Finite Group Databasebeta

← #26104615

Subgroups — #26104615

Order 12032.

518 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243456C211yesmaximal, characteristic
2C2×#15688878C211yesmaximal, characteristic
3#20241214C212yesmaximal, complemented
4#20241211C212yesmaximal, complemented
5C47×#2855C211yesmaximal, characteristic
6C23×C47⋊C8—21no—
7C2×C22⋊(C47⋊C8)—21no—
8#15688878—21no—
9#15688878—21no—
10#15688878C2212yes—
11#15688878C2212yes—
12C2×C22⋊Q8×C47C411yescharacteristic
13C22×C42×C47C2211yescharacteristic
14C47×#391C411yescharacteristic
15C47×#391C2211yescharacteristic
16C47×#389C411yescharacteristic
17C47×#389C2211yescharacteristic
18C47×#387—21no—
19C47×#387—21no—
20C47×#385—21no—
21C47×#381—21no—
22C47×#380C411yescharacteristic
23C22×C47⋊C8—24no—
24C22×C47⋊C8—22no—
25C22×C47⋊C8—22no—
26C22×C47⋊C8—42no—
27C22×C47⋊C8—42no—
28C22⋊(C47⋊C8)—42no—
29C22×Q8×C47C2×C411yescharacteristic
30C23×C4×C47C2×C411yescharacteristic
31C23×C4×C47D411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C22⋊Q8×C47—42no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47C2×C411yescharacteristic
38C2×C4⋊C4×C47C2×C411yescharacteristic
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C22⋊C4×C47—21no—
41C2×C22⋊C4×C47—21no—
42C2×C42×C47—42no—
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47C2311yescharacteristic
47#95×C47—21no—
48#95×C47—21no—
49#95×C47—21no—
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54C24×C47C22⋊C411yescharacteristic
55C2×Q8×C47—22no—
56C2×Q8×C47—41no—
57C2×Q8×C47—41no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47SD1612yes—
74C22×C4×C47Q1612yes—
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C4⋊C4×C47—22no—
79C4⋊C4×C47—42no—
80C4⋊C4×C47—42no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—42no—
84C22⋊C4×C47—42no—
85C22⋊C4×C47—42no—
86C42×C47—41no—
87C42×C47—41no—
88C42×C47—41no—
89C42×C47—41no—
90C2×C47⋊C8—24no—
91C2×C47⋊C8—48no—
92C2×C47⋊C8—42no—
93C2×C47⋊C8—42no—
94C2×C47⋊C8—42no—
95C2×C47⋊C8—42no—
96C2×C47⋊C8—42no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47#9912yes—
102C23×C47#10312yes—
103C23×C47#10412yes—
104C23×C47C2×C22⋊C411yescharacteristic
105Q8×C47—82no—
106C2×C4×C47—24no—
107C2×C4×C47—24no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—44no—
123C2×C4×C47—44no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47#10312yes—
137C2×C4×C47#13112yes—
138C2×C4×C47C2×SD1611yescharacteristic
139C2×C4×C47C2×Q1611yescharacteristic
140C2×C4×C47#13512yes—
141C47⋊C8—44no—
142C47⋊C8—82no—
143#7655—471nomaximal, Sylow
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—22no—
147C22×C47—22no—
148C22×C47—22no—
149C22×C47—22no—
150C22×C47#32814yes—
151C22×C47#40811yescharacteristic
152C22×C47C2×#9911yescharacteristic
153C22×C47#41311yescharacteristic
154C22×C47C2×#10311yescharacteristic
155C22×C47#41611yescharacteristic
156C22×C47C2×#10411yescharacteristic
157C22×C47#42111yescharacteristic
158C4×C47—22no—
159C4×C47—21no—
160C4×C47—22no—
161C4×C47—21no—
162C4×C47—22no—
163C4×C47—41no—
164C4×C47—41no—
165C4×C47—81no—
166C4×C47—81no—
167C4×C47—81no—
168C4×C47—81no—
169#2855D4711yescharacteristic
170#1817—471no—
171C2×#336—471no—
172#1272—472no—
173#1271—472no—
174C2×C47—24no—
175C2×C47#146811yescharacteristic
