Finite Group Databasebeta

← #26243647

Subgroups — #26243647

Order 12032.

1222 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251905C212yesmaximal
2#20245876C211yesmaximal, characteristic
3#20244546C214yesmaximal, complemented
4#20244541C214yesmaximal, complemented
5#20243127C211yesmaximal, characteristic, complemented
6C2×#15688882C212yesmaximal, complemented
7C47×#3458C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2212yes—
10C22×C4×Dic47C2211yescharacteristic
11#15689131C2214yes—
12#15689127C2214yes—
13#15689122—24no—
14#15689117—24no—
15#15689028C2212yes—
16#15689024C2212yes—
17#15689000—22no—
18C4×(C2×D47)⋊C4—22no—
19#15688892C2214yes—
20#15688882C2218yes—
21C47×#544—24no—
22C47×#544C2214yes—
23C47×#543—22no—
24C47×#533—24no—
25C47×#533—22no—
26C2×#117×C47C2212yes—
27C2×#117×C47C2211yescharacteristic
28C2×#10625350—22no—
29C2×#10625350—22no—
30C2×#10625350—22no—
31C2×#10625350—24no—
32C2×#10625350Q814yescomplemented
33C22×Dic94C2311yescharacteristic
34C22×D188C2311yescharacteristic
35C22×C22⋊D47—21no—
36C23×Dic47—21no—
37C22×C4×D47—21no—
38C2×#10625329—22no—
39C2×#10625329—24no—
40C2×#10625329—24no—
41C2×#10625329C2314yes—
42C22×C47⋊C8—22no—
43C2×(C2×D47)⋊C4—24no—
44C2×C4×Dic47—24no—
45C2×C4×Dic47—21no—
46C2×C4×Dic47C2312yes—
47#13093947—28no—
48#13093944—28no—
49C47⋊#95—24no—
50C23×C4×C47C2311yescharacteristic
51C4⋊Q8×C47—24no—
52C4⋊Q8×C47—28no—
53#125×C47—24no—
54#125×C47—28no—
55C22⋊Q8×C47—216no—
56C22⋊Q8×C47—28no—
57C4×Q8×C47—28no—
58#117×C47—22no—
59#117×C47—22no—
60#117×C47—22no—
61#117×C47—22no—
62#117×C47D414yes—
63#117×C47C2314yes—
64C2×C4⋊C4×C47D414yes—
65C2×C4⋊C4×C47C2312yes—
66C2×C22⋊C4×C47—22no—
67C2×C22⋊C4×C47D412yes—
68C2×C42×C47—22no—
69C2×C42×C47D412yes—
70C24×C47C2×D411yescharacteristic
71C23×D47—21no—
72C2×Q8×C47—28no—
73C22×C4×C47—24no—
74C22×C4×C47C2×D414yes—
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D412yes—
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×Q812yes—
79C22×C4×C47#4114yes—
80C22×C4×C47#4112yes—
81C22×C4×C47C2411yescharacteristic
82C2×C22⋊D47—24no—
83C2×C22⋊D47—42no—
84C2×C22⋊D47—42no—
85C22×Dic47—21no—
86C22×Dic47—22no—
87C22×Dic47—44no—
88C22×Dic47—42no—
89#10625350—28no—
90#10625350—28no—
91#10625350—44no—
92#10625350—44no—
93C2×D188—22no—
94C2×D188—22no—
95C2×D188—21no—
96C2×D188C2×Q812yes—
97C2×C4×D47—24no—
98C2×C4×D47—41no—
99C2×C4×D47—41no—
100C2×C4×D47—42no—
101C2×Dic94—22no—
102C2×Dic94—22no—
103C2×Dic94—21no—
104C2×Dic94C2×Q812yes—
105C4⋊C4×C47—216no—
106C4⋊C4×C47—28no—
107C4⋊C4×C47—24no—
108C4⋊C4×C47—24no—
109C4⋊C4×C47C2×D418yes—
110C22⋊C4×C47—24no—
111C22⋊C4×C47—24no—
112C22⋊C4×C47—24no—
113C42×C47—24no—
114C42×C47—22no—
115C42×C47—22no—
116C42×C47—24no—
117(C2×D47)⋊C4—48no—
118C4×Dic47—24no—
119C4×Dic47—42no—
120#10625329—24no—
121#10625329—24no—
122#10625329—44no—
123C2×C47⋊C8—24no—
124C2×C47⋊C8—44no—
