Finite Group Databasebeta

← #26187699

Subgroups — #26187699

Order 12032.

794 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D376C211yesmaximal, characteristic
2#20252123C212yesmaximal, complemented
3#20242272C212yesmaximal, complemented
4#20242216C214yesmaximal, complemented
5#20242182C212yesmaximal, complemented
6#20241812C212yesmaximal, complemented
7C8×#10625329C211yesmaximal, characteristic, complemented
8C47×#1567C211yesmaximal, characteristic, complemented
9C22×D376C411yescharacteristic
10C2×M16×D47—22no—
11C2×C4×D188C2212yes—
12C4×D376C411yescharacteristic
13C4×D376C412yes—
14C4×D376C411yescharacteristic
15C4×D376C2211yescharacteristic
16C4×D376C2212yes—
17C4×D376C2211yescharacteristic
18#15689200C2214yescomplemented
19#15689193—24no—
20#15689188C2214yescomplemented
21#15689176C2211yescharacteristic
22C2×#13093958C411yescharacteristic
23C2×#13093946C412yes—
24#15689086—22no—
25C4×#10625329C2211yescharacteristic
26#15688850C2214yescomplemented
27#15688850C2214yescomplemented
28#15688845C2212yescomplemented
29C8×C47⋊C8C2212yescomplemented
30C47×#423C2212yes—
31C2×C4×C8×C47C2211yescharacteristic
32C47×#335C2212yescomplemented
33C47×#335C2212yescomplemented
34C22×D188C2×C412yes—
35C22×C4×D47—22no—
36C2×D376—24no—
37C2×D376C2×C412yes—
38C2×D376C2×C411yescharacteristic
39C2×D376C2×C411yescharacteristic
40C2×#10625329D411yescharacteristic
41C2×#10625329D411yescharacteristic
42C2×#10625324—24no—
43M16×D47—48no—
44C2×C8×D47—24no—
45C4×D188—22no—
46C4×D188—22no—
47C4×D188C2×C412yes—
48C4×D188C2×C412yes—
49C4×D188C2312yes—
50C4×D188C2312yes—
51C2×C4⋊Dic47C2×C412yes—
52C2×(C2×D47)⋊C4—22no—
53#13093958C2×C412yes—
54#13093958C2×C411yescharacteristic
55#13093958C2×C411yescharacteristic
56C4⋊(C47⋊C8)—21no—
57#13093946C2×C412yes—
58#13093946C2×C412yes—
59#13093946C2×C412yes—
60#13093946C2×C412yes—
61C22⋊(C47⋊C8)—21no—
62(C2×D47)⋊C8—24no—
63(C2×D47)⋊C8—24no—
64#13093935D412yes—
65C4×C47⋊C8C2312yes—
66C4×C47⋊C8C2312yes—
67C2×M16×C47—22no—
68C22×C8×C47C2×C411yescharacteristic
69C22×C8×C47C2×C411yescharacteristic
70C2×C42×C47C2311yescharacteristic
71C4⋊C8×C47—22no—
72C4⋊C8×C47C2314yes—
73C4×C8×C47C2×C411yescharacteristic
74C4×C8×C47C2×C411yescharacteristic
75C4×C8×C47C2311yescharacteristic
76C4×C8×C47C2311yescharacteristic
77C23×D47—22no—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×Dic47—22no—
82C2×D188—24no—
83C2×D188—22no—
84C2×D188—42no—
85C2×D188C22×C412yes—
86C2×D188C22×C412yes—
87C2×C4×D47—22no—
88C2×C4×D47—22no—
89C2×C4×D47—22no—
90C2×C4×D47—22no—
91C2×C4×D47—44no—
92C2×C4×D47—44no—
93M16×C47—44no—
94C2×C8×C47—22no—
95C2×C8×C47—24no—
96C2×C8×C47—41no—
97C2×C8×C47C22×C411yescharacteristic
98C2×C8×C47C22×C411yescharacteristic
99C2×C8×C47C22×C411yescharacteristic
100C2×C8×C47C22×C411yescharacteristic
101C42×C47C22×C411yescharacteristic
102C42×C47C2×D411yescharacteristic
103C42×C47#4111yescharacteristic
104C42×C47C2411yescharacteristic
105(C2×D47)⋊C4—22no—
106(C2×D47)⋊C4—22no—
107(C2×D47)⋊C4—22no—
108(C2×D47)⋊C4—22no—
109C4⋊Dic47—22no—
110C4⋊Dic47C22×C412yes—
111C4⋊Dic47C22×C412yes—
112#10625329—22no—
113#10625329D814yescomplemented
114C2×C47⋊C8—22no—
115C2×C47⋊C8C2×D412yes—
116C2×C47⋊C8C2×D412yes—
117D376—24no—
118D376—22no—
