Finite Group Databasebeta

← #26104736

Subgroups — #26104736

Order 12032.

568 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251727C211yesmaximal, characteristic
2C2×#15688878C211yesmaximal, characteristic
3#20241213C212yesmaximal, complemented
4#20241210C212yesmaximal, complemented
5C47×#1728C211yesmaximal, characteristic
6C23×C47⋊C8—21no—
7C2×#13094003C411yescharacteristic
8#15689029C411yescharacteristic
9#15689029C2211yescharacteristic
10#15689021C411yescharacteristic
11#15689021C2211yescharacteristic
12#15689007—21no—
13#15689007—21no—
14#15688996—21no—
15#15688986—21no—
16#15688983C411yescharacteristic
17#15688878—21no—
18#15688878C2212yes—
19C22×M16×C47—21no—
20C22×C42×C47C2211yescharacteristic
21C47×#336—21no—
22C47×#336C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C22×C47⋊C8—24no—
25C22×C47⋊C8—22no—
26C22×C47⋊C8—42no—
27C22×C47⋊C8—42no—
28#13094003—42no—
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×C4⋊Dic47C2×C411yescharacteristic
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C2×M16×C47—24no—
48C2×M16×C47—42no—
49C2×M16×C47—42no—
50C22×C8×C47—22no—
51C2×C42×C47—42no—
52C2×C42×C47C2×C411yescharacteristic
53C2×C42×C47D411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C2×C42×C47C2311yescharacteristic
56C24×C47C22⋊C411yescharacteristic
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—41no—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47SD1612yes—
70C22×C4×C47Q1612yes—
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×Dic47—21no—
75C22×Dic47—21no—
76C22×Dic47—21no—
77C22×Dic47—21no—
78C2×Dic94—22no—
79C2×Dic94—41no—
80C2×Dic94—41no—
81M16×C47—48no—
82C2×C8×C47—24no—
83C2×C8×C47—42no—
84C2×C8×C47—42no—
85C42×C47—41no—
86C42×C47—41no—
87C42×C47—41no—
88C42×C47—41no—
89C22⋊Dic47—42no—
90C22⋊Dic47—42no—
91C4⋊Dic47—22no—
92C4⋊Dic47—41no—
93C4⋊Dic47—41no—
94Dic47⋊C4—42no—
95Dic47⋊C4—42no—
96Dic47⋊C4—41no—
97Dic47⋊C4—41no—
98C2×C47⋊C8—24no—
99C2×C47⋊C8—48no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—42no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47#9912yes—
107C23×C47#10312yes—
108C23×C47#10412yes—
109C23×C47C2×C22⋊C411yescharacteristic
110C2×C4×C47—24no—
111C2×C4×C47—24no—
112C2×C4×C47—24no—
113C2×C4×C47—24no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—44no—
128C2×C4×C47#10312yes—
129C2×C4×C47#10412yes—
130C2×C4×C47#13112yes—
131C2×C4×C47C2×SD1611yescharacteristic
132C2×C4×C47C2×Q1611yescharacteristic
133C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C47#13711yescharacteristic
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147Dic94—82no—
148C8×C47—44no—
149C47⋊C8—44no—
150#7681—471nomaximal, Sylow
151C22×C47—22no—
152C22×C47—22no—
153C22×C47—22no—
154C22×C47—22no—
155C22×C47—22no—
156C22×C47—22no—
157C22×C47#32814yes—
158C22×C47#40811yescharacteristic
159C22×C47C2×#9911yescharacteristic
160C22×C47#41311yescharacteristic
161C22×C47C2×#10311yescharacteristic
162C22×C47#41711yescharacteristic
163C22×C47C2×#10411yescharacteristic
164C22×C47C2×#10411yescharacteristic
165C4×C47—22no—
166C4×C47—21no—
167C4×C47—22no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—22no—
171C4×C47—21no—
172C4×C47—41no—
173Dic47—81no—
174Dic47—81no—
175Dic47—81no—
176Dic47—81no—
177#2855—471no—
178#1728D4711yescharacteristic
179C2×#336—471no—
180#1272—472no—
181#1271—472no—
182C2×C47—24no—
183C2×C47#146811yescharacteristic
184C2×C47#148311yescharacteristic
185C2×C47#149311yescharacteristic
186C2×C47C2×#32811yescharacteristic
187C2×C47#150411yescharacteristic
188C2×C47#152411yescharacteristic
189C2×C47#155011yescharacteristic
190C22×M16—21no—
191C23×C8—941no—
192C2×C22⋊Q8—471no—
