Finite Group Databasebeta

← #26107639

Subgroups — #26107639

Order 12032.

874 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C22×#13094017C211yesmaximal, characteristic
2#20243454C212yesmaximal, complemented
3C2×#15688848C212yesmaximal, complemented
4C2×#15688847C211yesmaximal, characteristic
5C2×C47×#323C211yesmaximal, characteristic
6C2×#13094017—24no—
7C2×#13094017—24no—
8C2×#13094017—22no—
9C2×#13094017—22no—
10C22×C4⋊Dic47C411yescharacteristic
11C22×Dic47⋊C4C411yescharacteristic
12C22×Dic47⋊C4C2211yescharacteristic
13C22×C22⋊Dic47C411yescharacteristic
14C22×C22⋊Dic47C411yescharacteristic
15C22×C4×Dic47C2211yescharacteristic
16C2×C22⋊(C47⋊C8)C2212yes—
17#15688877—24no—
18#15688848—24no—
19#15688847—22no—
20C22×C22⋊C4×C47C2211yescharacteristic
21C2×C22⋊C8×C47C2212yes—
22C47×#323—22no—
23C23×Dic47C2×C411yescharacteristic
24C23×Dic47C2×C411yescharacteristic
25C23×Dic47C2×C411yescharacteristic
26C23×Dic47C2×C411yescharacteristic
27C22×C47⋊C8—22no—
28#13094017—216no—
29#13094017—216no—
30C2×C4⋊Dic47—22no—
31C2×C4⋊Dic47—22no—
32C2×C4⋊Dic47—22no—
33C2×Dic47⋊C4—22no—
34C2×Dic47⋊C4—22no—
35C2×Dic47⋊C4—21no—
36C2×Dic47⋊C4—21no—
37C2×Dic47⋊C4—22no—
38C2×Dic47⋊C4—22no—
39C2×Dic47⋊C4D414yes—
40C2×C22⋊Dic47—22no—
41C2×C22⋊Dic47—22no—
42C2×C22⋊Dic47—22no—
43C2×C22⋊Dic47—22no—
44C2×C22⋊Dic47—22no—
45C2×C22⋊Dic47—22no—
46C2×C4×Dic47—21no—
47C2×C4×Dic47—22no—
48C2×C4×Dic47—21no—
49C2×C4×Dic47D414yes—
50C22⋊(C47⋊C8)—22no—
51C22⋊(C47⋊C8)—22no—
52C25×C47C2×C411yescharacteristic
53C23×C4×C47C2×C411yescharacteristic
54C23×C4×C47C2311yescharacteristic
55C22×C8×C47—22no—
56C2×C22⋊C4×C47—22no—
57C2×C22⋊C4×C47—22no—
58C2×C22⋊C4×C47D412yes—
59C2×C22⋊C4×C47D412yes—
60C22⋊C8×C47—22no—
61C22⋊C8×C47—22no—
62C24×C47—21no—
63C24×C47—21no—
64C24×C47—22no—
65C24×C47—22no—
66C24×C47—42no—
67C24×C47—42no—
68C24×C47C22⋊C412yes—
69C24×C47C22×C411yescharacteristic
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—42no—
73C22×C4×C47—42no—
74C22×C4×C47C22⋊C412yes—
75C22×C4×C47C2×D412yes—
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×D411yescharacteristic
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C22×Dic47—21no—
85C22×Dic47—21no—
86C22×Dic47—21no—
87C22×Dic47—22no—
88C22×Dic47—22no—
89C22×Dic47—22no—
90C22×Dic47—22no—
91C22×Dic47—42no—
92C22×Dic47—42no—
93C22×Dic47C22⋊C412yes—
94C22×Dic47C22⋊C412yes—
95C22×Dic47C22⋊C412yes—
96C22×Dic47C22⋊C412yes—
97C2×C8×C47—44no—
98C2×C8×C47—42no—
99C22⋊C4×C47—24no—
100C22⋊C4×C47—24no—
101C22⋊Dic47—28no—
102C22⋊Dic47—28no—
103C4⋊Dic47—28no—
104Dic47⋊C4—28no—
105Dic47⋊C4—24no—
106Dic47⋊C4—24no—
107C4×Dic47—24no—
108C4×Dic47—24no—
109C2×C47⋊C8—44no—
110C2×C47⋊C8—42no—
111C23×C47—22no—
112C23×C47—22no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—22no—
116C23×C47—42no—
117C23×C47—42no—
118C23×C47—42no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—42no—
122C23×C47—42no—
123C23×C47—42no—
124C23×C47—84no—
125C23×C47—84no—
126C23×C47C2×C22⋊C411yescharacteristic
127C23×C47C2×C22⋊C411yescharacteristic
128C23×C47C2×C22⋊C411yescharacteristic
129C23×C47#13112yes—
130C23×C47#13112yes—
131C2×C4×C47—24no—
132C2×C4×C47—22no—
133C2×C4×C47—41no—
134C2×C4×C47—42no—
135C2×C4×C47—41no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47#12012yes—
140C2×C4×C47#12012yes—
141C2×Dic47—22no—
142C2×Dic47—22no—
143C2×Dic47—22no—
144C2×Dic47—22no—
145C2×Dic47—24no—
146C2×Dic47—24no—
147C2×Dic47—24no—
148C2×Dic47—42no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—42no—
158C2×Dic47—42no—
159C8×C47—82no—
160C47⋊C8—82no—
161#8309—471nomaximal, Sylow
162C22×C47—24no—
163C22×C47—24no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—42no—
169C22×C47—42no—
170C22×C47—42no—
171C22×C47—82no—
172C22×C47—82no—
173C22×C47—84no—
174C22×C47—84no—
175C22×C47#37911yescharacteristic
176C22×C47#40812yes—
177C22×C47#40812yes—
178C22×C47#40811yescharacteristic
179C22×C47#40811yescharacteristic
180C22×C47#41012yes—
181C22×C47#41312yes—
182C4×C47—22no—
183C4×C47—22no—
184C4×C47—42no—
185Dic47—42no—
186Dic47—42no—
187Dic47—42no—
