Finite Group Databasebeta

← #26134942

Subgroups — #26134942

Order 12032.

488 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 99 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#20248731C211yesmaximal, characteristic
2C4×#13093939C211yesmaximal, characteristic
3#20241507C212yesmaximal
4#20241507C212yesmaximal
5C47×#1800C211yesmaximal, characteristic
6C22×Dic47⋊C4C411yescharacteristic, complemented
7C2×#13093939C2212yes—
8#15689056—21no—
9#15689046—21no—
10#15689007—21no—
11#15689003C2211yescharacteristic
12#15689003C2211yescharacteristic
13#15688996—21no—
14#15688995C411yescharacteristic, complemented
15#15688995C411yescharacteristic, complemented
16#15688983C411yescharacteristic, complemented
17C4×C22⋊Dic47—21no—
18#15688890—21no—
19C2×C4×D4×C47C2211yescharacteristic
20C47×#423—21no—
21C2×#100×C47C2212yes—
22C47×#343—21no—
23C23×Dic47C2×C411yescharacteristic
24C23×Dic47C2×C411yescharacteristic
25C2×C4⋊Dic47—21no—
26C2×Dic47⋊C4—21no—
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—41no—
31C2×Dic47⋊C4—41no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47C2×C411yescharacteristic
36C2×C22⋊Dic47C2×C411yescharacteristic
37C2×C4×Dic47—41no—
38#13093939—22no—
39#13093939Q814yes—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—41no—
46C22×D4×C47C2311yescharacteristic
47C23×C4×C47C2×C411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C2×M16×C47—22no—
50C4×D4×C47—41no—
51C4×D4×C47—41no—
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C22⋊C4×C47C2×C411yescharacteristic
54C2×C22⋊C4×C47D411yescharacteristic
55C2×C42×C47D411yescharacteristic
56C4⋊C8×C47—41no—
57C4⋊C8×C47—41no—
58#100×C47—24no—
59C24×C47C22×C411yescharacteristic
60C24×C47C2×D411yescharacteristic
61C2×D4×C47—41no—
62C2×D4×C47—41no—
63C2×D4×C47C2×Q812yes—
64C2×D4×C47#4112yes—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—81no—
70C22×C4×C47—81no—
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79C22×Dic47—21no—
80C22×Dic47—42no—
81C22×Dic47—41no—
82C22×Dic47—42no—
83C22×Dic47—41no—
84C22×Dic47—81no—
85M16×C47—42no—
86M16×C47—42no—
87C2×C8×C47—42no—
88C4⋊C4×C47—41no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—41no—
91C42×C47—21no—
92C42×C47—21no—
93C22⋊Dic47—24no—
94C22⋊Dic47—42no—
95C22⋊Dic47—42no—
96C22⋊Dic47—41no—
97C22⋊Dic47—41no—
98C22⋊Dic47—81no—
99C4⋊Dic47—81no—
100Dic47⋊C4—42no—
101Dic47⋊C4—42no—
102Dic47⋊C4—41no—
103Dic47⋊C4—41no—
104Dic47⋊C4—81no—
105C4×Dic47—82no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47C2×C22⋊C411yescharacteristic
115C23×C47C4×Q812yes—
116C23×C47C22⋊Q812yes—
117D4×C47—42no—
118D4×C47—42no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—41no—
125C2×C4×C47—42no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—81no—
130C2×C4×C47—81no—
131C2×C4×C47—81no—
132C2×C4×C47#9911yescharacteristic
133C2×C4×C47#9911yescharacteristic
134C2×C4×C47#12411yescharacteristic
135C2×C4×C47C4⋊Q811yescharacteristic
136C2×Dic47—22no—
137C2×Dic47—41no—
138C2×Dic47—42no—
139C2×Dic47—41no—
140C2×Dic47—42no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—44no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151C2×Dic47—82no—
152C2×Dic47—82no—
153C2×Dic47—82no—
154C8×C47—82no—
155#13297—471nomaximal, Sylow
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—22no—
159C22×C47—22no—
160C22×C47—42no—
161C22×C47—42no—
162C22×C47—42no—
163C22×C47—41no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47—81no—
168C22×C47#39111yescharacteristic
169C22×C47C2×#9911yescharacteristic
170C22×C47#41011yescharacteristic
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—41no—
176C4×C47—41no—
177C4×C47—41no—
178Dic47—42no—
179Dic47—81no—
180Dic47—81no—
181Dic47—84no—
182Dic47—81no—
183#2579—471no—
184#1800D4711yescharacteristic
185C4×#99—471no—
186#1390—472no—
