
90C4×Dic47—42no—
91C4×Dic47—42no—
92C2×C47⋊C8—22no—
93C2×C47⋊C8—24no—
94C2×C47⋊C8—41no—
95C2×C47⋊C8—42no—
96C2×C47⋊C8—41no—
97C2×C47⋊C8—41no—
98C2×C47⋊C8—41no—
99C2×C47⋊C8—41no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47C4⋊C812yescomplemented
107C23×C47C2×C4⋊C411yescharacteristic
108C23×C47C4×D412yes—
109C23×C47C22×C811yescharacteristic
110C23×C47C2×M1611yescharacteristic
111C23×C47#13112yes—
112D4×C47—24no—
113D4×C47—24no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—22no—
118C2×C4×C47—21no—
119C2×C4×C47—22no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—81no—
129C2×C4×C47C4⋊C812yescomplemented
130C2×C4×C47#12011yescharacteristic
131C2×C4×C47C22⋊Q812yes—
132C2×C4×C47#12311yescharacteristic
133C2×Dic47—24no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C8×C47—42no—
140C8×C47—42no—
141C47⋊C8—42no—
142C47⋊C8—81no—
143#8851—471nomaximal, Sylow
144C22×C47—22no—
145C22×C47—21no—
146C22×C47—22no—
147C22×C47—21no—
148C22×C47—22no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—42no—
152C22×C47—42no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47—81no—
156C22×C47#38511yescharacteristic
157C22×C47C2×C4⋊C811yescharacteristic
158C22×C47#42411yescharacteristic
159C22×C47C8×D412yes—
160C22×C47#43512yes—
161C4×C47—22no—
162C4×C47—21no—
163C4×C47—22no—
164C4×C47—41no—
165C4×C47—41no—
166C4×C47B(4,𝔽2)11yescharacteristic
167C4×C47#45811yescharacteristic
168Dic47—44no—
169Dic47—82no—
170#2972D4711yescharacteristic
171#1869—471no—
172#1850—472no—
173C4×#99—471no—
174#1282—472no—
175C2×C47—22no—
176C2×C47—22no—
177C2×C47—21no—
178C2×C47—21no—
179C2×C47—42no—
180C2×C47#185011yescharacteristic
181C2×C47#186711yescharacteristic
182C2×C47#186811yescharacteristic
183C23×C8—471no—
184C2×C4×D4C2×D4711yescharacteristic
185C8×D4—22no—
186C8×D4—22no—
187#431—21no—
188C2×C4⋊C8—942no—
189C2×#99—471no—
190C2×#99—471no—
191C2×C22⋊C8—21no—
192C2×C22⋊C8—471no—
193C2×C22⋊C8—941no—
194C2×C22⋊C8C2×D4712yes—
195C4×C22⋊C4—472no—
196#342—472no—
197#336—942no—
198#336—941no—
199C47#885111yescharacteristic, complemented, Sylow
200C22×D4C4×D4711yescharacteristic
201C23×C4C22⋊D4711yescharacteristic
202C23×C4C22×D4711yescharacteristic
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—941no—
206C22×C8—942no—
207C22×C8—941no—
208C22×C8—1882no—
209C22×C8C22⋊D4712yes—
210C22×C8C22⋊D4712yes—
211C4×D4—22no—
212C4×D4—22no—
213C2×C4⋊C4Dic9411yescharacteristic
214C2×C22⋊C4—472no—
215C2×C22⋊C4—472no—
216C2×C22⋊C4Dic9411yescharacteristic
217C2×C22⋊C4C4×D4711yescharacteristic
218C2×C42—942no—
219C2×C42C22⋊D4711yescharacteristic
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—1884no—
223#99—474no—
224#99—942no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—942no—
229C22⋊C8—1882no—
230C4×C8—22no—
231#95—942no—
232C24Dic47⋊C411yescharacteristic
233C24C2×C4×D4711yescharacteristic
234C2×D4—22no—
235C2×D4—22no—
236C2×D4C8×D4712yes—
237C2×D4#1062532412yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—942no—
245C22×C4—942no—
246C22×C4Dic47⋊C411yescharacteristic
247C22×C4Dic47⋊C411yescharacteristic
248C22×C4Dic47⋊C411yescharacteristic
249C22×C4C2×Dic9411yescharacteristic
250C22×C4#1062535012yes—
251C22×C4C2×C22⋊D4712yes—
252C22×C4C2×C22⋊D4711yescharacteristic
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—81no—
258C2×C8—81no—
259C2×C8—942no—
260C2×C8—944no—
261C2×C8—1881no—
262C2×C8—1881no—
263C2×C8—1881no—
264C2×C8—1882no—
265C2×C8—1881no—
266C2×C8—1881no—
267C4⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—474no—
271C22⋊C4—942no—
272C22⋊C4—942no—
273C22⋊C4—942no—
274C42—21no—
275C42—21no—
276C42—1882no—
277C42—1882no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23Dic47⋊C812yescomplemented
285C23C2×Dic47⋊C411yescharacteristic
286C23C4×C22⋊D4712yes—
287C23C2×C8×D4711yescharacteristic
288C23C2×#1062532411yescharacteristic
289C23#1309403012yes—
290D4—24no—
291D4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—944no—
308C2×C4—1882no—
309C2×C4—1882no—
310C2×C4—1882no—
311C2×C4—1882no—
312C2×C4—1882no—
313C2×C4Dic47⋊C812yescomplemented
314C2×C4#1309399311yescharacteristic
315C2×C4#1309400312yes—
316C2×C4#1309400711yescharacteristic
317C8—42no—
318C8—42no—
319C8—3761no—
320C8—1882no—
321C22—21no—
322C22—22no—
323C22—21no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22#1568899611yescharacteristic
334C22C2×Dic47⋊C811yescharacteristic
335C22#1568914911yescharacteristic
336C22C8×C22⋊D4712yes—
337C22#1568919412yes—
338C4—22no—
339C4—21no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—3762no—
344C4—1884no—
345C4#1568931011yescharacteristic
346C4#1568931311yescharacteristic
347C2—22no—
348C2—22no—
349C2—21no—
350C2—21no—
351C2—42no—
352C2#2024356611yescharacteristic
353C2#2024366911yescharacteristic
354C2#2024367411yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC23×C47C22×C8
FrattiniC23×C4C22×D47
FittingC47×#2972C2
socleC22×C47#385
prime radicalC22×D4×C47C2×C4
cosocleC23×C4×C47C23
hypercenter#2972D47
hypocenterC47#8851
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20243565C2semidirect
4. #20241235C2semidirect
31. #13093939C8semidirect
89. C22⋊Dic47C2×C8semidirect
106. C23×C47C4⋊C8semidirect
129. C2×C4×C47C4⋊C8semidirect
199. C47#8851semidirect
284. C23Dic47⋊C8semidirect
313. C2×C4Dic47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202436812
#202435652
C4×#130939392
#202412352
C47×#29722
#885147