188Dic47—42no—
189C22×#126—471no—
190#1817—472no—
191C2×#324—472no—
192C2×#323—471no—
193C2×#323D4711yescharacteristic
194C2×C47—22no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47—82no—
198C2×C47#147314yes—
199C2×C47#178011yescharacteristic
200C2×C47#178211yescharacteristic
201C2×C47#178511yescharacteristic
202C2×#126—944no—
203C2×#126—944no—
204C2×#126—942no—
205C2×#126—942no—
206C22×C4⋊C4—471no—
207C22×C4⋊C4—471no—
208C22×C4⋊C4—471no—
209C22×C22⋊C4—471no—
210C22×C22⋊C4—471no—
211C22×C22⋊C4C2×D4711yescharacteristic
212C22×C42—471no—
213C2×C22⋊C8—472no—
214C2×C22⋊C8C2×D4712yes—
215#336—944no—
216#324—944no—
217#323—22no—
218#323—942no—
219C47#830911yescharacteristic, complemented, Sylow
220C25C4×D4711yescharacteristic
221C23×C4—471no—
222C23×C4—471no—
223C23×C4—471no—
224C23×C4—471no—
225C23×C4C4×D4711yescharacteristic
226C23×C4C22×D4711yescharacteristic
227C22×C8—22no—
228C22×C8—942no—
229#126—9416no—
230#126—9416no—
231C2×C4⋊C4—474no—
232C2×C4⋊C4—942no—
233C2×C4⋊C4—942no—
234C2×C4⋊C4—942no—
235C2×C4⋊C4—942no—
236C2×C4⋊C4—942no—
237C2×C4⋊C4—942no—
238C2×C4⋊C4—941no—
239C2×C4⋊C4—942no—
240C2×C4⋊C4—941no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—942no—
244C2×C22⋊C4—942no—
245C2×C22⋊C4—942no—
246C2×C22⋊C4—942no—
247C2×C22⋊C4—942no—
248C2×C22⋊C4—942no—
249C2×C22⋊C4D18812yes—
250C2×C22⋊C4C22⋊D4712yes—
251C2×C42—474no—
252C2×C42—941no—
253C2×C42—942no—
254C2×C42—941no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—942no—
258C22⋊C8—942no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—22no—
263C24—42no—
264C24—42no—
265C24(C2×D47)⋊C412yes—
266C24C2×C4×D4711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—472no—
272C22×C4—472no—
273C22×C4—472no—
274C22×C4—472no—
275C22×C4—941no—
276C22×C4—941no—
277C22×C4—941no—
278C22×C4—941no—
279C22×C4—941no—
280C22×C4—941no—
281C22×C4—941no—
282C22×C4—941no—
283C22×C4—942no—
284C22×C4—942no—
285C22×C4—942no—
286C22×C4—942no—
287C22×C4—1882no—
288C22×C4—1882no—
289C22×C4(C2×D47)⋊C412yes—
290C22×C4C2×D18811yescharacteristic
291C22×C4D4×D4712yes—
292C22×C4D4×D4712yes—
293C22×C4C2×C22⋊D4711yescharacteristic
294C2×C8—44no—
295C2×C8—42no—
296C2×C8—1884no—
297C2×C8—1882no—
298C4⋊C4—948no—
299C4⋊C4—948no—
300C4⋊C4—944no—
301C4⋊C4—944no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—948no—
305C22⋊C4—948no—
306C42—944no—
307C42—944no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—84no—
322C23—84no—
323C23C22⋊C4×D4711yescharacteristic
324C23C22⋊C4×D4711yescharacteristic
325C23C2×(C2×D47)⋊C411yescharacteristic
326C23#1309403012yes—
327C23#1309403212yes—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—942no—
337C2×C4—942no—
338C2×C4—942no—
339C2×C4—942no—
340C2×C4—944no—
341C2×C4—944no—
342C2×C4—944no—
343C2×C4—1882no—
344C2×C4—1882no—
345C2×C4—1882no—
346C2×C4—1882no—
347C2×C4—1882no—
348C2×C4—1881no—
349C2×C4—1881no—
350C2×C4—1881no—
351C2×C4—1881no—
352C2×C4—1882no—
353C2×C4—1882no—
354C2×C4#1309399212yes—
355C2×C4#1309399412yes—
356C8—82no—
357C8—3762no—
358C22—24no—
359C22—24no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—82no—
368C22—82no—
369C22—84no—
370C22—84no—
371C22#1568897811yescharacteristic
372C22#1568908812yes—
373C22#1568908812yes—
374C22#1568908911yescharacteristic
375C22#1568909111yescharacteristic
376C22#1568909512yes—
377C22#1568910412yes—
378C4—22no—
379C4—22no—
380C4—42no—
381C4—1882no—
382C4—1882no—
383C4—1882no—
384C4—1882no—
385C2—22no—
386C2—21no—
387C2—21no—
388C2—82no—
389C2#2024175414yes—
390C2#2024329911yescharacteristic
391C2#2024330411yescharacteristic
392C2#2024331311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#323C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47C2×C4
cosocleC23×C4×C47C23
hypercenterC2×#323D47
hypocenterC47#8309
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. #20243454C2semidirect
3. C2×#15688848C2semidirect
219. C47#8309semidirect

Maximal subgroups

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

subgroupindex
C22×#130940172
#202434542
C2×#156888482
C2×#156888472
C2×C47×#3232
#830947