Finite Group Databasebeta

← #26201577

Subgroups — #26201577

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#20253189C211yesmaximal, characteristic
2C2×#15689405C211yesmaximal, characteristic
3#20252468C211yesmaximal, characteristic
4#20242843C214yesmaximal, complemented
5#20242755C214yesmaximal, complemented
6#20242163C211yesmaximal, characteristic
7#20242073C211yesmaximal, characteristic
8#20242045C211yesmaximal, characteristic
9C47×#3209C211yesmaximal, characteristic
10C2×C4×Dic94C2211yescharacteristic
11#15689427—22no—
12#15689415C2212yes—
13#15689405—22no—
14#15689405C2212yes—
15#15689405C2214yes—
16C8×Dic94C2214yes—
17#15689247—22no—
18#15689176C2211yescharacteristic
19#15689173—22no—
20#15689166—21no—
21C2×#13093955—21no—
22#15689147C2211yescharacteristic
23C2×C4⋊(C47⋊C8)C2211yescharacteristic
24#15689131—21no—
25C2×#13093947—21no—
26#15688876C2214yes—
27#15688872C2212yes—
28#15688869C2212yes—
29#15688869C2214yes—
30C47×#533C2211yescharacteristic
31C2×C4⋊Q8×C47C2211yescharacteristic
32C47×#494C2212yes—
33C47×#487C2212yes—
34C47×#462—22no—
35C47×#451—22no—
36C47×#431C2211yescharacteristic
37C22×Dic94D411yescharacteristic
38C2×#10625329—21no—
39C22×C47⋊C8—21no—
40C4×Dic94—21no—
41C4×Dic94—21no—
42C4×Dic94D412yes—
43C4×Dic94C2312yes—
44C2×C4⋊Dic47D411yescharacteristic
45C2×Dic47⋊C4—21no—
46C2×C4×Dic47—21no—
47#13093957—22no—
48#13093955—24no—
49#13093955—22no—
50C4⋊(C47⋊C8)—21no—
51C4⋊(C47⋊C8)—21no—
52C4⋊(C47⋊C8)C2312yes—
53C4⋊(C47⋊C8)C2312yes—
54C4⋊(C47⋊C8)C2312yes—
55Dic47⋊C8—24no—
56#13093947—24no—
57#13093947—22no—
58#13093947—22no—
59C22⋊(C47⋊C8)—21no—
60C4×C47⋊C8—21no—
61C8×Dic47—24no—
62C22×Q8×C47—21no—
63C2×Q16×C47—24no—
64C4⋊Q8×C47—21no—
65C4⋊Q8×C47—21no—
66C4⋊Q8×C47—21no—
67C4⋊Q8×C47C2311yescharacteristic
68C4⋊Q8×C47C2312yes—
69C4⋊Q8×C47C2311yescharacteristic
70#125×C47—21no—
71C22⋊Q8×C47—21no—
72C22⋊Q8×C47—21no—
73C4×Q8×C47—21no—
74C4×Q8×C47—21no—
75#117×C47—21no—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—21no—
78C2×C42×C47C2311yescharacteristic
79C4⋊C8×C47D412yes—
80#103×C47—24no—
81C22⋊C8×C47D412yes—
82C4×C8×C47C2312yes—
83C2×Q8×C47—22no—
84C2×Q8×C47—21no—
85C2×Q8×C47—21no—
86C2×Q8×C47—41no—
87C2×Q8×C47—82no—
88C22×C4×C47—41no—
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47#4111yescharacteristic
92C22×Dic47—21no—
93C22×Dic47—21no—
94C2×Dic94—21no—
95C2×Dic94—22no—
96C2×Dic94Q1614yescomplemented
97C2×Dic94C2×D412yes—
98Q16×C47—84no—
99C2×C8×C47C2×D414yes—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47—41no—
109C4⋊C4×C47—41no—
110C22⋊C4×C47—41no—
111C42×C47—41no—
112C42×C47C2×D411yescharacteristic
113C42×C47C2×D411yescharacteristic
114C42×C47#4111yescharacteristic
115C42×C47C2411yescharacteristic
116C4⋊Dic47—21no—
117C4⋊Dic47C2×D412yes—
118Dic47⋊C4—22no—
119Dic47⋊C4—22no—
120C4×Dic47—22no—
121C4×Dic47—22no—
122#10625329—42no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—42no—
127C23×C47#12111yescharacteristic
128Q8×C47—42no—
129Q8×C47—44no—
130Q8×C47—81no—
131Q8×C47—82no—
132C2×C4×C47—22no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—42no—
137C2×C4×C47—82no—
138C2×C4×C47#12111yescharacteristic
139C2×C4×C47#12111yescharacteristic
140C2×C4×C47#12111yescharacteristic
141C2×C4×C47C2×Q1612yes—
142C2×C4×C47#13512yes—
143C2×C4×C47C22×D411yescharacteristic
144C2×C4×C47C22×D411yescharacteristic
145C2×C4×C47C2×#4111yescharacteristic
146C2×Dic47—22no—
147C2×Dic47—24no—
148C2×Dic47—22no—
149C2×Dic47—81no—
150Dic94—22no—
151Dic94—24no—
152Dic94C2×Q1614yes—
153C8×C47—24no—
154C47⋊C8—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—21no—
164C4×C47—21no—
165C4×C47—82no—
166C4×C47—81no—
167C4×C47—81no—
