Finite Group Databasebeta

← #26201578

Subgroups — #26201578

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#20254336C211yesmaximal, characteristic
2#20253193C211yesmaximal, characteristic
3#20252469C211yesmaximal, characteristic
4#20242844C214yesmaximal, complemented
5#20242756C214yesmaximal, complemented
6#20242162C211yesmaximal, characteristic
7#20242073C211yesmaximal, characteristic
8#20242046C211yesmaximal, characteristic
9C2×C47×#475C211yesmaximal, characteristic
10#15689680C2211yescharacteristic
11#15689502C2212yes—
12#15689473C2212yes—
13#15689430—22no—
14#15689418C2212yes—
15#15689408C2212yes—
16#15689330—22no—
17#15689279—22no—
18Q8×C47⋊C8—22no—
19Q8×C47⋊C8C2214yes—
20#15689176—22no—
21#15689176C2211yescharacteristic
22#15689176C2211yescharacteristic
23#15689163—21no—
24#15689147C2211yescharacteristic
25#15689132—21no—
26#15688875C2214yes—
27#15688871C2212yes—
28#15688869C2214yes—
29#15688868C2212yes—
30C2×C4⋊Q8×C47C2211yescharacteristic
31C2×C4×Q8×C47C2211yescharacteristic
32C47×#475—22no—
33C47×#475C2214yes—
34C2×#106×C47—21no—
35C2×C4⋊C8×C47C2211yescharacteristic
36C2×#103×C47—21no—
37C2×#10625337—24no—
38C2×#10625329—21no—
39#13094024C2311yescharacteristic
40#13094024C2311yescharacteristic
41#13094016—21no—
42#13094003—21no—
43#13094003—21no—
44C4×Dic94—21no—
45C4×Dic94—21no—
46#13093983—21no—
47#13093957—22no—
48#13093955—22no—
49C4⋊(C47⋊C8)—21no—
50C4⋊(C47⋊C8)—24no—
51C4⋊(C47⋊C8)D412yes—
52C4⋊(C47⋊C8)C2312yes—
53C4⋊(C47⋊C8)C2312yes—
54#13093948—22no—
55#13093948—22no—
56#13093948—24no—
57C22⋊(C47⋊C8)—21no—
58C22⋊(C47⋊C8)D412yes—
59C4×C47⋊C8—21no—
60C4×C47⋊C8—24no—
61C4×C47⋊C8C2312yes—
62C22×Q8×C47—21no—
63C22×Q8×C47D411yescharacteristic
64C22×C8×C47—21no—
65C4⋊Q8×C47—21no—
66C4⋊Q8×C47—21no—
67C4⋊Q8×C47—21no—
68C4⋊Q8×C47C2312yes—
69C4×Q8×C47—21no—
70C4×Q8×C47—21no—
71C4×Q8×C47D412yes—
72C4×Q8×C47C2312yes—
73C2×C4⋊C4×C47—21no—
74C2×C4⋊C4×C47—21no—
75C2×C4⋊C4×C47—21no—
76C2×C4⋊C4×C47D411yescharacteristic
77C2×C42×C47—21no—
78C2×C42×C47C2311yescharacteristic
79#106×C47—24no—
80C4⋊C8×C47—21no—
81C4⋊C8×C47C2312yes—
82#103×C47—24no—
83C2×Q8×C47—21no—
84C2×Q8×C47—22no—
85C2×Q8×C47—22no—
86C2×Q8×C47—41no—
87C2×Q8×C47—82no—
88C2×Q8×C47Q1614yescomplemented
89C2×Q8×C47C2×D412yes—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—41no—
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47#4111yescharacteristic
96C2×Dic94—21no—
97C2×Dic94—21no—
98C2×C8×C47—22no—
99C2×C8×C47—42no—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47C2×D412yes—
108C42×C47—22no—
109C42×C47—22no—
110C42×C47C2×D411yescharacteristic
111C42×C47C2×D411yescharacteristic
112C42×C47#4111yescharacteristic
113C42×C47C2411yescharacteristic
114C22⋊Dic47—41no—
115#10625337—84no—
116C4⋊Dic47—21no—
117C4⋊Dic47—21no—
118C4⋊Dic47—21no—
119C4⋊Dic47—21no—
120Dic47⋊C4—41no—
121Dic47⋊C4—41no—
122C4×Dic47—41no—
123#10625329—42no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8C2×D414yes—
127C23×C47#12111yescharacteristic
128Q8×C47—22no—
129Q8×C47—24no—
130Q8×C47—44no—
131Q8×C47—82no—
132Q8×C47C2×Q1614yes—
133C2×C4×C47—22no—
134C2×C4×C47—24no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—41no—
138C2×C4×C47—42no—
139C2×C4×C47—81no—
140C2×C4×C47—82no—
141C2×C4×C47#12111yescharacteristic
142C2×C4×C47#12111yescharacteristic
143C2×C4×C47#12111yescharacteristic
144C2×C4×C47C2×Q1612yes—
145C2×C4×C47#13512yes—
146C2×C4×C47C22×D411yescharacteristic
147C2×C4×C47C22×D411yescharacteristic
148C2×C4×C47C2×#4111yescharacteristic
149C2×Dic47—41no—
150C2×Dic47—41no—
151Dic94—42no—
152Dic94—81no—
153C8×C47—42no—
154C47⋊C8—24no—
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—82no—
168C4×C47#46712yes—
169C4×C47C22×Q1611yescharacteristic
170C4×C47C2×#13511yescharacteristic
171C4×C47#57712yes—
