Finite Group Databasebeta

← #26188771

Subgroups — #26188771

Order 12032.

930 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×#15689754C211yesmaximal, characteristic
2#20253336C212yesmaximal
3#20242429C214yesmaximal, complemented
4#20242395C214yesmaximal, complemented
5#20241920C212yesmaximal
6#20241718C211yesmaximal, characteristic
7C47×#2484C211yesmaximal, characteristic
8#15689754—24no—
9#15689754—22no—
10#15689754C2214yes—
11C2×#13094023—22no—
12C2×#13094015—22no—
13C2×#13094003—21no—
14C2×#13094003C2212yes—
15C2×#13093983—21no—
16#15689464—24no—
17#15689458C2214yes—
18#15689453C2214yes—
19#15689166—22no—
20#15689131—22no—
21#15689087C2212yes—
22#15688859C2218yes—
23#15688849C2214yes—
24C2×#117×C47C2211yescharacteristic
25C22×C4⋊C4×C47C2211yescharacteristic
26C47×#400C2212yes—
27C47×#398C2212yes—
28C47×#387—22no—
29C47×#385—21no—
30C47×#381—21no—
31C47×#381C2211yescharacteristic
32C22×Dic94—21no—
33C22×Dic94D412yes—
34C2×#10625329—22no—
35#13094023—28no—
36#13094023—28no—
37#13094015—28no—
38#13094015—28no—
39#13094003—24no—
40#13094003—22no—
41#13094003—22no—
42#13094003—24no—
43#13094003D414yes—
44#13094003C2314yes—
45#13093983—24no—
46#13093983—24no—
47C2×C4⋊Dic47—21no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47—21no—
50C2×C4⋊Dic47D412yes—
51C2×Dic47⋊C4—22no—
52C2×Dic47⋊C4—22no—
53C2×C22⋊Dic47—22no—
54C2×C4×Dic47—22no—
55#13093957—24no—
56#13093955—24no—
57#13093947—24no—
58#13093947—24no—
59C22⋊(C47⋊C8)—22no—
60C22⋊(C47⋊C8)C2314yes—
61C23×C4×C47D411yescharacteristic
62C23×C4×C47D411yescharacteristic
63C23×C4×C47C2311yescharacteristic
64#117×C47—22no—
65#117×C47—22no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47—22no—
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47D412yes—
70C2×C4⋊C4×C47C2311yescharacteristic
71C2×C4⋊C4×C47C2311yescharacteristic
72C2×C4⋊C4×C47C2312yes—
73C2×C22⋊C4×C47—21no—
74C2×C22⋊C4×C47—21no—
75C2×C42×C47—21no—
76#95×C47—22no—
77#95×C47—22no—
78#95×C47—21no—
79#95×C47—22no—
80#95×C47C2312yes—
81C24×C47C2×D411yescharacteristic
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—24no—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—22no—
90C22×C4×C47—42no—
91C22×C4×C47—42no—
92C22×C4×C47C2×D412yes—
93C22×C4×C47C2×D412yes—
94C22×C4×C47C2×D412yes—
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47C2×D412yes—
97C22×C4×C47C2411yescharacteristic
98C22×Dic47—22no—
99C22×Dic47—22no—
100C2×Dic94—24no—
101C2×Dic94—22no—
102C2×Dic94—44no—
103C2×Dic94—82no—
104C2×Dic94C2×D414yes—
105C4⋊C4×C47—24no—
106C4⋊C4×C47—24no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—22no—
109C4⋊C4×C47—44no—
110C22⋊C4×C47—42no—
111C22⋊C4×C47—41no—
112C22⋊C4×C47—41no—
113C42×C47—42no—
114C22⋊Dic47—24no—
115C22⋊Dic47—24no—
116C4⋊Dic47—24no—
117C4⋊Dic47—24no—
118C4⋊Dic47—24no—
119C4⋊Dic47—22no—
120C4⋊Dic47C2×D414yes—
121Dic47⋊C4—28no—
122Dic47⋊C4—24no—
123Dic47⋊C4—24no—
124C4×Dic47—28no—
125#10625329—44no—
126C2×C47⋊C8—24no—
127C23×C47—22no—
128C23×C47—22no—
129C23×C47—41no—
130C23×C47#12011yescharacteristic
131C23×C47#12011yescharacteristic
132C23×C47C22×D411yescharacteristic
133C2×C4×C47—24no—
134C2×C4×C47—24no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—24no—
138C2×C4×C47—22no—
139C2×C4×C47—42no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—42no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—42no—
150C2×C4×C47—42no—
151C2×C4×C47—81no—
152C2×C4×C47#12012yes—
153C2×C4×C47#13714yes—
154C2×C4×C47C22×D412yes—
155C2×Dic47—24no—
156C2×Dic47—24no—
157C2×Dic47—44no—
158C2×Dic47—44no—
159Dic94—44no—
160Dic94—82no—
161Dic94—84no—
162C47⋊C8—44no—
163#18148—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—44no—
172C22×C47#45812yes—
173C22×C47C2×#12011yescharacteristic
174C22×C47C2×#13712yes—
175C4×C47—24no—
