Finite Group Databasebeta

← #26371025

Subgroups — #26371025

Order 12032.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253916C211yesmaximal, characteristic, complemented
2#20252305C211yesmaximal, characteristic
3#20245562C214yesmaximal, complemented
4#20245557C214yesmaximal, complemented
5#20245161C211yesmaximal, characteristic, complemented
6C2×#15688945C211yesmaximal, characteristic, complemented
7#20245153C211yesmaximal, characteristic, complemented
8#20245124C211yesmaximal, characteristic
9C2×C47×#360C211yesmaximal, characteristic
10#15689890—21no—
11#15689754—21no—
12C2×#13093964C2211yescharacteristic
13#15689530—22no—
14#15689523C2212yescomplemented
15#15689520C2212yescomplemented
16#15689245—22no—
17#15689240C2214yes—
18C2×#13093960C2211yescharacteristic
19#15689163—21no—
20#15689161C2211yescharacteristic
21#15689140—21no—
22#15689078C2211yescharacteristic
23#15688946C2212yescomplemented
24#15688946C2212yescomplemented
25#15688945C2214yescomplemented
26#15688943C2214yescomplemented
27#15688933C2212yes—
28#15688933C2212yes—
29C2×#135×C47C2211yescharacteristic
30C2×M32×C47C2211yescharacteristic
31C2×#106×C47C2211yescharacteristic
32C47×#360C2214yes—
33#13094031—24no—
34#13094031—42no—
35C2×#10625329—21no—
36C2×#10625329—42no—
37C2×#10625329D411yescharacteristic
38#13094023—22no—
39#13094023—41no—
40#13094015—22no—
41#13094015—41no—
42#13094003—42no—
43#13093983—21no—
44#13093983D411yescharacteristic
45#13093964—21no—
46#13093964C2312yes—
47C2×C47⋊C16—21no—
48#13093960—21no—
49#13093960C2312yes—
50#13093958—21no—
51#13093957—22no—
52#13093956C2311yescharacteristic
53#13093956C2311yescharacteristic
54#13093955—22no—
55#13093951—24no—
56#13093934D412yes—
57C8×Dic47C2312yes—
58C2×#41×C47—21no—
59#135×C47—21no—
60#135×C47—21no—
61#135×C47Q812yescomplemented
62#135×C47C2312yes—
63C2×Q16×C47Q811yescharacteristic, complemented
64C2×SD16×C47—21no—
65C2×D8×C47Q811yescharacteristic, complemented
66C22×C8×C47C2311yescharacteristic
67C2×C4⋊C4×C47—21no—
68M32×C47—21no—
69M32×C47C2312yes—
70C2×C16×C47—21no—
71#106×C47—21no—
72#106×C47C2312yes—
73#41×C47—22no—
74#41×C47—22no—
75#41×C47—41no—
76#41×C47—41no—
77C2×Q8×C47—21no—
78C2×D4×C47—21no—
79C2×D4×C47—41no—
80C22×C4×C47—41no—
81C22×C4×C47—41no—
82C22×C4×C47C2×D411yescharacteristic
83C2×Dic94—42no—
84Q16×C47—21no—
85Q16×C47C2×Q812yes—
86SD16×C47—22no—
87SD16×C47—22no—
88D8×C47—21no—
89D8×C47C2×Q812yes—
90C2×C8×C47C2×D411yescharacteristic
91C2×C8×C47C2×Q811yescharacteristic
92C2×C8×C47C2×Q811yescharacteristic
93C2×C8×C47#4111yescharacteristic
94C2×C8×C47#4111yescharacteristic
95C2×C8×C47C2411yescharacteristic
96C4⋊C4×C47—22no—
97C4⋊C4×C47—41no—
98C22⋊Dic47—21no—
99C22⋊Dic47—41no—
100C4⋊Dic47—21no—
101C4⋊Dic47—21no—
102C4⋊Dic47—21no—
103C4⋊Dic47—42no—
104Dic47⋊C4—42no—
105Dic47⋊C4—42no—
106C4×Dic47—41no—
107C4×Dic47C2×D412yes—
108#10625329—22no—
