Finite Group Databasebeta

← #26113012

Subgroups — #26113012

Order 12032.

508 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254509C211yesmaximal, characteristic
2#20244220C211yesmaximal, characteristic
3#20242060C211yesmaximal, characteristic, complemented
4#20241947C211yesmaximal, characteristic
5#20241246C211yesmaximal, characteristic, complemented
6#20241217C211yesmaximal, characteristic, complemented
7C47×#2327C211yesmaximal, characteristic, complemented
8#15689673C2211yescharacteristic
9#15689526C412yescomplemented
10#15689471C412yescomplemented
11#15689436—22no—
12#15689400—22no—
13#15689176C2211yescharacteristic
14#15689176C2211yescharacteristic
15#15689147C2211yescharacteristic
16C2×#13093952—21no—
17C2×#13093952—21no—
18#15688869—22no—
19#15688856—22no—
20C2×C4⋊Q8×C47C2211yescharacteristic
21C2×C4×Q8×C47—21no—
22C47×#391—21no—
23C47×#387—22no—
24C47×#387—22no—
25C47×#382C2211yescharacteristic
26C47×#376C2211yescharacteristic
27C2×#10625350—21no—
28C2×#10625329—21no—
29#13094024C2×C411yescharacteristic
30#13094021C2×C411yescharacteristic
31#13094012—21no—
32#13093998—21no—
33#13093998—21no—
34C4×D188—21no—
35C4×D188—21no—
36#13093957—24no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41#13093952—42no—
42#13093952—42no—
43#13093944—24no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—21no—
46C4×C47⋊C8—21no—
47C4×C47⋊C8C2×C412yes—
48C22×Q8×C47—21no—
49C4⋊Q8×C47—21no—
50C4⋊Q8×C47—21no—
51C4⋊Q8×C47C2×C412yes—
52C4⋊Q8×C47D412yes—
53C4×Q8×C47—42no—
54C4×Q8×C47—42no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—22no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47D412yes—
59C2×C42×C47—22no—
60C2×C42×C47D411yescharacteristic
61C2×C42×C47D411yescharacteristic
62C2×C42×C47C2311yescharacteristic
63#95×C47—24no—
64#95×C47—21no—
65#95×C47—21no—
66#95×C47—21no—
67#95×C47D412yes—
68C2×Q8×C47—22no—
69C2×Q8×C47—22no—
70C2×Q8×C47—41no—
71C2×Q8×C47—41no—
72C2×Q8×C47—41no—
73C2×Q8×C47—41no—
74C22×C4×C47—21no—
75C22×C4×C47—22no—
76C22×C4×C47—22no—
77C22×C4×C47—21no—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C2×C22⋊D47—41no—
82#10625350—41no—
83#10625350—41no—
84#10625350—41no—
85#10625350—41no—
86C2×D188—21no—
87C2×D188—21no—
88C2×D188—21no—
89C2×C4×D47—41no—
90C2×Dic94—21no—
91C4⋊C4×C47—22no—
92C4⋊C4×C47—22no—
93C4⋊C4×C47—22no—
94C4⋊C4×C47—44no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C4⋊C4×C47C22⋊C414yes—
98C42×C47—21no—
99C42×C47—21no—
100C42×C47—21no—
101C42×C47—44no—
102C42×C47D812yes—
103C42×C47C22×C411yescharacteristic
104C42×C47C2×D411yescharacteristic
105C42×C47#4111yescharacteristic
106C42×C47#4111yescharacteristic
107(C2×D47)⋊C4—41no—
108(C2×D47)⋊C4—41no—
109C4⋊Dic47—21no—
110C4⋊Dic47—21no—
111#10625329—41no—
112#10625329—41no—
113C2×C47⋊C8—22no—
114C2×C47⋊C8—41no—
115C2×C47⋊C8—41no—
116C23×C47#12011yescharacteristic
117C22×D47—41no—
118Q8×C47—42no—
119Q8×C47—42no—
120Q8×C47—42no—
121Q8×C47—42no—
122C2×C4×C47—24no—
123C2×C4×C47—22no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—41no—
132C2×C4×C47—42no—
133C2×C4×C47—41no—
134C2×C4×C47—42no—
135C2×C4×C47—41no—
136C2×C4×C47C2×C22⋊C411yescharacteristic
137C2×C4×C47C4×D411yescharacteristic
138C2×C4×C47C4×D411yescharacteristic
139C2×C4×C47#12111yescharacteristic
140C2×C4×C47#12111yescharacteristic
141C2×C4×C47#12311yescharacteristic
142C2×C4×C47C2×D811yescharacteristic
143C2×C4×C47#13511yescharacteristic
144C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
146C22⋊D47—81no—
147C22⋊D47—81no—
148C2×Dic47—41no—
149D188—42no—
150D188—41no—
151D188—41no—
152D188—81no—
153C4×D47—81no—
154C4×D47—81no—
155Dic94—41no—
156Dic94—41no—
157C47⋊C8—42no—
158C47⋊C8—81no—
159C47⋊C8—81no—
160#9300—471nomaximal, Sylow
161C22×C47—21no—
162C22×C47—21no—
163C22×C47#38611yescharacteristic
164C22×C47#44711yescharacteristic
165C22×C47#44911yescharacteristic
166C2×D47—81no—
167C2×D47—81no—
168C2×D47—81no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—42no—
174C4×C47—44no—
175C4×C47—82no—
176C4×C47—81no—
177C4×C47C4×D811yescharacteristic
178C4×C47#44211yescharacteristic
179C4×C47#45911yescharacteristic
180C4×C47#46311yescharacteristic
