Finite Group Databasebeta

← #26113064

Subgroups — #26113064

Order 12032.

464 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253181C211yesmaximal, characteristic
2#20252428C211yesmaximal, characteristic
3#20243399C211yesmaximal, characteristic
4#20241952C211yesmaximal, characteristic, complemented
5#20241285C211yesmaximal, characteristic, complemented
6#20241247C211yesmaximal, characteristic, complemented
7C47×#1481C211yesmaximal, characteristic, complemented
8C2×C4×D188C2211yescharacteristic
9#15689420—22no—
10#15689410C412yescomplemented
11#15689400C412yescomplemented
12#15689202—22no—
13#15689200—22no—
14#15689176C2211yescharacteristic
15#15689166—21no—
16#15689147C2211yescharacteristic
17#15689127—21no—
18C2×#13093946—21no—
19#15689085—22no—
20#15688989C2211yescharacteristic
21#15688886—21no—
22#15688860—22no—
23C47×#533C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C2×C4⋊C8×C47C2211yescharacteristic
26C47×Q8⋊C8—22no—
27C22×D188D411yescharacteristic
28C22×C4×D47—21no—
29C2×#10625329—21no—
30C2×#10625324—24no—
31C4×D188—21no—
32C4×D188—21no—
33C4×D188C2×C412yes—
34C4×D188D412yes—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47D411yescharacteristic
37C2×(C2×D47)⋊C4—21no—
38#13093957—22no—
39#13093955—22no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)C2×C412yes—
43#13093946—42no—
44#13093944—22no—
45#13093944—22no—
46C22⋊(C47⋊C8)D412yes—
47(C2×D47)⋊C8—24no—
48C4×C47⋊C8—21no—
49C47⋊#95—21no—
50C47⋊#95—21no—
51C22×C8×C47—21no—
52C4⋊Q8×C47C2×C411yescharacteristic
53C4⋊Q8×C47C2×C411yescharacteristic
54#125×C47—21no—
55C22⋊Q8×C47—21no—
56C22⋊Q8×C47—21no—
57C4×Q8×C47—21no—
58C4×Q8×C47—21no—
59#117×C47—21no—
60C2×C42×C47C2311yescharacteristic
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C22⋊C8×C47—21no—
65C4×C8×C47D412yes—
66C23×D47—21no—
67C2×Q8×C47—21no—
68C2×Q8×C47—21no—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47#4111yescharacteristic
72C22×Dic47—21no—
73C22×Dic47—41no—
74C2×D188—21no—
75C2×D188—21no—
76C2×D188—21no—
77C2×D188—21no—
78C2×D188—21no—
79C2×D188C22⋊C412yes—
80C2×C4×D47—22no—
81C2×C4×D47—22no—
82C2×C4×D47—82no—
83C2×C8×C47—22no—
84C2×C8×C47—41no—
85C2×C8×C47—41no—
86C2×C8×C47—41no—
87C4⋊C4×C47—21no—
88C4⋊C4×C47—21no—
89C4⋊C4×C47—21no—
90C4⋊C4×C47—21no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—41no—
93C22⋊C4×C47—41no—
94C42×C47—41no—
95C42×C47C22×C411yescharacteristic
96C42×C47C2×D411yescharacteristic
97C42×C47C2×D411yescharacteristic
98C42×C47#4111yescharacteristic
99(C2×D47)⋊C4—22no—
100(C2×D47)⋊C4—22no—
101C4⋊Dic47—21no—
102C4⋊Dic47—41no—
103C4⋊Dic47—41no—
104C4⋊Dic47C22⋊C412yes—
105#10625329—41no—
106#10625329—41no—
107C2×C47⋊C8—22no—
108C2×C47⋊C8—22no—
109#10625324—84no—
110C23×C47#12111yescharacteristic
111C22×D47—22no—
112C22×D47—41no—
113C22×D47—41no—
114C22×D47—81no—
115C22×D47—81no—
116Q8×C47—42no—
117Q8×C47—81no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47C2×C22⋊C411yescharacteristic
123C2×C4×C47C4×D411yescharacteristic
124C2×C4×C47C4×D411yescharacteristic
125C2×C4×C47#12011yescharacteristic
126C2×C4×C47#12111yescharacteristic
127C2×C4×C47#12311yescharacteristic
128C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C47#13711yescharacteristic
130C2×C4×C47#13711yescharacteristic
131C2×C4×C47#13711yescharacteristic
132C2×Dic47—22no—
133C2×Dic47—81no—
134C2×Dic47—81no—
135C2×Dic47—81no—
136C2×Dic47—81no—
137D188—22no—
138D188—22no—
139D188—41no—
140D188—41no—
141C4×D47—44no—
142C4×D47—82no—
143C8×C47—42no—
144C8×C47—81no—
145C47⋊C8—42no—
146C47⋊C8—42no—
147#9307—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47#38611yescharacteristic
151C22×C47#46811yescharacteristic
152C22×C47#47011yescharacteristic
153C2×D47—42no—
154C2×D47—42no—
155C2×D47—42no—
156C2×D47—82no—
157C2×D47—82no—
158C4×C47—21no—
159C4×C47—21no—
160C4×C47—21no—
161C4×C47—21no—
162C4×C47—81no—
163C4×C47—81no—
164C4×C47#41711yescharacteristic
165C4×C47#41911yescharacteristic
166C4×C47#46811yescharacteristic
167C4×C47#47011yescharacteristic
168Dic47—42no—
169Dic47—161no—
170#3077—471no—
