Finite Group Databasebeta

← #26201581

Subgroups — #26201581

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254338C211yesmaximal, characteristic
2#20253193C211yesmaximal, characteristic
3C2×#15689408C211yesmaximal, characteristic
4#20242846C214yesmaximal, complemented
5#20242758C214yesmaximal, complemented
6#20242163C211yesmaximal, characteristic
7#20242072C211yesmaximal, characteristic
8#20242043C211yesmaximal, characteristic
9C47×#2982C211yesmaximal, characteristic
10#15689680C2211yescharacteristic
11C2×#13094024C2211yescharacteristic
12#15689503C2212yes—
13#15689474C2212yes—
14#15689430—22no—
15#15689418C2212yes—
16#15689408—22no—
17#15689408C2212yes—
18#15689408C2214yes—
19#15689334—22no—
20#15689281—22no—
21#15689163—21no—
22C2×#13093955—21no—
23#15689147C2211yescharacteristic
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25#15689132—21no—
26C2×#13093948—21no—
27#15688876C2214yes—
28#15688871C2212yes—
29#15688870C2212yes—
30#15688870C2214yes—
31C2×C4×Q8×C47C2211yescharacteristic
32C8×Q8×C47—22no—
33C8×Q8×C47C2214yes—
34C47×#431—22no—
35C47×#431C2211yescharacteristic
36C47×#431C2211yescharacteristic
37C22×Dic94—21no—
38C2×Dic188—24no—
39C2×#10625329—21no—
40C22×C47⋊C8—21no—
41#13094024—21no—
42#13094024—21no—
43#13094024—21no—
44#13094024C2311yescharacteristic
45#13094024C2312yes—
46#13094024C2311yescharacteristic
47#13094016—21no—
48#13094003—21no—
49#13094003—21no—
50C4×Dic94—21no—
51C4×Dic94—21no—
52#13093983—21no—
53C2×C4⋊Dic47—21no—
54C2×C4⋊Dic47—21no—
55#13093957—22no—
56#13093955—24no—
57#13093955—22no—
58C4⋊(C47⋊C8)—21no—
59C4⋊(C47⋊C8)—21no—
60C4⋊(C47⋊C8)C2312yes—
61C4⋊(C47⋊C8)C2312yes—
62#13093949—24no—
63#13093948—24no—
64#13093948—22no—
65#13093948—22no—
66C22×Q8×C47D411yescharacteristic
67C4×Q8×C47—21no—
68C4×Q8×C47—21no—
69C4×Q8×C47D412yes—
70C4×Q8×C47C2312yes—
71C2×C4⋊C4×C47—21no—
72C2×C4⋊C4×C47D411yescharacteristic
73C2×C42×C47—21no—
74C2×C42×C47C2311yescharacteristic
75C4⋊C8×C47—24no—
76C4⋊C8×C47D412yes—
77C4⋊C8×C47C2312yes—
78C22⋊C8×C47—21no—
79C22⋊C8×C47D412yes—
80C4×C8×C47—24no—
81C4×C8×C47—21no—
82C4×C8×C47C2312yes—
83C2×Q8×C47—21no—
84C2×Q8×C47—22no—
85C2×Q8×C47Q1614yescomplemented
86C2×Q8×C47C2×D412yes—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47#4111yescharacteristic
92C22×Dic47—41no—
93C2×Dic94—22no—
94C2×Dic94—21no—
95C2×Dic94—21no—
96C2×Dic94—41no—
97C2×Dic94—82no—
98C2×C8×C47—22no—
99C2×C8×C47C2×D414yes—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47C2×D412yes—
104C42×C47—22no—
105C42×C47—22no—
106C42×C47C2×D411yescharacteristic
107C42×C47C2×D411yescharacteristic
108C42×C47#4111yescharacteristic
109C42×C47C2411yescharacteristic
110C22⋊Dic47—41no—
111C4⋊Dic47—22no—
112C4⋊Dic47—21no—
113C4⋊Dic47—21no—
114C4⋊Dic47—21no—
115C4⋊Dic47—22no—
116C4⋊Dic47—21no—
117C4⋊Dic47—41no—
118C4⋊Dic47—41no—
119Dic47⋊C4—41no—
120Dic47⋊C4—41no—
121C4×Dic47—41no—
122#10625329—42no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—42no—
126Dic188—84no—
127C23×C47#12111yescharacteristic
128Q8×C47—22no—
129Q8×C47—24no—
130Q8×C47C2×Q1614yes—
131C2×C4×C47—22no—
132C2×C4×C47—24no—
133C2×C4×C47—22no—
134C2×C4×C47—22no—
135C2×C4×C47—41no—
136C2×C4×C47—81no—
137C2×C4×C47#12111yescharacteristic
138C2×C4×C47#12111yescharacteristic
139C2×C4×C47#12111yescharacteristic
140C2×C4×C47C2×Q1612yes—
141C2×C4×C47#13512yes—
142C2×C4×C47C22×D411yescharacteristic
143C2×C4×C47C22×D411yescharacteristic
144C2×C4×C47C2×#4111yescharacteristic
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—42no—
148C2×Dic47—82no—
149Dic94—42no—
150Dic94—44no—
151Dic94—82no—
152Dic94—81no—
153C8×C47—24no—
154C8×C47—42no—
155C47⋊C8—42no—
156C47⋊C8—42no—
157#19399—471nomaximal, Sylow
158C22×C47—21no—
159C22×C47—21no—
160C22×C47#46712yes—
161C22×C47C2×#12111yescharacteristic
162C4×C47—21no—
163C4×C47—24no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—42no—
167C4×C47#46712yes—
168C4×C47C22×Q1611yescharacteristic
169C4×C47C2×#13511yescharacteristic
170C4×C47#57712yes—
171Dic47—82no—
172Dic47—81no—
173Dic47—81no—
174#3209—471no—
175#3078—471no—
