Finite Group Databasebeta

← #26224146

Subgroups — #26224146

Order 12032.

974 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689247C211yesmaximal, characteristic, complemented
2C2×D4×C47⋊C8C211yesmaximal, characteristic
3C4×#13094011C211yesmaximal, characteristic
4#20244324C211yesmaximal, characteristic, complemented
5#20243991C214yesmaximal, complemented
6#20243504C211yesmaximal, characteristic, complemented
7#20243162C214yesmaximal, complemented
8C4×C8×Dic47C211yesmaximal, characteristic, complemented
9C47×#1954C211yesmaximal, characteristic
10C23×C47⋊C8C2212yes—
11C2×#13094011C411yescharacteristic
12C2×C4×Dic94C2211yescharacteristic
13#15689247—24no—
14D4×C47⋊C8—24no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×Dic47⋊C8C2212yes—
17C2×C22⋊(C47⋊C8)C2212yes—
18C2×C4×C47⋊C8C2211yescharacteristic
19C2×C8×Dic47C2212yes—
20C2×C8×Dic47C2212yes—
21C2×C8×Dic47C2212yes—
22#15689028C411yescharacteristic
23#15689026C411yescharacteristic
24#15689017C414yes—
25#15688988C411yescharacteristic
26C4×C22⋊Dic47C2214yes—
27C42×Dic47C2211yescharacteristic
28#15688877C2214yes—
29#15688877C2214yes—
30C2×C4×D4×C47C2211yescharacteristic
31C2×C4⋊C8×C47C2211yescharacteristic
32C2×C22⋊C8×C47C2214yes—
33C2×C4×C8×C47C2211yescharacteristic
34C22×Dic94C2×C411yescharacteristic
35C22×C47⋊C8—24no—
36C22×C47⋊C8—28no—
37C22×C47⋊C8C2312yes—
38C22×C47⋊C8C2311yescharacteristic
39C22×C47⋊C8C2311yescharacteristic
40#13094011C818yescomplemented
41C4×Dic94—22no—
42C4×Dic94—22no—
43C2×C4⋊Dic47C2×C411yescharacteristic
44C2×Dic47⋊C4C2×C412yes—
45C2×C22⋊Dic47C2×C414yes—
46C2×C22⋊Dic47C2×C414yes—
47C2×C4×Dic47C2×C412yes—
48C2×C4×Dic47C2×C411yescharacteristic
49C2×C4×Dic47C2×C411yescharacteristic
50C2×C4×Dic47C2312yes—
51C2×C4×Dic47C2312yes—
52C4⋊(C47⋊C8)—22no—
53Dic47⋊C8—24no—
54C22⋊(C47⋊C8)—24no—
55C4×C47⋊C8D414yes—
56C8×Dic47—24no—
57C8×Dic47—24no—
58C8×Dic47—24no—
59C47⋊#95C2×C414yes—
60C47⋊#95C2×C414yes—
61C22×D4×C47C2×C411yescharacteristic
62C23×C4×C47C2312yes—
63C22×C8×C47C2312yes—
64C22×C8×C47C2312yes—
65C4×D4×C47—22no—
66C4×D4×C47—22no—
67C2×C4⋊C4×C47C2×C411yescharacteristic
68C2×C22⋊C4×C47C2×C412yes—
69C2×C42×C47C2311yescharacteristic
70C4⋊C8×C47—22no—
71C22⋊C8×C47—28no—
72C4×C8×C47—22no—
73C24×C47C22×C412yes—
74C2×D4×C47—22no—
75C2×D4×C47—22no—
76C2×D4×C47C2×C814yes—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47—24no—
80C22×C4×C47—24no—
81C22×C4×C47C22×C412yes—
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C2411yescharacteristic
85C22×Dic47C22×C412yes—
86C22×Dic47C22×C412yes—
87C22×Dic47C22×C412yes—
88C22×Dic47C22×C412yes—
89C2×Dic94—22no—
90C2×Dic94—22no—
91C2×Dic94C2×C814yescomplemented
92C2×C8×C47—22no—
93C2×C8×C47—22no—
94C2×C8×C47—22no—
95C2×C8×C47—24no—
96C2×C8×C47#4114yes—
97C4⋊C4×C47—22no—
98C22⋊C4×C47—24no—
99C42×C47C2×D412yes—
100C42×C47#4112yes—
101C22⋊Dic47—28no—
102C22⋊Dic47C2×C8116yescomplemented
103C4⋊Dic47—22no—
104Dic47⋊C4—24no—
105C4×Dic47—22no—
106C4×Dic47—22no—
107C4×Dic47—22no—
108C4×Dic47—22no—
109C4×Dic47—22no—