109#10625329—41no—
110#10625329—44no—
111#10625329SD1612yes—
112#10625329SD1612yes—
113C2×C47⋊C8—41no—
114C2×C47⋊C8—44no—
115C2×C47⋊C8C2×D412yes—
116C16×C47—22no—
117C47⋊C16—22no—
118C23×C47—41no—
119C23×C47AGL(1,ℤ/8)11yescharacteristic
120Q8×C47—22no—
121Q8×C47—41no—
122D4×C47—22no—
123D4×C47—42no—
124D4×C47—42no—
125D4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—161no—
131C2×C4×C47C22⋊Q811yescharacteristic
132C2×C4×C47C22⋊Q811yescharacteristic
133C2×C4×C47C2×SD1611yescharacteristic
134C2×C4×C47C2×SD1611yescharacteristic
135C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C47C22×D411yescharacteristic
137C2×Dic47—22no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140Dic94—162no—
141C8×C47C22⋊Q811yescharacteristic
142C8×C47C22⋊Q811yescharacteristic
143C8×C47C22×Q811yescharacteristic
144C8×C47C2×#4111yescharacteristic
145C47⋊C8—42no—
146C47⋊C8C2×SD1614yes—
147#33638—471nomaximal, Sylow
148C22×C47—41no—
149C22×C47—42no—
150C22×C47—161no—
151C22×C47#47611yescharacteristic
152C22×C47#47611yescharacteristic
153C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
154C4×C47—42no—
155C4×C47—82no—
156C4×C47#47611yescharacteristic
157C4×C47#47611yescharacteristic
158C4×C47C2×C22⋊Q811yescharacteristic
159C4×C47C22×SD1611yescharacteristic
160Dic47—42no—
161Dic47—81no—
162Dic47—81no—
163#3163—471no—
164#2957—471no—
165#2231—474no—
166#2230—474no—
167#2155—471no—
168C2×#365—471no—
169#2153—471no—
170#2145—471no—
171C2×#360D4711yescharacteristic
172C2×C47—21no—
173C2×C47—21no—
174C2×C47—21no—
175C2×C47—82no—
176C2×C47C2×#47611yescharacteristic
177C2×#135C2×D4711yescharacteristic
178#568—941no—
179#544—941no—
180C2×M32—471no—
181C2×M32C2×D4711yescharacteristic
182#501—942no—
183#499—472no—
184#498—472no—
185#444—942no—
186#443—474no—
187C2×#108—471no—
188#428—941no—
189#427—471no—
190C2×#106C2×D4711yescharacteristic
191#421—941no—
192#405—471no—
193#365—472no—
194#365—472no—
195#365—474no—
196#364—474no—
197#360—472no—
198#360—472no—
199#360C2×D4714yes—
200C47#3363811yescharacteristic, complemented, Sylow
201C2×#41—21no—
202#135—21no—
203#135—21no—
204#135Dic9412yescomplemented
205#135C22×D4712yes—
206C2×Q16Dic9411yescharacteristic, complemented
207C2×SD16—21no—
208C2×D8Dic9411yescharacteristic, complemented
209#131—944no—
210#131—1882no—
211C2×M16—471no—
212C2×M16—941no—
213C2×M16—1882no—
214C22×C8C22×D4711yescharacteristic
215C4⋊Q8—942no—
216C4⋊Q8—1881no—
217#125—942no—
218#125—1881no—
219C22⋊Q8—1882no—
220#117—471no—
221#117—941no—
222C2×C4⋊C4—21no—
223M32—21no—
224M32—472no—
225M32—941no—
226M32C22×D4712yes—
227C2×C16—21no—
228C2×C16—941no—
229#108—472no—
230#108—941no—
231#107—942no—
232#107—941no—
233#106—21no—
234#106—471no—
235#106—471no—
236#106—942no—
237#106C22×D4712yes—
238#104—944no—
239#97—472no—
240C4×C8—472no—
241#41—22no—
242#41—22no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8—1882no—