125C23×C47—42no—
126C23×C47#12012yes—
127C23×C47C22⋊Q812yes—
128C23×C47#12412yes—
129C23×C47C22×D411yescharacteristic
130C22×D47—22no—
131C22×D47—44no—
132C22×D47—41no—
133C22×D47—41no—
134Q8×C47—88no—
135C2×C4×C47—24no—
136C2×C4×C47—28no—
137C2×C4×C47—44no—
138C2×C4×C47—84no—
139C2×C4×C47#12011yescharacteristic
140C2×C4×C47#12011yescharacteristic
141C2×C4×C47C22⋊Q814yes—
142C2×C4×C47C22⋊Q814yes—
143C2×C4×C47C22⋊Q812yes—
144C2×C4×C47#12414yes—
145C2×C4×C47#12414yes—
146C2×C4×C47#12412yes—
147C2×C4×C47C22×D412yes—
148C2×C4×C47C22×Q811yescharacteristic
149C2×C4×C47C2×#4112yes—
150C2×C4×C47C2×#4111yescharacteristic
151C22⋊D47—48no—
152C2×Dic47—24no—
153C2×Dic47—22no—
154C2×Dic47—42no—
155C2×Dic47—48no—
156C2×Dic47—41no—
157C2×Dic47—42no—
158C2×Dic47—41no—
159D188—24no—
160D188—42no—
161C4×D47—44no—
162C4×D47—44no—
163Dic94—24no—
164Dic94—42no—
165C47⋊C8—44no—
166#23461—471nomaximal, Sylow
167C22×C47—22no—
168C22×C47—24no—
169C22×C47—42no—
170C22×C47—42no—
171C22×C47#39314yes—
172C22×C47C2×#12011yescharacteristic
173C22×C47C2×C22⋊Q811yescharacteristic
174C22×C47C2×#12411yescharacteristic
175C2×D47—44no—
176C2×D47—42no—
177C2×D47—82no—
178C2×D47—82no—
179C4×C47—44no—
180C4×C47—48no—
181C4×C47#39312yes—
182C4×C47#39312yes—
183C4×C47C2×C22⋊Q812yes—
184C4×C47C2×#12412yes—
185Dic47—44no—
186Dic47—42no—
187#3458D4711yescharacteristic, complemented
188#2896—472no—
189#2297—471no—
190#2028—474no—
191#2027—474no—
192#1742—471no—
193C2×#339—472no—
194C2×C47—21no—
195C2×C47—21no—
196C2×C47—24no—
197C2×C47C2×#39311yescharacteristic
198C2×C47#301812yes—
199D47—82no—
200C22×#41—471no—
201C22×M16—472no—
202#544—24no—
203#544C2×D4714yes—
204#543—22no—
205#533—24no—
206#533—22no—
207C2×#117C2×D4712yes—
208C2×#117C2×D4711yescharacteristic
209C22×C42—471no—
210#419—474no—
211#418—474no—
212#417—944no—
213#416—944no—
214#391—472no—
215#390—472no—
216#386—942no—
217C4×C22⋊C4—942no—
218#343—474no—
219#339—478no—
220C47#2346111yescharacteristic, complemented, Sylow
221C2×#41—474no—
222C2×#41—942no—
223C2×#41—944no—
224C2×#41—942no—
225C2×#41—942no—
226C22×Q8—471no—
227C22×D4—471no—
228C22×D4—941no—
229C23×C4—941no—
230C23×C4—941no—
231C23×C4C22×D4711yescharacteristic
232C2×M16—474no—
233C2×M16—942no—
234C2×M16—944no—
235C2×M16—944no—
236C22×C8—942no—
237C4⋊Q8—24no—
238C4⋊Q8—28no—
239#125—24no—
240#125—28no—
241C22⋊Q8—216no—
242C22⋊Q8—28no—
243C4×Q8—28no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248#117C22⋊D4714yes—
249#117C22×D4714yes—
250C2×C4⋊C4C22⋊D4714yes—
251C2×C4⋊C4C22×D4712yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—944no—
254C2×C22⋊C4C22⋊D4712yes—
255C2×C42—22no—
256C2×C42—472no—
257C2×C42—944no—
258C2×C42—941no—
259C2×C42C22⋊D4712yes—
260#103—948no—
261#102—948no—
262#95—944no—
263C24—941no—
264C24C2×C22⋊D4711yescharacteristic
265#41—948no—
266#41—948no—
267#41—1884no—
268#41—1884no—
