Finite Group Databasebeta

← #26244393

Subgroups — #26244393

Order 12032.

1074 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251977C212yesmaximal
2#20246012C211yesmaximal, characteristic
3#20244735C214yesmaximal, complemented
4#20244730C214yesmaximal, complemented
5C2×#15688892C211yesmaximal, characteristic
6#20243097C212yesmaximal
7C2×C47×#544C211yesmaximal, characteristic
8C22×#10625329C2212yes—
9C2×#13093983C2211yescharacteristic
10C2×#13093983C2211yescharacteristic
11#15689166C2214yes—
12#15689166C2214yes—
13#15689163—22no—
14#15689163—22no—
15#15689159—24no—
16#15689019—22no—
17#15688996—22no—
18#15688992C2212yes—
19#15688988C2212yes—
20#15688892C2214yes—
21#15688882C2218yes—
22C47×#544—24no—
23C47×#544—22no—
24C47×#544C2214yes—
25C2×C4⋊Q8×C47—22no—
26C2×#125×C47—22no—
27C2×C22⋊Q8×C47—22no—
28C2×#117×C47C2212yes—
29C22×C4⋊C4×C47C2211yescharacteristic
30C2×#10625329—22no—
31C2×#10625329—24no—
32C2×#10625329Q818yescomplemented
33C2×#10625329C2314yes—
34C22×C47⋊C8—22no—
35#13093983—22no—
36#13093983—22no—
37#13093983—22no—
38#13093983Q814yescomplemented
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47C2311yescharacteristic
41C2×C4⋊Dic47C2311yescharacteristic
42C2×Dic47⋊C4—24no—
43C2×C22⋊Dic47—21no—
44C2×C22⋊Dic47—21no—
45C2×C4×Dic47—21no—
46C2×C4×Dic47C2312yes—
47#13093957—24no—
48#13093957—24no—
49#13093955—24no—
50#13093955—24no—
51C47⋊#95—24no—
52C22×Q8×C47—22no—
53C23×C4×C47D412yes—
54C23×C4×C47C2311yescharacteristic
55C4⋊Q8×C47—28no—
56C4⋊Q8×C47—28no—
57#125×C47—28no—
58#125×C47—28no—
59C22⋊Q8×C47—28no—
60C22⋊Q8×C47—28no—
61#117×C47—22no—
62#117×C47—22no—
63#117×C47D414yes—
64#117×C47C2314yes—
65C2×C4⋊C4×C47—22no—
66C2×C4⋊C4×C47—24no—
67C2×C4⋊C4×C47—22no—
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47D412yes—
70C2×C4⋊C4×C47D412yes—
71C2×C4⋊C4×C47D412yes—
72C2×C4⋊C4×C47C2312yes—
73C2×C22⋊C4×C47—22no—
74C2×C42×C47—22no—
75C24×C47C2×D411yescharacteristic
76C2×Q8×C47—28no—
77C2×Q8×C47—84no—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47—22no—
81C22×C4×C47—22no—
82C22×C4×C47—82no—
83C22×C4×C47C2×D414yes—
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D412yes—
86C22×C4×C47C2×D412yes—
87C22×C4×C47C2×Q814yes—
88C22×C4×C47C2×Q812yes—
89C22×C4×C47#4112yes—
90C22×C4×C47C2411yescharacteristic
91C22×Dic47—22no—
92C22×Dic47—21no—
93C22×Dic47—21no—
94C4⋊C4×C47—28no—
95C4⋊C4×C47—216no—
96C4⋊C4×C47—24no—
97C4⋊C4×C47—22no—
98C4⋊C4×C47—22no—
99C4⋊C4×C47—24no—
100C4⋊C4×C47C2×D414yes—
101C4⋊C4×C47C2×D414yes—
102C22⋊C4×C47—24no—
103C22⋊C4×C47—24no—
104C42×C47—24no—
105C42×C47—24no—
106C22⋊Dic47—24no—
107C22⋊Dic47—41no—
108C22⋊Dic47—41no—
109C4⋊Dic47—24no—
110C4⋊Dic47—22no—
111C4⋊Dic47—22no—
112Dic47⋊C4—84no—
113C4×Dic47—22no—
114C4×Dic47—42no—
115C4×Dic47C2×Q814yescomplemented