172Dic47—81no—
173Dic47—81no—
174#3209—471no—
175#3078—471no—
176C2×#475D4711yescharacteristic
177#2982—471no—
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⋊Q8C2×D4711yescharacteristic
189C2×C4×Q8C2×D4711yescharacteristic
190#494—472no—
191#487—472no—
192#479—942no—
193#477—472no—
194#475—22no—
195#475—472no—
196#475C2×D4714yes—
197#462—942no—
198#451—942no—
199C8×Q8—474no—
200C8×Q8—942no—
201#431—471no—
202#431—471no—
203#431—942no—
204#428—941no—
205C2×#106—21no—
206#423—471no—
207C2×C4⋊C8C2×D4711yescharacteristic
208#419—941no—
209C2×#103—21no—
210#335—474no—
211#333—472no—
212#332—474no—
213#332—472no—
214C47#1939911yescharacteristic, complemented, Sylow
215C22×Q8—21no—
216C22×Q8C22⋊D4711yescharacteristic
217C2×Q16—944no—
218C2×M16—941no—
219C22×C8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—471no—
224C4⋊Q8—471no—
225C4⋊Q8C22×D4712yes—
226#125—941no—
227C22⋊Q8—941no—
228C22⋊Q8—941no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—941no—
232C4×Q8—941no—
233C4×Q8C22⋊D4712yes—
234C4×Q8C22×D4712yes—
235#117—941no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C22⋊D4711yescharacteristic
240C2×C42—21no—
241C2×C42C22×D4711yescharacteristic
242#107—942no—
243#106—24no—
244#106—942no—
245C4⋊C8—21no—
246C4⋊C8—472no—
247C4⋊C8—472no—
248C4⋊C8—472no—
249C4⋊C8—944no—
250C4⋊C8—941no—
251C4⋊C8C22×D4712yes—
252#103—24no—
253#103—944no—
254#103—942no—
255#103—942no—
256C22⋊C8—472no—
257C22⋊C8—941no—
258C4×C8—472no—
259C4×C8—944no—
260C4×C8—941no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—82no—
266C2×Q8—941no—
267C2×Q8—941no—
268C2×Q8#1062533714yescomplemented
269C2×Q8C2×C22⋊D4712yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4D4×D4711yescharacteristic
274C22×C4#1062535211yescharacteristic
275C22×C4C2×C22⋊D4711yescharacteristic
276Q16—3764no—
277M16—1882no—
278C2×C8—22no—
279C2×C8—42no—
280C2×C8—474no—
281C2×C8—942no—
282C2×C8—942no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—941no—
291C4⋊C4—941no—
292C4⋊C4—941no—
293C4⋊C4—941no—
294C4⋊C4—1881no—
295C4⋊C4—1881no—
296C4⋊C4C2×C22⋊D4712yes—
297C22⋊C4—1881no—
298C42—22no—
299C42—22no—
300C42—1881no—
301C42D4×D4711yescharacteristic
302C42#1062535211yescharacteristic
303C42C2×C22⋊D4711yescharacteristic
304C42C23×D4711yescharacteristic
305C23#1309399611yescharacteristic
306Q8—24no—
307Q8—22no—
308Q8—44no—
309Q8—82no—
310Q8—3761no—
311Q8—1882no—
312Q8C2×#1062533714yes—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—1881no—
322C2×C4—1881no—
323C2×C4#1309399611yescharacteristic
324C2×C4#1309399611yescharacteristic
325C2×C4#1309399611yescharacteristic
326C2×C4C2×#1062533712yes—
327C2×C4#1309404412yes—
328C2×C4C2×D4×D4711yescharacteristic
329C2×C4C22×C22⋊D4711yescharacteristic
330C2×C4C2×#1062535211yescharacteristic
331C8—42no—
332C8—944no—
333C8—1882no—
334C8—1882no—
335C22—21no—
336C22—21no—
337C22#1568936212yes—
338C22C2×#1309399611yescharacteristic
339C4—24no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—82no—
345C4—3761no—
346C4—3761no—
347C4#1568936212yes—
348C4C22×#1062533711yescharacteristic
349C4C2×#1309404411yescharacteristic
350C4#1568992912yes—
351C2—22no—
352C2#2025290611yescharacteristic
353C2C2×#1568936211yescharacteristic
354C2#2025312211yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#475C2
socleC22×C47C2×#121
prime radicalC23×C47#121
hypercenterC2×#475D47
hypocenterC47#19399
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
4. #20242844C2semidirect
5. #20242756C2semidirect
88. C2×Q8×C47Q16semidirect
214. C47#19399semidirect
268. C2×Q8#10625337semidirect

Maximal subgroups

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

subgroupindex
#202543362
#202531932
#202524692
#202428442
#202427562
#202421622
#202420732
#202420462
C2×C47×#4752
#1939947