193C22×C42C2×D4711yescharacteristic
194#391—471no—
195#391—471no—
196#389—471no—
197#389—471no—
198#387—941no—
199#387—941no—
200#385—941no—
201#381—941no—
202#380—471no—
203#336—21no—
204#336—472no—
205#336—941no—
206#336C2×D4712yes—
207C47#768111yescharacteristic, complemented, Sylow
208C22×Q8—471no—
209C23×C4C4×D4711yescharacteristic
210C23×C4D18811yescharacteristic
211C23×C4C22⋊D4711yescharacteristic
212C2×M16—24no—
213C2×M16—42no—
214C2×M16—42no—
215C22×C8—22no—
216C22×C8—944no—
217C22×C8—942no—
218C22×C8—1882no—
219C22×C8—1882no—
220C22⋊Q8—1882no—
221C2×C4⋊C4—471no—
222C2×C4⋊C4—471no—
223C2×C4⋊C4—471no—
224C2×C4⋊C4—941no—
225C2×C4⋊C4—941no—
226C2×C4⋊C4—941no—
227C2×C22⋊C4—941no—
228C2×C22⋊C4—941no—
229C2×C42—42no—
230C2×C42C4×D4711yescharacteristic
231C2×C42D18811yescharacteristic
232C2×C42C22⋊D4711yescharacteristic
233C2×C42C22×D4711yescharacteristic
234#95—941no—
235#95—941no—
236#95—941no—
237#95—941no—
238#95—941no—
239#95—941no—
240#95—941no—
241C24(C2×D47)⋊C411yescharacteristic
242C2×Q8—942no—
243C2×Q8—1881no—
244C2×Q8—1881no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—941no—
255C22×C4—941no—
256C22×C4—941no—
257C22×C4—941no—
258C22×C4(C2×D47)⋊C411yescharacteristic
259C22×C4(C2×D47)⋊C411yescharacteristic
260C22×C4(C2×D47)⋊C411yescharacteristic
261C22×C4#1062533512yes—
262C22×C4#1062533712yes—
263C22×C4C2×C4×D4711yescharacteristic
264C22×C4C2×D18811yescharacteristic
265C22×C4C2×C22⋊D4711yescharacteristic
266M16—48no—
267C2×C8—24no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—944no—
271C2×C8—1888no—
272C2×C8—1882no—
273C2×C8—1882no—
274C4⋊C4—942no—
275C4⋊C4—1881no—
276C4⋊C4—1881no—
277C4⋊C4—1882no—
278C4⋊C4—1882no—
279C4⋊C4—1881no—
280C4⋊C4—1881no—
281C22⋊C4—1882no—
282C22⋊C4—1882no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42—41no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23#1309393812yes—
292C23#1309394712yes—
293C23#1309395212yes—
294C23C2×(C2×D47)⋊C411yescharacteristic
295Q8—3762no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—44no—
314C2×C4—1881no—
315C2×C4—1881no—
316C2×C4—1881no—
317C2×C4—1881no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4—1881no—
321C2×C4—1881no—
322C2×C4—1881no—
323C2×C4—1881no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4#1309394712yes—
327C2×C4#1309395012yes—
328C2×C4#1309403212yes—
329C2×C4C2×#1062533511yescharacteristic
330C2×C4C2×#1062533711yescharacteristic
331C2×C4#1309405211yescharacteristic
332C2×C4#1309405711yescharacteristic
333C8—44no—
334C8—1884no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22#1568885914yes—
342C22#1568909111yescharacteristic
343C22C2×#1309393811yescharacteristic
344C22#1568910511yescharacteristic
345C22C2×#1309394711yescharacteristic
346C22#1568912511yescharacteristic
347C22C2×#1309395011yescharacteristic
348C22C2×#1309395211yescharacteristic
349C4—22no—
350C4—21no—
351C4—22no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—21no—
356C4—41no—
357C4—3761no—
358C4—3761no—
359C4—3761no—
360C4—3761no—
361C2—24no—
362C2#2024173711yescharacteristic
363C2#2024180211yescharacteristic
364C2#2024184211yescharacteristic
365C2C2×#1568885911yescharacteristic
366C2#2024189311yescharacteristic
367C2#2024197711yescharacteristic
368C2#2024207911yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1728C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1728D47
hypocenterC47#7681
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
3. #20241213C2semidirect
4. #20241210C2semidirect
207. C47#7681semidirect

Maximal subgroups

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

subgroupindex
#202517272
C2×#156888782
#202412132
#202412102
C47×#17282
#768147