116#10625329—28no—
117#10625329—24no—
118#10625329C2×Q818yescomplemented
119C2×C47⋊C8—28no—
120C2×C47⋊C8—24no—
121C23×C47—41no—
122C23×C47—41no—
123C23×C47#12011yescharacteristic
124C23×C47#12011yescharacteristic
125C23×C47C22⋊Q812yes—
126C23×C47C4⋊Q812yes—
127C23×C47C22×D411yescharacteristic
128Q8×C47—88no—
129C2×C4×C47—24no—
130C2×C4×C47—24no—
131C2×C4×C47—24no—
132C2×C4×C47—44no—
133C2×C4×C47—44no—
134C2×C4×C47—44no—
135C2×C4×C47—44no—
136C2×C4×C47—82no—
137C2×C4×C47#12012yes—
138C2×C4×C47C22⋊Q814yes—
139C2×C4×C47C22⋊Q814yes—
140C2×C4×C47C22⋊Q812yes—
141C2×C4×C47C4⋊Q814yes—
142C2×C4×C47C4⋊Q814yes—
143C2×C4×C47C4⋊Q812yes—
144C2×C4×C47C22×D412yes—
145C2×C4×C47C22×Q812yes—
146C2×C4×C47C22×Q811yescharacteristic
147C2×C4×C47C2×#4111yescharacteristic
148C2×Dic47—24no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C2×Dic47—42no—
152C2×Dic47—42no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C47⋊C8—28no—
156#23561—471nomaximal, Sylow
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—42no—
164C22×C47—42no—
165C22×C47#39512yes—
166C22×C47#39512yes—
167C22×C47C2×#12011yescharacteristic
168C22×C47C2×C22⋊Q811yescharacteristic
169C22×C47C2×C4⋊Q811yescharacteristic
170C4×C47—44no—
171C4×C47—44no—
172C4×C47—44no—
173C4×C47#39514yes—
174C4×C47C2×C22⋊Q812yes—
175C4×C47C2×C4⋊Q812yes—
176Dic47—44no—
177Dic47—81no—
178Dic47—81no—
179C2×#544D4711yescharacteristic
180#2911—472no—
181#2317—471no—
182#2061—474no—
183#2060—474no—
184C2×#343—471no—
185#1733—472no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—22no—
189C2×C47—22no—
190C2×C47C2×#39511yescharacteristic
191C2×C47#301911yescharacteristic
192C2×C47#302011yescharacteristic
193C22×M16—472no—
194#544—24no—
195#544—22no—
196#544C2×D4714yes—
197C2×C4⋊Q8—22no—
198C2×#125—22no—
199C2×C22⋊Q8—22no—
200C2×#117—471no—
201C2×#117—471no—
202C2×#117C2×D4712yes—
203C22×C4⋊C4C2×D4711yescharacteristic
204#428—474no—
205#428—474no—
206#428—942no—
207#428—942no—
208#427—944no—
209#389—942no—
210#385—942no—
211#384—472no—
212#382—472no—
213#343—474no—
214#339—478no—
215C47#2356111yescharacteristic, complemented, Sylow
216C22×Q8—22no—
217C23×C4C22⋊D4712yes—
218C23×C4C22×D4711yescharacteristic
219C2×M16—478no—
220C2×M16—474no—
221C2×M16—944no—
222C2×M16—942no—
223C22×C8—942no—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226#125—28no—
227#125—28no—
228C22⋊Q8—28no—
229C22⋊Q8—28no—
230#117—22no—
231#117—22no—
232#117—474no—
233#117—942no—
234#117—942no—
235#117—942no—
236#117C22⋊D4714yes—
237#117C22×D4714yes—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—471no—
243C2×C4⋊C4—471no—
244C2×C4⋊C4—941no—
245C2×C4⋊C4—944no—
246C2×C4⋊C4C22⋊D4712yes—
247C2×C4⋊C4C22⋊D4712yes—
248C2×C4⋊C4C22⋊D4712yes—