176C4×C47—22no—
177C4×C47—22no—
178C4×C47—42no—
179C4×C47—82no—
180C4×C47—81no—
181C4×C47—81no—
182Dic47—44no—
183Dic47—44no—
184C2×#544—471no—
185#3095—472no—
186#2484D4711yescharacteristic
187#1617—474no—
188#1611—474no—
189#1511—472no—
190#1463—471no—
191C2×C47—22no—
192C2×C47—22no—
193C2×C47—41no—
194C2×C47#300411yescharacteristic
195C2×C47#302011yescharacteristic
196C2×C47C2×#45811yescharacteristic
197#544—474no—
198#544—944no—
199#544—942no—
200C2×C4⋊Q8—942no—
201C2×#125—942no—
202C2×C22⋊Q8—472no—
203C2×C22⋊Q8—941no—
204C2×#117—941no—
205C2×#117C2×D4711yescharacteristic
206C22×C4⋊C4C2×D4711yescharacteristic
207#485—944no—
208#484—474no—
209#483—474no—
210#428—942no—
211#419—942no—
212#407—472no—
213#400C2×D4712yes—
214#398C2×D4712yes—
215#387—22no—
216#385—21no—
217#381—21no—
218#381C2×D4711yescharacteristic
219#328—478no—
220#324—474no—
221C47#1814811yescharacteristic, complemented, Sylow
222C22×Q8—472no—
223C22×Q8—941no—
224C23×C4C22⋊D4711yescharacteristic
225C23×C4C22⋊D4711yescharacteristic
226C23×C4C22×D4711yescharacteristic
227C2×M16—942no—
228C4⋊Q8—948no—
229C4⋊Q8—948no—
230#125—948no—
231#125—948no—
232C22⋊Q8—474no—
233C22⋊Q8—474no—
234C22⋊Q8—944no—
235C22⋊Q8—942no—
236C22⋊Q8—942no—
237C22⋊Q8—944no—
238#117—22no—
239#117—22no—
240#117—944no—
241#117—944no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—472no—
246C2×C4⋊C4—941no—
247C2×C4⋊C4—941no—
248C2×C4⋊C4—941no—
249C2×C4⋊C4—942no—
250C2×C4⋊C4—942no—
251C2×C4⋊C4C22⋊D4712yes—
252C2×C4⋊C4C22×D4711yescharacteristic
253C2×C4⋊C4C22×D4711yescharacteristic
254C2×C4⋊C4C22×D4712yes—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—942no—
258C2×C42—21no—
259C2×C42—942no—
260#107—944no—
261#106—944no—
262#103—944no—
263#103—944no—
264C22⋊C8—474no—
265C22⋊C8—942no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—22no—
270#95C22×D4712yes—
271C24C2×C22⋊D4711yescharacteristic
272C2×Q8—474no—
273C2×Q8—944no—
274C2×Q8—942no—
275C2×Q8—3762no—
276C2×Q8—1884no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—942no—
288C22×C4—942no—
289C22×C4D4×D4712yes—
290C22×C4D4×D4712yes—
291C22×C4C2×C22⋊D4712yes—
292C22×C4C2×C22⋊D4712yes—
293C22×C4C2×C22⋊D4711yescharacteristic
294C22×C4C23×D4711yescharacteristic
295M16—1884no—
296C2×C8—944no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—44no—
302C4⋊C4—474no—
303C4⋊C4—948no—
304C4⋊C4—944no—
305C4⋊C4—944no—
306C4⋊C4—944no—
307C4⋊C4—944no—
308C4⋊C4—942no—
309C4⋊C4—944no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—944no—
314C22⋊C4—944no—
315C42—42no—
316C42—948no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23#1309399211yescharacteristic
321C23#1309399211yescharacteristic
322C23C22×C22⋊D4711yescharacteristic
323Q8—3762no—
324Q8—3764no—
325Q8—1884no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—944no—
346C2×C4—944no—
347C2×C4—1884no—
348C2×C4—1884no—
349C2×C4#1309399212yes—
350C2×C4#1309405414yes—
351C2×C4C2×D4×D4712yes—
352C8—1884no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22#1568931212yes—
362C22C2×#1309399211yescharacteristic
363C22C2×#1309405412yes—
364C4—22no—
365C4—22no—
366C4—24no—
367C4—42no—
368C4—82no—
369C4—81no—
370C4—81no—
371C4—1884no—
372C4—1884no—
373C2—22no—
374C2—22no—
375C2—41no—
376C2#2025262111yescharacteristic
377C2#2025275111yescharacteristic
378C2C2×#1568931211yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2484C2
socleC22×C47C2×#120
prime radicalC24×C47C2×D4
hypercenter#2484D47
hypocenterC47#18148
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
3. #20242429C2semidirect
4. #20242395C2semidirect
221. C47#18148semidirect

Maximal subgroups

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

subgroupindex
C2×#156897542
#202533362
#202424292
#202423952
#202419202
#202417182
C47×#24842
#1814847