Finite Group Databasebeta

← #26104735

Subgroups — #26104735

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#20243082C211yesmaximal, characteristic
2C2×#15688878C211yesmaximal, characteristic
3#20241214C212yesmaximal, complemented
4#20241211C212yesmaximal, complemented
5C47×#2855C211yesmaximal, characteristic
6C22×#10625329—21no—
7C23×C47⋊C8—21no—
8#15688878—21no—
9#15688878—21no—
10#15688878C2212yes—
11#15688878C2212yes—
12C2×C22⋊Q8×C47C411yescharacteristic
13C22×C42×C47C2211yescharacteristic
14C47×#391C411yescharacteristic
15C47×#391C2211yescharacteristic
16C47×#389C411yescharacteristic
17C47×#389C2211yescharacteristic
18C47×#387—21no—
19C47×#387—21no—
20C47×#385—21no—
21C47×#381—21no—
22C47×#380C411yescharacteristic
23C2×#10625329—24no—
24C2×#10625329—42no—
25C2×#10625329—42no—
26C22×C47⋊C8—24no—
27C22×C47⋊C8—22no—
28C22×C47⋊C8—22no—
29C22×C47⋊C8—42no—
30C22×C47⋊C8—42no—
31C22×Q8×C47C2×C411yescharacteristic
32C23×C4×C47C2×C411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C22⋊Q8×C47—42no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—21no—
44C2×C42×C47—42no—
45C2×C42×C47C2×C411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C2×C42×C47C2311yescharacteristic
49#95×C47—21no—
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56C24×C47C22⋊C411yescharacteristic
57C2×Q8×C47—22no—
58C2×Q8×C47—41no—
59C2×Q8×C47—41no—
60C22×C4×C47—22no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—41no—
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47SD1612yes—
77C22×C4×C47Q1612yes—
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C4⋊C4×C47—22no—
82C4⋊C4×C47—42no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—42no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C22⋊C4×C47—42no—
89C22⋊C4×C47—42no—
90C42×C47—41no—
91C42×C47—41no—
92C42×C47—41no—
93C42×C47—41no—
94#10625329—48no—
95C2×C47⋊C8—24no—
96C2×C47⋊C8—24no—
97C2×C47⋊C8—48no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
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
110Q8×C47—82no—
111C2×C4×C47—24no—
112C2×C4×C47—24no—
113C2×C4×C47—24no—
114C2×C4×C47—24no—
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—21no—
128C2×C4×C47—44no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47#10312yes—
142C2×C4×C47#10412yes—
143C2×C4×C47#13112yes—
144C2×C4×C47C2×SD1611yescharacteristic
145C2×C4×C47C2×Q1611yescharacteristic
146C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C47#13711yescharacteristic
148C47⋊C8—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—
173C4×C47—81no—
174C4×C47—81no—
175C4×C47—81no—
176C4×C47—81no—
177#2855D4711yescharacteristic
178#1728—471no—
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—941no—
191C23×C8—941no—
192C2×C22⋊Q8Dic4711yescharacteristic
193C22×C42C2×D4711yescharacteristic
194#391Dic4711yescharacteristic
195#391C2×D4711yescharacteristic
196#389Dic4711yescharacteristic
197#389C2×D4711yescharacteristic
198#387—21no—
199#387—21no—
200#385—21no—
201#381—21no—
202#380Dic4711yescharacteristic
203#336—472no—
204#336—472no—
205#336—941no—
206#336—941no—
207C47#768111yescharacteristic, complemented, Sylow
208C22×Q8C2×Dic4711yescharacteristic
209C23×C4C2×Dic4711yescharacteristic
210C23×C4C22⋊D4711yescharacteristic
211C23×C4C22⋊D4711yescharacteristic
212C2×M16—944no—
213C2×M16—1882no—
214C2×M16—1882no—
215C22×C8—944no—
216C22×C8—942no—
217C22×C8—942no—
218C22×C8—1882no—
219C22×C8—1882no—
220C22⋊Q8—42no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C2×Dic4711yescharacteristic
225C2×C4⋊C4C2×Dic4711yescharacteristic
226C2×C4⋊C4C2×Dic4711yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C42—42no—
230C2×C42C2×Dic4711yescharacteristic
231C2×C42C22⋊D4711yescharacteristic
232C2×C42C22⋊D4711yescharacteristic
233C2×C42C22×D4711yescharacteristic
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241C24C22⋊Dic4711yescharacteristic
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—41no—
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—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4#1062533612yes—
259C22×C4#1062533712yes—
260C22×C4C22⋊Dic4711yescharacteristic
261C22×C4C22⋊Dic4711yescharacteristic
262C22×C4C22⋊Dic4711yescharacteristic
263C22×C4C22×Dic4711yescharacteristic
264C22×C4C2×C22⋊D4711yescharacteristic
265C22×C4C2×C22⋊D4711yescharacteristic
266M16—1888no—
267C2×C8—944no—
268C2×C8—944no—
269C2×C8—1888no—
270C2×C8—1882no—
271C2×C8—1882no—
272C2×C8—1882no—
273C2×C8—1882no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42—41no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23#1309393912yes—
292C23#1309394812yes—
293C23#1309395112yes—
294C23C2×C22⋊Dic4711yescharacteristic
295Q8—82no—
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—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4#1309394812yes—
327C2×C4#1309395112yes—
328C2×C4#1309403112yes—
329C2×C4C2×#1062533611yescharacteristic
330C2×C4C2×#1062533711yescharacteristic
331C2×C4#1309404911yescharacteristic
332C2×C4#1309405411yescharacteristic
333C8—1884no—
334C8—1884no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22#1568886014yes—
342C22#1568909011yescharacteristic
343C22C2×#1309393911yescharacteristic
344C22#1568910711yescharacteristic
345C22C2×#1309394811yescharacteristic
346C22#1568912411yescharacteristic
347C22C2×#1309395111yescharacteristic
348C22C2×#1309395111yescharacteristic
349C4—22no—
350C4—21no—
351C4—22no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—21no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C2—24no—
362C2#2024173611yescharacteristic
363C2#2024180111yescharacteristic
364C2#2024184311yescharacteristic
365C2C2×#1568886011yescharacteristic
366C2#2024189211yescharacteristic
367C2#2024197611yescharacteristic
368C2#2024207811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2855C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2855D47
hypocenterC47#7681
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
3. #20241214C2semidirect
4. #20241211C2semidirect
207. C47#7681semidirect

Maximal subgroups

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

subgroupindex
#202430822
C2×#156888782
#202412142
#202412112
C47×#28552
#768147