176C2×C47#148111yescharacteristic
177C2×C47#149411yescharacteristic
178C2×C47C2×#32811yescharacteristic
179C2×C47#151611yescharacteristic
180C2×C47#153211yescharacteristic
181C2×C47#154211yescharacteristic
182C23×C8—941no—
183C2×C22⋊Q8Dic4711yescharacteristic
184C22×C42C2×D4711yescharacteristic
185C2×C22⋊C8—941no—
186#391Dic4711yescharacteristic
187#391C2×D4711yescharacteristic
188#389Dic4711yescharacteristic
189#389C2×D4711yescharacteristic
190#387—21no—
191#387—21no—
192#385—21no—
193#381—21no—
194#380Dic4711yescharacteristic
195#336—472no—
196#336—472no—
197#336—941no—
198#336—941no—
199C47#765511yescharacteristic, complemented, Sylow
200C22×Q8C2×Dic4711yescharacteristic
201C23×C4C2×Dic4711yescharacteristic
202C23×C4C22⋊D4711yescharacteristic
203C23×C4C22⋊D4711yescharacteristic
204C22×C8—944no—
205C22×C8—942no—
206C22×C8—942no—
207C22×C8—1882no—
208C22×C8—1882no—
209C22⋊Q8—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C2×Dic4711yescharacteristic
214C2×C4⋊C4C2×Dic4711yescharacteristic
215C2×C4⋊C4C2×Dic4711yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C42—42no—
219C2×C42C2×Dic4711yescharacteristic
220C2×C42C22⋊D4711yescharacteristic
221C2×C42C22⋊D4711yescharacteristic
222C2×C42C22×D4711yescharacteristic
223C22⋊C8—1882no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231C24C22⋊Dic4711yescharacteristic
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4#1062533612yes—
248C22×C4#1062533712yes—
249C22×C4C22⋊Dic4711yescharacteristic
250C22×C4C22⋊Dic4711yescharacteristic
251C22×C4C22⋊Dic4711yescharacteristic
252C22×C4C22×Dic4711yescharacteristic
253C22×C4C2×C22⋊D4711yescharacteristic
254C22×C4C2×C22⋊D4711yescharacteristic
255C2×C8—944no—
256C2×C8—1882no—
257C2×C8—1882no—
258C2×C8—1882no—
259C2×C8—1882no—
260C2×C8—1888no—
261C2×C8—1882no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23#1309393912yes—
279C23#1309394812yes—
280C23#1309395112yes—
281C23C2×C22⋊Dic4711yescharacteristic
282Q8—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4#1309394812yes—
314C2×C4#1309403112yes—
315C2×C4C2×#1062533611yescharacteristic
316C2×C4C2×#1062533711yescharacteristic
317C2×C4#1309404412yes—
318C8—3762no—
319C8—1884no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22#1568886014yes—
327C22#1568909011yescharacteristic
328C22C2×#1309393911yescharacteristic
329C22#1568910711yescharacteristic
330C22C2×#1309394811yescharacteristic
331C22#1568911911yescharacteristic
332C22C2×#1309395111yescharacteristic
333C22#1568914011yescharacteristic
334C4—22no—
335C4—22no—
336C4—21no—
337C4—22no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C2—24no—
346C2#2024173611yescharacteristic
347C2#2024179111yescharacteristic
348C2#2024184811yescharacteristic
349C2C2×#1568886011yescharacteristic
350C2#2024194411yescharacteristic
351C2#2024200811yescharacteristic
352C2#2024204511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2855C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2855D47
hypocenterC47#7655
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
3. #20241214C2semidirect
4. #20241211C2semidirect
199. C47#7655semidirect

Maximal subgroups

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

subgroupindex
#202434562
C2×#156888782
#202412142
#202412112
C47×#28552
#765547