168C4×C47#46712yes—
169C4×C47C22×Q1611yescharacteristic
170C4×C47C2×#13511yescharacteristic
171C4×C47#57712yes—
172Dic47—24no—
173Dic47—42no—
174#3209D4711yescharacteristic
175#3078—471no—
176C2×#475—471no—
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#533C2×D4711yescharacteristic
188C2×C4⋊Q8C2×D4711yescharacteristic
189C2×C4×Q8—471no—
190#494C2×D4712yes—
191#487C2×D4712yes—
192#479—942no—
193#477—472no—
194#475—474no—
195#475—472no—
196#475—942no—
197#462—22no—
198#451—22no—
199C8×Q8—474no—
200C8×Q8—942no—
201#431—471no—
202#431—942no—
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—21no—
216C22×Q8—471no—
217C2×Q16—24no—
218C2×M16—941no—
219C22×C8—941no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8C22×D4712yes—
224C4⋊Q8C22×D4711yescharacteristic
225C4⋊Q8C22×D4711yescharacteristic
226#125—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—472no—
232C4×Q8—472no—
233C4×Q8—941no—
234C4×Q8—941no—
235#117—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—471no—
239C2×C4⋊C4—941no—
240C2×C42—941no—
241C2×C42C22×D4711yescharacteristic
242#107—942no—
243#106—944no—
244#106—942no—
245C4⋊C8—472no—
246C4⋊C8—472no—
247C4⋊C8—472no—
248C4⋊C8—944no—
249C4⋊C8—941no—
250C4⋊C8—941no—
251C4⋊C8C22⋊D4712yes—
252#103—24no—
253#103—944no—
254#103—942no—
255#103—942no—
256C22⋊C8—941no—
257C22⋊C8C22⋊D4712yes—
258C4×C8—944no—
259C4×C8—941no—
260C4×C8C22×D4712yes—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—41no—
265C2×Q8—82no—
266C2×Q8—474no—
267C2×Q8—472no—
268C2×Q8—941no—
269C2×Q8—942no—
270C22×C4—41no—
271C22×C4—941no—
272C22×C4—941no—
273C22×C4D4×D4711yescharacteristic
274C22×C4#1062535211yescharacteristic
275C22×C4C2×C22⋊D4711yescharacteristic
276Q16—84no—
277M16—1882no—
278C2×C8—942no—
279C2×C8—942no—
280C2×C8—942no—
281C2×C8—1882no—
282C2×C8C2×C22⋊D4714yes—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—472no—
294C4⋊C4—942no—
295C4⋊C4—941no—
296C4⋊C4—942no—
297C22⋊C4—41no—
298C42—41no—
299C42—942no—
300C42—942no—
301C42D4×D4711yescharacteristic
302C42#1062535211yescharacteristic
303C42C2×C22⋊D4711yescharacteristic
304C42C23×D4711yescharacteristic
305C23#1309399511yescharacteristic
306Q8—42no—
307Q8—44no—
308Q8—81no—
309Q8—82no—
310Q8—474no—
311Q8—944no—
312Q8—942no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—82no—
319C2×C4—942no—
320C2×C4—944no—
321C2×C4—942no—
322C2×C4—3761no—
323C2×C4#1309399511yescharacteristic
324C2×C4#1309399511yescharacteristic
325C2×C4#1309399511yescharacteristic
326C2×C4Q16×D4712yes—
327C2×C4#1309404712yes—
328C2×C4C2×D4×D4711yescharacteristic
329C2×C4C22×C22⋊D4711yescharacteristic
330C2×C4C2×#1062535211yescharacteristic
331C8—24no—
332C8—1882no—
333C8—1882no—
334C8—1882no—
335C22—21no—
336C22—21no—
337C22#1568936112yes—
338C22C2×#1309399511yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—81no—
343C4—82no—
344C4—81no—
345C4—944no—
346C4—1882no—
347C4#1568936112yes—
348C4C2×Q16×D4711yescharacteristic
349C4C2×#1309404711yescharacteristic
350C4#1568993412yes—
351C2—22no—
352C2#2025290511yescharacteristic
353C2C2×#1568936111yescharacteristic
354C2#2025312111yescharacteristic
subgroupG/H
centerC22C2×#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3209C2
socleC22×C47C2×#121
prime radicalC23×C47#121
hypercenter#3209D47
hypocenterC47#19399
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
4. #20242843C2semidirect
5. #20242755C2semidirect
96. C2×Dic94Q16semidirect
214. C47#19399semidirect

Maximal subgroups

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

subgroupindex
#202531892
C2×#156894052
#202524682
#202428432
#202427552
#202421632
#202420732
#202420452
C47×#32092
#1939947