187#1390—472no—
188C2×C47—21no—
189C2×C47—21no—
190C2×C47—42no—
191C2×C47—42no—
192C2×C47#199611yescharacteristic
193C2×C47#212011yescharacteristic
194C2×C47#212111yescharacteristic
195C2×C4×D4C2×D4711yescharacteristic
196C22×C4⋊C4—471no—
197#423—21no—
198C2×#100C2×D4712yes—
199C2×#99—472no—
200#398—941no—
201#396—941no—
202#387—941no—
203#386—471no—
204#386—471no—
205#385—471no—
206#385—471no—
207#385—941no—
208#380—471no—
209C4×C22⋊C4—941no—
210#343—21no—
211#342—941no—
212C47#1329711yescharacteristic, complemented, Sylow
213C22×D4C22×D4711yescharacteristic
214C23×C4—471no—
215C23×C4—471no—
216C23×C4C4×D4711yescharacteristic
217C23×C4D18811yescharacteristic
218C2×M16—22no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—941no—
222C2×C4⋊C4—941no—
223C2×C4⋊C4—941no—
224C2×C4⋊C4—941no—
225C2×C4⋊C4—941no—
226C2×C4⋊C4—1881no—
227C2×C4⋊C4—1881no—
228C2×C4⋊C4C22⋊D4711yescharacteristic
229C2×C22⋊C4—471no—
230C2×C22⋊C4—471no—
231C2×C22⋊C4—942no—
232C2×C22⋊C4—941no—
233C2×C22⋊C4—941no—
234C2×C22⋊C4C4×D4711yescharacteristic
235C2×C22⋊C4D18811yescharacteristic
236C2×C42—1881no—
237C2×C42C22⋊D4711yescharacteristic
238C4⋊C8—41no—
239C4⋊C8—41no—
240#100—24no—
241#99—474no—
242#99—942no—
243#95—941no—
244#95—941no—
245#95—941no—
246#95—941no—
247#95—941no—
248#95—1881no—
249C24C2×C4×D4711yescharacteristic
250C24C2×D18811yescharacteristic
251C2×D4—41no—
252C2×D4—41no—
253C2×D4#1062535212yes—
254C2×D4Q8×D4712yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—941no—
262C22×C4—941no—
263C22×C4—941no—
264C22×C4—941no—
265C22×C4—3761no—
266C22×C4—1882no—
267C22×C4—1882no—
268C22×C4—1881no—
269C22×C4—1881no—
270C22×C4(C2×D47)⋊C411yescharacteristic
271C22×C4(C2×D47)⋊C411yescharacteristic
272C22×C4(C2×D47)⋊C411yescharacteristic
273C22×C4(C2×D47)⋊C411yescharacteristic
274C22×C4C2×C22⋊D4711yescharacteristic
275M16—42no—
276M16—42no—
277C2×C8—42no—
278C4⋊C4—41no—
279C4⋊C4—3761no—
280C4⋊C4—3761no—
281C4⋊C4—1882no—
282C4⋊C4—1882no—
283C4⋊C4—1881no—
284C4⋊C4—1881no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—944no—
288C22⋊C4—3761no—
289C22⋊C4—1882no—
290C22⋊C4—1882no—
291C22⋊C4—1881no—
292C22⋊C4—1881no—
293C42—21no—
294C42—21no—
295C42—3762no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23C2×(C2×D47)⋊C411yescharacteristic
305C23#1309398912yes—
306C23#1309400412yes—
307D4—42no—
308D4—42no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—942no—
323C2×C4—3762no—
324C2×C4—3762no—
325C2×C4—3762no—
326C2×C4—1884no—
327C2×C4—1881no—
328C2×C4—1881no—
329C2×C4—1881no—
330C2×C4—1881no—
331C2×C4—1881no—
332C2×C4—1881no—
333C2×C4—1881no—
334C2×C4—1881no—
335C2×C4—1881no—
336C2×C4—1881no—
337C2×C4—1882no—
338C2×C4—1881no—
339C2×C4—1882no—
340C2×C4#1309393811yescharacteristic
341C2×C4#1309393811yescharacteristic
342C2×C4#1309401111yescharacteristic
343C2×C4#1309402211yescharacteristic
344C8—82no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22#1568902811yescharacteristic
358C22C2×#1309393811yescharacteristic
359C22#1568909611yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—3761no—
368C4—3761no—
369C4—3764no—
370C4—3761no—
371C4—1882no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2#2024437211yescharacteristic
377C2#2024503011yescharacteristic
378C2#2024503511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC24×C47C22×C4
FrattiniC22×D4C22×D47
FittingC47×#1800C2
socleC22×C47C2×#99
prime radical, cosocleC22×D4×C47C23
hypercenter#1800D47
hypocenterC47#13297
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
6. C22×Dic47⋊C4C4semidirect
14. #15688995C4semidirect
15. #15688995C4semidirect
16. #15688983C4semidirect
212. C47#13297semidirect

Maximal subgroups

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

subgroupindex
#202487312
C4×#130939392
#202415072
C47×#18002
#1329747