176C2×#475—471no—
177#2982D4711yescharacteristic
178#1684—474no—
179#1670—474no—
180#1571—471no—
181#1549—471no—
182#1542—471no—
183C2×C47—22no—
184C2×C47#304311yescharacteristic
185C2×C47C2×#46711yescharacteristic
186C2×C47#306911yescharacteristic
187#533—471no—
188C2×C4⋊Q8—471no—
189C2×C4×Q8C2×D4711yescharacteristic
190#494—472no—
191#487—472no—
192#479—942no—
193#477—472no—
194#475—474no—
195#475—472no—
196#475—942no—
197#462—942no—
198#451—942no—
199C8×Q8—22no—
200C8×Q8C2×D4714yes—
201#431—22no—
202#431C2×D4711yescharacteristic
203#431C2×D4711yescharacteristic
204#428—941no—
205C2×#106—941no—
206#423—471no—
207C2×C4⋊C8—471no—
208#419—941no—
209C2×#103—941no—
210#335—474no—
211#333—472no—
212#332—472no—
213#332—474no—
214C47#1939911yescharacteristic, complemented, Sylow
215C22×Q8—941no—
216C22×Q8D18811yescharacteristic
217C2×Q16—944no—
218C2×M16—941no—
219C22×C8—941no—
220C4⋊Q8—472no—
221C4⋊Q8—471no—
222C4⋊Q8—471no—
223C4⋊Q8—941no—
224C4⋊Q8—941no—
225C4⋊Q8—941no—
226#125—941no—
227C22⋊Q8—941no—
228C22⋊Q8—941no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—941no—
232C4×Q8—941no—
233C4×Q8D18812yes—
234C4×Q8C22×D4712yes—
235#117—941no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—941no—
238C2×C4⋊C4—941no—
239C2×C4⋊C4D18811yescharacteristic
240C2×C42—21no—
241C2×C42C22×D4711yescharacteristic
242#107—942no—
243#106—944no—
244#106—942no—
245C4⋊C8—24no—
246C4⋊C8—472no—
247C4⋊C8—472no—
248C4⋊C8—941no—
249C4⋊C8—941no—
250C4⋊C8C22⋊D4712yes—
251C4⋊C8C22×D4712yes—
252#103—944no—
253#103—942no—
254#103—942no—
255#103—944no—
256C22⋊C8—21no—
257C22⋊C8C22⋊D4712yes—
258C4×C8—24no—
259C4×C8—21no—
260C4×C8C22×D4712yes—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—942no—
264C2×Q8—941no—
265C2×Q8—941no—
266C2×Q8—3762no—
267C2×Q8—1881no—
268C2×Q8Dic18814yescomplemented
269C2×Q8C2×D18812yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—1881no—
273C22×C4C2×D18811yescharacteristic
274C22×C4#1062535011yescharacteristic
275C22×C4C2×C22⋊D4711yescharacteristic
276Q16—3764no—
277M16—1882no—
278C2×C8—22no—
279C2×C8—942no—
280C2×C8—942no—
281C2×C8—1882no—
282C2×C8C2×C22⋊D4714yes—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—941no—
287C4⋊C4—942no—
288C4⋊C4—941no—
289C4⋊C4—941no—
290C4⋊C4—942no—
291C4⋊C4—941no—
292C4⋊C4—1881no—
293C4⋊C4—1881no—
294C4⋊C4—1881no—
295C4⋊C4—1881no—
296C4⋊C4C2×D18812yes—
297C22⋊C4—1881no—
298C42—22no—
299C42—22no—
300C42—1881no—
301C42C2×D18811yescharacteristic
302C42#1062535011yescharacteristic
303C42C2×C22⋊D4711yescharacteristic
304C42C23×D4711yescharacteristic
305C23#1309399811yescharacteristic
306Q8—24no—
307Q8—22no—
308Q8—3761no—
309Q8—3762no—
310Q8—1882no—
311Q8—1884no—
312Q8C2×Dic18814yes—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—3762no—
320C2×C4—1881no—
321C2×C4—1881no—
322C2×C4—1882no—
323C2×C4#1309399811yescharacteristic
324C2×C4#1309399811yescharacteristic
325C2×C4#1309399811yescharacteristic
326C2×C4C2×Dic18812yes—
327C2×C4#1309404612yes—
328C2×C4C22×C22⋊D4711yescharacteristic
329C2×C4C22×D18811yescharacteristic
330C2×C4C2×#1062535011yescharacteristic
331C8—24no—
332C8—42no—
333C8—1882no—
334C8—1882no—
335C22—21no—
336C22—21no—
337C22#1568936412yes—
338C22C2×#1309399811yescharacteristic
339C4—24no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—3761no—
345C4—3762no—
346C4—3761no—
347C4#1568936412yes—
348C4C22×Dic18811yescharacteristic
349C4C2×#1309404611yescharacteristic
350C4#1568993112yes—
351C2—22no—
352C2#2025290911yescharacteristic
353C2C2×#1568936411yescharacteristic
354C2#2025312511yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2982C2
socleC22×C47C2×#121
prime radicalC23×C47#121
hypercenter#2982D47
hypocenterC47#19399
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
4. #20242846C2semidirect
5. #20242758C2semidirect
85. C2×Q8×C47Q16semidirect
214. C47#19399semidirect
268. C2×Q8Dic188semidirect

Maximal subgroups

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

subgroupindex
#202543382
#202531932
C2×#156894082
#202428462
#202427582
#202421632
#202420722
#202420432
C47×#29822
#1939947