247C2×D4—21no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4D4×D4711yescharacteristic
252Q16—21no—
253Q16C2×Dic9412yes—
254SD16—22no—
255SD16—22no—
256D8—21no—
257D8C2×Dic9412yes—
258M16—472no—
259M16—472no—
260M16—942no—
261M16—1884no—
262M16—1881no—
263C2×C8—472no—
264C2×C8—1884no—
265C2×C8—1881no—
266C2×C8C2×Dic9411yescharacteristic
267C2×C8C2×Dic9411yescharacteristic
268C2×C8D4×D4711yescharacteristic
269C2×C8#1062535211yescharacteristic
270C2×C8#1062535211yescharacteristic
271C2×C8C23×D4711yescharacteristic
272C4⋊C4—22no—
273C4⋊C4—41no—
274C4⋊C4—941no—
275C4⋊C4—941no—
276C4⋊C4—941no—
277C4⋊C4—1882no—
278C4⋊C4—1882no—
279C4⋊C4—1882no—
280C22⋊C4—941no—
281C22⋊C4—1881no—
282C42—472no—
283C42—1881no—
284C16—22no—
285C16—942no—
286C23—41no—
287C23#1309404811yescharacteristic
288Q8—22no—
289Q8—41no—
290Q8—7522no—
291D4—22no—
292D4—42no—
293D4—42no—
294D4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—161no—
300C2×C4—942no—
301C2×C4—1881no—
302C2×C4—1881no—
303C2×C4#1309400111yescharacteristic
304C2×C4#1309400111yescharacteristic
305C2×C4SD16×D4711yescharacteristic
306C2×C4SD16×D4711yescharacteristic
307C2×C4#1309404811yescharacteristic
308C2×C4C2×D4×D4711yescharacteristic
309C8—474no—
310C8—1882no—
311C8#1309400111yescharacteristic
312C8#1309400111yescharacteristic
313C8C22×Dic9411yescharacteristic
314C8C2×#1062535211yescharacteristic
315C22—41no—
316C22—42no—
317C22—161no—
318C22#1568941111yescharacteristic
319C22#1568941111yescharacteristic
320C22C2×#1309404811yescharacteristic
321C4—42no—
322C4—82no—
323C4—3761no—
324C4—3761no—
325C4—1882no—
326C4#1568941111yescharacteristic
327C4#1568941111yescharacteristic
328C4C2×#1309400111yescharacteristic
329C4C2×SD16×D4711yescharacteristic
330C2—21no—
331C2—21no—
332C2—21no—
333C2—82no—
334C2C2×#1568941111yescharacteristic
subgroupG/H
centerC2C2×#15689411
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC2×C47×#360C2
socleC2×C47C2×#476
prime radicalC2×#135×C47C22
hypercenterC2×#360D47
hypocenterC47#33638
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#360

Complements

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

NH ≅ G/Ntype
1. #20253916C2semidirect
3. #20245562C2semidirect
4. #20245557C2semidirect
5. #20245161C2semidirect
6. C2×#15688945C2semidirect
7. #20245153C2semidirect
14. #15689523C22semidirect
15. #15689520C22semidirect
23. #15688946C22semidirect
24. #15688946C22semidirect
25. #15688945C22semidirect
26. #15688943C22semidirect
61. #135×C47Q8semidirect
63. C2×Q16×C47Q8semidirect
65. C2×D8×C47Q8semidirect
200. C47#33638semidirect
204. #135Dic94semidirect
206. C2×Q16Dic94semidirect
208. C2×D8Dic94semidirect

Maximal subgroups

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

subgroupindex
#202539162
#202523052
#202455622
#202455572
#202451612
C2×#156889452
#202451532
#202451242
C2×C47×#3602
#3363847