119D376M1614yescomplemented
120#10625324—48no—
121C8×D47—48no—
122C23×C47C4×D411yescharacteristic
123C22×D47—22no—
124C22×D47—22no—
125C22×D47—44no—
126C22×D47—44no—
127C22×D47—82no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—41no—
131C2×C4×C47C4×D411yescharacteristic
132C2×C4×C47C4×D411yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×C4×C47C2×D812yes—
135C2×C4×C47#13512yes—
136C2×C4×C47C23×C411yescharacteristic
137C2×C4×C47C22×D411yescharacteristic
138C2×C4×C47C2×#4111yescharacteristic
139C2×Dic47—22no—
140C2×Dic47—22no—
141C2×Dic47—44no—
142D188—22no—
143D188—22no—
144D188—22no—
145D188—44no—
146D188C2×M1614yescomplemented
147C4×D47—42no—
148C4×D47—42no—
149C4×D47—42no—
150C4×D47—42no—
151C4×D47—44no—
152C4×D47—44no—
153C8×C47—21no—
154C8×C47—21no—
155C8×C47—21no—
156C8×C47—44no—
157C8×C47C2×M1612yescomplemented
158C47⋊C8—22no—
159C47⋊C8—42no—
160C47⋊C8C2×D814yescomplemented
161M16×D8—471nomaximal, Sylow
162C22×C47—21no—
163C22×C47—21no—
164C22×C47C4×D812yes—
165C22×C47C2×C4×D411yescharacteristic
166C2×D47—24no—
167C2×D47—42no—
168C2×D47—42no—
169C2×D47—42no—
170C2×D47—42no—
171C2×D47—42no—
172C2×D47—88no—
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—21no—
176C4×C47C4×D812yes—
177C4×C47C22×M1611yescharacteristic
178C4×C47#56811yescharacteristic
179C4×C47C22×D811yescharacteristic
180C4×C47C2×#13511yescharacteristic
181Dic47—42no—
182Dic47—42no—
183C2×C4×D8—471no—
184D4×M16—472no—
185#1589—472no—
186#1581—474no—
187C8×D8—472no—
188#1567D4711yescharacteristic, complemented
189#1486—472no—
190C8×M16—471no—
191C2×C47—22no—
192C2×C47D4×M1611yescharacteristic
193C2×C47C2×C4×D811yescharacteristic
194C2×C47#295711yescharacteristic
195D47—44no—
196D47—82no—
197C22×D8—471no—
198C22×M16—942no—
199C2×C4×D4—472no—
200C4×D8—471no—
201C4×D8—472no—
202C4×D8—471no—
203C4×D8—471no—
204C4×D8—472no—
205C4×D8—471no—
206#436—474no—
207#435—944no—
208C8×D4—474no—
209#431—471no—
210C2×#107—471no—
211#423C2×D4712yes—
212C2×#102—472no—
213#407—942no—
214C4×M16—471no—
215C2×C4×C8C2×D4711yescharacteristic
216#335C2×D4712yescomplemented
217#335C2×D4712yescomplemented
218D4⋊C8—474no—
219D4⋊C8—474no—
220#322—472no—
221C82—472no—
222C47M16×D811yescharacteristic, complemented, Sylow
223C22×D4—472no—
224C23×C4—942no—
225C2×D8—471no—
226C2×D8—472no—
227C2×D8—471no—
228C2×D8—944no—
229C2×M16—22no—
230C2×M16—471no—
231C2×M16—471no—
232C2×M16—944no—
233C2×M16—1888no—
234C22×C8—944no—
235C22×C8C4×D4711yescharacteristic
236C22×C8C4×D4711yescharacteristic
237C4×D4—472no—
238C4×D4—472no—
239C4×D4—472no—
240C4×D4—472no—
241C4×D4—942no—
242C4×D4—942no—
243C2×C4⋊C4—472no—
244C2×C22⋊C4—942no—
245C2×C42C22×D4711yescharacteristic
246#107—472no—
247#107—471no—
248#107—471no—
249C4⋊C8—22no—
250C4⋊C8—941no—
251C4⋊C8C22×D4714yes—
252#102—472no—
253#102—472no—
254#102—472no—
255#102—472no—
256C22⋊C8—944no—
257C22⋊C8—944no—
258C22⋊C8—941no—
259#97—472no—
260C4×C8—472no—
261C4×C8—472no—
262C4×C8C4×D4711yescharacteristic
263C4×C8C4×D4711yescharacteristic
264C4×C8C22×D4711yescharacteristic
265C4×C8C22×D4711yescharacteristic
266C24—942no—
267C2×D4—472no—
268C2×D4—472no—
269C2×D4—944no—
270C2×D4—942no—
271C2×D4—1882no—
272C22×C4—942no—
273C22×C4—942no—
274C22×C4—942no—