181Dic47—81no—
182#3226—471no—
183#2327D4711yescharacteristic, complemented
184#1963—471no—
185#1546—471no—
186#1517—471no—
187#1286—471no—
188#1274—471no—
189C2×C47—21no—
190C2×C47—21no—
191C2×C47#187911yescharacteristic
192C2×C47#190511yescharacteristic
193C2×C47#191211yescharacteristic
194D47—161no—
195#532—471no—
196C2×C4⋊Q8C2×D4711yescharacteristic
197C2×C4×Q8—21no—
198#500—472no—
199#486—472no—
200#480—942no—
201#474—942no—
202#431—471no—
203#431—471no—
204#423—471no—
205C2×#104—941no—
206C2×#104—941no—
207#391—21no—
208#387—22no—
209#387—22no—
210#382C2×D4711yescharacteristic
211#376C2×D4711yescharacteristic
212#332—942no—
213#327—942no—
214C47#930011yescharacteristic, complemented, Sylow
215C2×#41—941no—
216C22×Q8—21no—
217C2×M16—941no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—471no—
221C4⋊Q8C4×D4712yes—
222C4⋊Q8D18812yes—
223#127—471no—
224#124—941no—
225#121—941no—
226#121—941no—
227C4×Q8—42no—
228C4×Q8—42no—
229C4×D4—941no—
230C4×D4—941no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22⋊D4712yes—
235C2×C42—22no—
236C2×C42C22⋊D4711yescharacteristic
237C2×C42C22⋊D4711yescharacteristic
238C2×C42C22×D4711yescharacteristic
239#107—944no—
240C4⋊C8—941no—
241C4⋊C8—941no—
242C4⋊C8—941no—
243C4⋊C8—941no—
244#104—1882no—
245#104—1882no—
246#102—944no—
247C22⋊C8—941no—
248C22⋊C8—941no—
249C4×C8—472no—
250C4×C8—941no—
251#95—24no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95C22⋊D4712yes—
256#41—1881no—
257#41—1881no—
258#41—1881no—
259#41—1881no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—941no—
267C2×D4—941no—
268C2×D4—941no—
269C2×D4—941no—
270C2×D4—1881no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—1881no—
276C22×C4C2×C22⋊D4711yescharacteristic
277C22×C4C2×C22⋊D4711yescharacteristic
278C22×C4C2×C22⋊D4711yescharacteristic
279M16—1881no—
280M16—1881no—
281C2×C8—942no—
282C2×C8—1881no—
283C2×C8—1881no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—44no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—941no—
291C4⋊C4—941no—
292C4⋊C4(C2×D47)⋊C414yes—
293C22⋊C4—1881no—
294C22⋊C4—1881no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—44no—
299C42#1062533412yes—
300C42C2×C4×D4711yescharacteristic
301C42C2×D18811yescharacteristic
302C42#1062535011yescharacteristic
303C42#1062535011yescharacteristic
304C23—1881no—
305C23#1309399311yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—42no—
309Q8—42no—
310Q8—1881no—
311Q8—1881no—
312D4—3761no—
313D4—3761no—
314D4—3761no—
315D4—1882no—
316D4—1881no—
317D4—1881no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—3761no—
333C2×C4—3761no—
334C2×C4—1881no—
335C2×C4C2×(C2×D47)⋊C411yescharacteristic
336C2×C4C4×C22⋊D4711yescharacteristic
337C2×C4C4×C22⋊D4711yescharacteristic
338C2×C4#1309399811yescharacteristic
339C2×C4#1309399811yescharacteristic
340C2×C4#1309400711yescharacteristic
341C2×C4C2×#1062533411yescharacteristic
342C2×C4#1309404411yescharacteristic
343C2×C4#1309404911yescharacteristic
344C2×C4#1309405011yescharacteristic
345C8—3761no—
346C8—3761no—
347C8—1882no—
348C22—21no—
349C22—21no—
350C22—3761no—
351C22—3761no—
352C22—3761no—
353C22#1568900211yescharacteristic
354C22#1568925711yescharacteristic
355C22#1568926811yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—44no—
362C4—82no—
363C4—81no—
364C4—3761no—
365C4C4×#1062533411yescharacteristic
366C4#1568923311yescharacteristic
367C4#1568931811yescharacteristic
368C4#1568933811yescharacteristic
369C2—21no—
370C2—21no—
371C2—7521no—
372C2#2024373611yescharacteristic
373C2#2024388211yescharacteristic
374C2#2024392511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2327C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#2327D47
hypocenterC47#9300
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
3. #20242060C2semidirect
5. #20241246C2semidirect
6. #20241217C2semidirect
7. C47×#2327C2semidirect
9. #15689526C4semidirect
10. #15689471C4semidirect
183. #2327D47semidirect
214. C47#9300semidirect

Maximal subgroups

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

subgroupindex
#202545092
#202442202
#202420602
#202419472
#202412462
#202412172
C47×#23272
#930047