110C4×Dic47—24no—
111C4×Dic47C2×C814yescomplemented
112C2×C47⋊C8—22no—
113C2×C47⋊C8—22no—
114C2×C47⋊C8—24no—
115C2×C47⋊C8—48no—
116C2×C47⋊C8C2×D412yes—
117C2×C47⋊C8C2×D412yes—
118C2×C47⋊C8#4114yes—
119C23×C47—22no—
120C23×C47—24no—
121C23×C47—24no—
122C23×C47—22no—
123C23×C47C22×C814yes—
124C23×C47C23×C411yescharacteristic
125D4×C47—28no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—24no—
129C2×C4×C47—42no—
130C2×C4×C47—44no—
131C2×C4×C47—42no—
132C2×C4×C47C4×D412yes—
133C2×C4×C47C4×D412yes—
134C2×C4×C47C22×C812yes—
135C2×C4×C47#13112yes—
136C2×C4×C47C22×D411yescharacteristic
137C2×C4×C47C2×#4112yes—
138C2×C4×C47C2×#4111yescharacteristic
139C2×C4×C47C2×#4112yes—
140C2×Dic47—22no—
141C2×Dic47—22no—
142C2×Dic47—22no—
143C2×Dic47—22no—
144C2×Dic47—22no—
145C2×Dic47—22no—
146C2×Dic47—22no—
147C2×Dic47—24no—
148C2×Dic47C22×C814yes—
149C2×Dic47C22×C814yes—
150C2×Dic47#13114yes—
151Dic94—28no—
152C8×C47—24no—
153C8×C47—42no—
154C47⋊C8—22no—
155C47⋊C8—42no—
156C47⋊C8#12414yescomplemented
157C8×#124—471nomaximal, Sylow
158C22×C47—24no—
159C22×C47—28no—
160C22×C47—42no—
161C22×C47—44no—
162C22×C47—42no—
163C22×C47C2×C4×D411yescharacteristic
164C22×C47C4×#4112yes—
165C22×C47C23×C811yescharacteristic
166C22×C47C2×#13112yes—
167C22×C47C2×#13111yescharacteristic
168C4×C47—22no—
169C4×C47—42no—
170C4×C47C8×D414yes—
171C4×C47C2×#12411yescharacteristic
172C4×C47#52912yes—
173C4×C47#53211yescharacteristic
174Dic47—24no—
175Dic47—24no—
176Dic47—24no—
177Dic47—42no—
178C2×C8×Q8—471no—
179C2×C8×D4—471no—
180C4×#124—471no—
181#1984—471no—
182#1954D4711yescharacteristic
183#1923—474no—
184#1831—471no—
185C8×C22⋊C4—474no—
186C42×C8—471no—
187C2×C47—44no—
188C2×C47C4×#12411yescharacteristic
189C2×C47C2×C8×D411yescharacteristic
190C2×C47#293111yescharacteristic
191C2×C47C8×#4112yes—
192C2×C47#296612yes—
193C23×C8—472no—
194C2×#124—471no—
195C2×C4×Q8—471no—
196C2×C4×D4C2×D4711yescharacteristic
197C8×Q8—944no—
198C8×D4—944no—
199C2×C4⋊C8—472no—
200C2×C4⋊C8—471no—
201C2×C4⋊C8C2×D4711yescharacteristic
202C2×C22⋊C8—472no—
203C2×C22⋊C8C2×D4714yes—
204C2×C4×C8—471no—
205C2×C4×C8—472no—
206C2×C4×C8—472no—
207C2×C4×C8—472no—
208C2×C4×C8C2×D4711yescharacteristic
209#391—471no—
210#390—471no—
211#388—474no—
212#382—471no—
213C4×C22⋊C4—474no—
214C43—471no—
215#336—474no—
216#336—474no—
217C47C8×#12411yescharacteristic, complemented, Sylow
218C22×Q8—471no—
219C22×D4C4×D4711yescharacteristic
220C23×C4C22×D4712yes—
221C22×C8—471no—
222C22×C8—471no—
223C22×C8—472no—
224C22×C8—948no—
225C22×C8—944no—
226C22×C8C22×D4712yes—
227C22×C8C22×D4712yes—
228#124—478no—
229C4×Q8—942no—
230C4×Q8—942no—
231C4×D4—22no—
232C4×D4—22no—
233C2×C4⋊C4—472no—
234C2×C4⋊C4—471no—
235C2×C4⋊C4C4×D4711yescharacteristic
236C2×C22⋊C4—474no—
237C2×C22⋊C4—474no—
238C2×C22⋊C4C4×D4712yes—
239C2×C42—471no—
240C2×C42—472no—
241C2×C42—471no—
242C2×C42—472no—
243C2×C42—472no—
244C2×C42C22×D4711yescharacteristic
245C4⋊C8—22no—
246C4⋊C8—944no—
247C4⋊C8—942no—
248C22⋊C8—28no—
249C22⋊C8—944no—
250C4×C8—22no—