269C2×Q8—28no—
270C2×Q8—472no—
271C2×Q8—941no—
272C2×Q8—942no—
273C2×Q8—942no—
274C2×D4—472no—
275C2×D4—944no—
276C2×D4—942no—
277C2×D4—942no—
278C2×D4—941no—
279C2×D4—1882no—
280C2×D4—1882no—
281C22×C4—24no—
282C22×C4—941no—
283C22×C4—942no—
284C22×C4—944no—
285C22×C4—1884no—
286C22×C4—1882no—
287C22×C4—1881no—
288C22×C4—1881no—
289C22×C4—1882no—
290C22×C4Q8×D4712yes—
291C22×C4#1062535414yes—
292C22×C4#1062535412yes—
293C22×C4C2×C22⋊D4714yes—
294C22×C4C2×C22⋊D4711yescharacteristic
295C22×C4C2×C22⋊D4712yes—
296C22×C4C2×C22⋊D4712yes—
297C22×C4C23×D4711yescharacteristic
298M16—944no—
299M16—944no—
300M16—1884no—
301C2×C8—944no—
302C2×C8—1884no—
303C4⋊C4—216no—
304C4⋊C4—28no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4C2×C22⋊D4718yes—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—1888no—
312C42—22no—
313C42—24no—
314C42—22no—
315C42—24no—
316C42—944no—
317C42—1882no—
318C23—42no—
319C23—942no—
320C23—1884no—
321C23—1881no—
322C23—1881no—
323C23#1309399312yes—
324C23#1309400012yes—
325C23#1309401312yes—
326C23C22×C22⋊D4711yescharacteristic
327Q8—88no—
328Q8—944no—
329Q8—1882no—
330D4—944no—
331D4—1888no—
332D4—1882no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—44no—
336C2×C4—84no—
337C2×C4—944no—
338C2×C4—942no—
339C2×C4—1882no—
340C2×C4—1888no—
341C2×C4—1881no—
342C2×C4—1882no—
343C2×C4—1881no—
344C2×C4—1884no—
345C2×C4—1884no—
346C2×C4#1309399311yescharacteristic
347C2×C4#1309399311yescharacteristic
348C2×C4#1309400012yes—
349C2×C4#1309400014yes—
350C2×C4#1309400014yes—
351C2×C4#1309401312yes—
352C2×C4#1309401314yes—
353C2×C4#1309401314yes—
354C2×C4C22×C22⋊D4712yes—
355C2×C4C2×Q8×D4711yescharacteristic
356C2×C4C2×#1062535412yes—
357C2×C4C2×#1062535411yescharacteristic
358C8—1884no—
359C22—22no—
360C22—24no—
361C22—42no—
362C22—42no—
363C22—3762no—
364C22—3762no—
365C22—1884no—
366C22—1882no—
367C22#1568903414yes—
368C22C2×#1309399311yescharacteristic
369C22C2×#1309400011yescharacteristic
370C22C2×#1309401311yescharacteristic
371C4—44no—
372C4—48no—
373C4—1884no—
374C4—1882no—
375C4#1568903412yes—
376C4#1568903412yes—
377C4C2×#1309400012yes—
378C4C2×#1309401312yes—
379C2—21no—
380C2—21no—
381C2—24no—
382C2—3762no—
383C2C2×#1568903411yescharacteristic
384C2#2025273712yes—
subgroupG/H
centerC2×C4#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#3458C2
socleC22×C47C2×#120
prime radicalC22×#10625350C22
hypercenter#3458D47
hypocenterC47#23461
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
3. #20244546C2semidirect
4. #20244541C2semidirect
5. #20243127C2semidirect
6. C2×#15688882C2semidirect
7. C47×#3458C2semidirect
32. C2×#10625350Q8semidirect
187. #3458D47semidirect
220. C47#23461semidirect

Maximal subgroups

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

subgroupindex
#202519052
#202458762
#202445462
#202445412
#202431272
C2×#156888822
C47×#34582
#2346147