275C22×C4—942no—
276C22×C4—942no—
277C22×C4—1884no—
278C22×C4—1884no—
279C22×C4C2×C4×D4711yescharacteristic
280C22×C4D4×D4711yescharacteristic
281C22×C4#1062535411yescharacteristic
282D8—474no—
283D8—944no—
284D8—942no—
285M16—44no—
286M16—474no—
287M16—942no—
288M16—1888no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—41no—
292C2×C8—472no—
293C2×C8—472no—
294C2×C8—942no—
295C2×C8—1888no—
296C2×C8C2×C4×D4711yescharacteristic
297C2×C8C2×C4×D4711yescharacteristic
298C2×C8C2×C4×D4711yescharacteristic
299C2×C8C2×C4×D4711yescharacteristic
300C4⋊C4—472no—
301C4⋊C4—472no—
302C4⋊C4—942no—
303C22⋊C4—942no—
304C22⋊C4—942no—
305C22⋊C4—942no—
306C22⋊C4—942no—
307C42C2×C4×D4711yescharacteristic
308C42D4×D4711yescharacteristic
309C42#1062535411yescharacteristic
310C42C23×D4711yescharacteristic
311C23—942no—
312C23—942no—
313C23—3762no—
314C23—1884no—
315C23—1884no—
316C23#1309398611yescharacteristic
317D4—474no—
318D4—942no—
319D4—942no—
320D4—942no—
321D4—1884no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—942no—
326C2×C4—942no—
327C2×C4—1884no—
328C2×C4—1882no—
329C2×C4—1882no—
330C2×C4—1882no—
331C2×C4—1882no—
332C2×C4—1884no—
333C2×C4—1884no—
334C2×C4#1309398611yescharacteristic
335C2×C4#1309398611yescharacteristic
336C2×C4#1309398611yescharacteristic
337C2×C4D8×D4712yes—
338C2×C4#1309404712yes—
339C2×C4C22×C4×D4711yescharacteristic
340C2×C4C2×D4×D4711yescharacteristic
341C2×C4C2×#1062535411yescharacteristic
342C8—21no—
343C8—21no—
344C8—21no—
345C8—44no—
346C8—474no—
347C8—942no—
348C8—1882no—
349C8M16×D4712yescomplemented
350C22—21no—
351C22—21no—
352C22—944no—
353C22—3768no—
354C22—1882no—
355C22—1882no—
356C22—1882no—
357C22—1882no—
358C22—1882no—
359C22#1568920512yes—
360C22C2×#1309398611yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—1882no—
365C4—1882no—
366C4#1568920512yes—
367C4C2×M16×D4711yescharacteristic
368C4#1568988911yescharacteristic
369C4C2×D8×D4711yescharacteristic
370C4C2×#1309404711yescharacteristic
371C2—22no—
372C2—3762no—
373C2—1884no—
374C2#2025212311yescharacteristic
375C2C2×#1568920511yescharacteristic
376C2#2025230411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1567C2
socleC22×C47C2×C4×D4
prime radicalC22×D376C4
cosocleC42×C47C24
hypercenter#1567D47
hypocenterC47M16×D8
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1567

Complements

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

NH ≅ G/Ntype
2. #20252123C2semidirect
3. #20242272C2semidirect
4. #20242216C2semidirect
5. #20242182C2semidirect
6. #20241812C2semidirect
7. C8×#10625329C2semidirect
8. C47×#1567C2semidirect
18. #15689200C22semidirect
20. #15689188C22semidirect
26. #15688850C22semidirect
27. #15688850C22semidirect
28. #15688845C22semidirect
29. C8×C47⋊C8C22semidirect
32. C47×#335C22semidirect
33. C47×#335C22semidirect
113. #10625329D8semidirect
119. D376M16semidirect
146. D188C2×M16semidirect
157. C8×C47C2×M16semidirect
160. C47⋊C8C2×D8semidirect
188. #1567D47semidirect
216. #335C2×D47semidirect
217. #335C2×D47semidirect
222. C47M16×D8semidirect
349. C8M16×D47semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3762
#202521232
#202422722
#202422162
#202421822
#202418122
C8×#106253292
C47×#15672
M16×D847