251C4×C8—474no—
252C4×C8—944no—
253C4×C8—944no—
254C4×C8—944no—
255#95—474no—
256#95—474no—
257C24C2×C4×D4712yes—
258C2×Q8—474no—
259C2×Q8—942no—
260C2×Q8—942no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4C8×D4714yes—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—472no—
269C22×C4—472no—
270C22×C4—472no—
271C22×C4—472no—
272C22×C4C2×C4×D4712yes—
273C22×C4C2×C4×D4711yescharacteristic
274C22×C4C2×C4×D4711yescharacteristic
275C22×C4C23×D4711yescharacteristic
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—472no—
281C2×C8—474no—
282C2×C8—472no—
283C2×C8—942no—
284C2×C8—942no—
285C2×C8—944no—
286C2×C8—1888no—
287C2×C8#1062535214yes—
288C4⋊C4—22no—
289C4⋊C4—944no—
290C4⋊C4—942no—
291C22⋊C4—24no—
292C22⋊C4—4716no—
293C22⋊C4—948no—
294C42—474no—
295C42—944no—
296C42—942no—
297C42—942no—
298C42—942no—
299C42—942no—
300C42—942no—
301C42D4×D4712yes—
302C42#1062535212yes—
303C23—22no—
304C23—24no—
305C23—24no—
306C23—22no—
307C23C2×C8×D4714yes—
308C23C22×C4×D4711yescharacteristic
309Q8—948no—
310D4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—474no—
318C2×C4—474no—
319C2×C4—474no—
320C2×C4—942no—
321C2×C4—942no—
322C2×C4—942no—
323C2×C4—942no—
324C2×C4—942no—
325C2×C4—942no—
326C2×C4—942no—
327C2×C4—944no—
328C2×C4(C22⋊D47)⋊C412yes—
329C2×C4(C22⋊D47)⋊C412yes—
330C2×C4C2×C8×D4712yes—
331C2×C4#1309403212yes—
332C2×C4C2×D4×D4711yescharacteristic
333C2×C4#41×D4712yes—
334C2×C4C2×#1062535211yescharacteristic
335C2×C4C2×#1062535212yes—
336C8—24no—
337C8—42no—
338C8—474no—
339C8—942no—
340C8—1882no—
341C22—24no—
342C22—28no—
343C22—42no—
344C22—42no—
345C22—44no—
346C22C2×(C22⋊D47)⋊C411yescharacteristic
347C22C4×#1062535212yes—
348C22C22×C8×D4711yescharacteristic
349C22#131×D4712yes—
350C22C2×#1309403211yescharacteristic
351C4—22no—
352C4—42no—
353C4—944no—
354C4—944no—
355C4—944no—
356C4—1882no—
357C4(C22⋊D47)⋊C814yes—
358C4#124×D4711yescharacteristic
359C4#1568965412yes—
360C4#1568967111yescharacteristic
361C2—44no—
362C2#2024593511yescharacteristic
363C2C2×(C22⋊D47)⋊C811yescharacteristic
364C2#2025209611yescharacteristic
365C2C8×#1062535212yes—
366C2#2025236612yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutatorC22×C47C23×C8
FittingC47×#1954C2
socleC23×C47C23×C4
prime radicalC22×D4×C47C2×C4
cosocleC22×C4×C47C24
hypercenter#1954D47
hypocenterC47C8×#124
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1954

Complements

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

NH ≅ G/Ntype
1. C2×#15689247C2semidirect
4. #20244324C2semidirect
5. #20243991C2semidirect
6. #20243504C2semidirect
7. #20243162C2semidirect
8. C4×C8×Dic47C2semidirect
40. #13094011C8semidirect
91. C2×Dic94C2×C8semidirect
102. C22⋊Dic47C2×C8semidirect
111. C4×Dic47C2×C8semidirect
156. C47⋊C8#124semidirect
217. C47C8×#124semidirect

Maximal subgroups

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

subgroupindex
C2×#156892472
C2×D4×C47⋊C82
C4×#130940112
#202443242
#202439912
#202435042
#202431622
C4×C8×Dic472
C47×#19542
C8×#12447