171#2975—471no—
172#1804—471no—
173#1518—471no—
174#1481D4711yescharacteristic, complemented
175#1301—471no—
176#1286—471no—
177C2×C47—21no—
178C2×C47—21no—
179C2×C47#188111yescharacteristic
180C2×C47#189411yescharacteristic
181C2×C47#190411yescharacteristic
182D47—82no—
183#533C2×D4711yescharacteristic
184C2×C4×D4—471no—
185#478—942no—
186#476—472no—
187#474—472no—
188#436—942no—
189#436—942no—
190#431—471no—
191#431C2×D4711yescharacteristic
192#428—941no—
193#423—471no—
194C2×C4⋊C8C2×D4711yescharacteristic
195#418—941no—
196C2×#102—941no—
197#407—942no—
198#382—471no—
199#340—941no—
200#328—942no—
201Q8⋊C8—22no—
202C47#930711yescharacteristic, complemented, Sylow
203C22×D4—471no—
204C23×C4—941no—
205C2×M16—944no—
206C2×M16—941no—
207C22×C8—21no—
208C4⋊Q8C4×D4711yescharacteristic
209C4⋊Q8C4×D4711yescharacteristic
210#125—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×D4—472no—
216C4×D4—472no—
217C4×D4—941no—
218C4×D4—941no—
219#117—21no—
220C2×C4⋊C4—471no—
221C2×C4⋊C4—941no—
222C2×C22⋊C4—941no—
223C2×C42C22×D4711yescharacteristic
224#107—942no—
225#106—942no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—472no—
230C4⋊C8—941no—
231C4⋊C8—941no—
232#102—942no—
233#102—942no—
234#102—1882no—
235C22⋊C8—21no—
236C22⋊C8—472no—
237C22⋊C8—944no—
238C4×C8—941no—
239C4×C8C22⋊D4712yes—
240#95—941no—
241#95—941no—
242C24—941no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×D4—472no—
246C2×D4—941no—
247C2×D4—941no—
248C2×D4—941no—
249C2×D4—941no—
250C2×D4—941no—
251C22×C4—941no—
252C22×C4—942no—
253C22×C4—942no—
254C22×C4—3762no—
255C22×C4—1881no—
256C22×C4#1062535011yescharacteristic
257C22×C4D4×D4711yescharacteristic
258C22×C4D4×D4711yescharacteristic
259M16—3764no—
260M16—1881no—
261M16—1881no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—942no—
267C2×C8—942no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—472no—
275C4⋊C4—941no—
276C4⋊C4—1881no—
277C4⋊C4—1881no—
278C22⋊C4—41no—
279C22⋊C4—942no—
280C22⋊C4—942no—
281C42—41no—
282C42C2×C4×D4711yescharacteristic
283C42D4×D4711yescharacteristic
284C42#1062535211yescharacteristic
285C42C2×C22⋊D4711yescharacteristic
286C23—942no—
287C23—3761no—
288C23—3761no—
289C23—1881no—
290C23—1881no—
291C23#1309399711yescharacteristic
292Q8—42no—
293Q8—81no—
294D4—942no—
295D4—942no—
296D4—1881no—
297D4—1881no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—942no—
303C2×C4—3762no—
304C2×C4—3761no—
305C2×C4—3761no—
306C2×C4—3761no—
307C2×C4—3761no—
308C2×C4—1884no—
309C2×C4C22⋊C4×D4711yescharacteristic
310C2×C4(C22⋊D47)⋊C411yescharacteristic
311C2×C4C4×C22⋊D4711yescharacteristic
312C2×C4#1309399211yescharacteristic
313C2×C4#1309399511yescharacteristic
314C2×C4#1309400611yescharacteristic
315C2×C4#1309405111yescharacteristic
316C2×C4#1309405311yescharacteristic
317C2×C4#1309405611yescharacteristic
318C2×C4#1309405611yescharacteristic
319C8—42no—
320C8—81no—
321C8—1882no—
322C8—1882no—
323C22—21no—
324C22—21no—
325C22—3762no—
326C22—3762no—
327C22—1882no—
328C22—1882no—
329C22—1882no—
330C22#1568899911yescharacteristic
331C22#1568937211yescharacteristic
332C22#1568938011yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—81no—
338C4—81no—
339C4—1882no—
340C4—7521no—
341C4#1568912311yescharacteristic
342C4#1568913011yescharacteristic
343C4#1568936611yescharacteristic
344C4#1568937811yescharacteristic
345C2—21no—
346C2—21no—
347C2—3762no—
348C2#2024375311yescharacteristic
349C2#2024381811yescharacteristic
350C2#2024387211yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1481C2
socleC22×C47#386
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1481D47
hypocenterC47#9307
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
4. #20241952C2semidirect
5. #20241285C2semidirect
6. #20241247C2semidirect
7. C47×#1481C2semidirect
10. #15689410C4semidirect
11. #15689400C4semidirect
174. #1481D47semidirect
202. C47#9307semidirect

Maximal subgroups

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

subgroupindex
#202531812
#202524282
#202433992
#202419522
#202412852
#202412472
C47×#14812
#930747