249C2×C4⋊C4C22×D4712yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—941no—
252C2×C22⋊C4—941no—
253C2×C42—22no—
254C2×C42—472no—
255C2×C42—941no—
256#107—944no—
257#107—944no—
258#106—944no—
259#106—944no—
260#95—944no—
261C24C2×C22⋊D4711yescharacteristic
262C2×Q8—28no—
263C2×Q8—84no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—82no—
269C22×C4—942no—
270C22×C4—941no—
271C22×C4—941no—
272C22×C4Q8×D4714yes—
273C22×C4Q8×D4712yes—
274C22×C4#1062535412yes—
275C22×C4C2×C22⋊D4714yes—
276C22×C4C2×C22⋊D4711yescharacteristic
277C22×C4C2×C22⋊D4712yes—
278C22×C4C2×C22⋊D4712yes—
279C22×C4C23×D4711yescharacteristic
280M16—478no—
281M16—948no—
282M16—944no—
283C2×C8—948no—
284C2×C8—944no—
285C4⋊C4—28no—
286C4⋊C4—216no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—944no—
292C4⋊C4—942no—
293C4⋊C4—942no—
294C4⋊C4—3764no—
295C4⋊C4C2×C22⋊D4714yes—
296C4⋊C4C2×C22⋊D4714yes—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—944no—
300C22⋊C4—1881no—
301C22⋊C4—1881no—
302C42—24no—
303C42—24no—
304C42—474no—
305C42—942no—
306C42—1882no—
307C23—41no—
308C23—41no—
309C23#1309399311yescharacteristic
310C23#1309399311yescharacteristic
311C23#1309400012yes—
312C23#1309402212yes—
313C23C22×C22⋊D4711yescharacteristic
314Q8—88no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—82no—
323C2×C4—944no—
324C2×C4—1882no—
325C2×C4—1882no—
326C2×C4—1881no—
327C2×C4—1881no—
328C2×C4—1882no—
329C2×C4—1882no—
330C2×C4#1309399312yes—
331C2×C4#1309400012yes—
332C2×C4#1309400014yes—
333C2×C4#1309400014yes—
334C2×C4#1309402212yes—
335C2×C4#1309402214yes—
336C2×C4#1309402214yes—
337C2×C4C22×C22⋊D4712yes—
338C2×C4C2×Q8×D4712yes—
339C2×C4C2×Q8×D4711yescharacteristic
340C2×C4C2×#1062535411yescharacteristic
341C8—948no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22#1568904212yes—
351C22#1568904212yes—
352C22C2×#1309399311yescharacteristic
353C22C2×#1309400011yescharacteristic
354C22C2×#1309402211yescharacteristic
355C4—44no—
356C4—44no—
357C4—44no—
358C4—3761no—
359C4—3761no—
360C4—1884no—
361C4#1568904214yes—
362C4C2×#1309400012yes—
363C4C2×#1309402212yes—
364C2—21no—
365C2—22no—
366C2—22no—
367C2—21no—
368C2C2×#1568904211yescharacteristic
369C2#2025274511yescharacteristic
370C2#2025275211yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#544C2
socleC22×C47C2×#120
prime radicalC24×C47C2×D4
hypercenterC2×#544D47
hypocenterC47#23561
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #20244735C2semidirect
4. #20244730C2semidirect
32. C2×#10625329Q8semidirect
38. #13093983Q8semidirect
115. C4×Dic47C2×Q8semidirect
118. #10625329C2×Q8semidirect
215. C47#23561semidirect

Maximal subgroups

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

subgroupindex
#202519772
#202460122
#202447352
#202447302
C2×#156888922
#202430972
C2×C47×#5442
#2356147