Finite Group Databasebeta

← #26373164

Subgroups — #26373164

Order 12032.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20255781C211yesmaximal, characteristic
2#20245615C211yesmaximal, characteristic, complemented
3#20245588C212yesmaximal, complemented
4#20245540C211yesmaximal, characteristic, complemented
5#20245477C212yesmaximal, complemented
6#20245435C211yesmaximal, characteristic, complemented
7#20245322C212yesmaximal, complemented
8C4×#13093970C211yesmaximal, characteristic, complemented
9D4×C47⋊C16C211yesmaximal, characteristic
10C2×#15688927C212yesmaximal, complemented
11D4×Q16×C47C211yesmaximal, characteristic
12#15689830—21no—
13C2×#13093970C2211yescharacteristic
14C2×#13093970C2212yes—
15C22×C47⋊C16C2212yes—
16#15689528C2211yescharacteristic
17#15689459—22no—
18#15689419—21no—
19#15689364C2212yes—
20C4×Dic188C2211yescharacteristic
21C2×#13093958C2212yes—
22#15688948C2212yes—
23#15688948C2211yescharacteristic
24C4⋊(C47⋊C16)C2211yescharacteristic
25#15688927C2212yes—
26#15688927C2212yes—
27#15688927C2212yes—
28#15688927C2211yescharacteristic
29#15688927C2212yes—
30#15688926C2211yescharacteristic
31C22⋊(C47⋊C16)C2212yes—
32C4×C47⋊C16C2211yescharacteristic
33C22×Q16×C47C2212yes—
34D4×Q8×C47—21no—
35C47×#494C2211yescharacteristic
36C47×#467C2212yes—
37C47×#462—21no—
38C47×#451—22no—
39C4×Q16×C47C2211yescharacteristic
40C8×D4×C47C2211yescharacteristic
41C2×Dic188C2311yescharacteristic
42#13094024—21no—
43#13094015—41no—
44#13094003—22no—
45#13094001—42no—
46C4×Dic94—21no—
47D4×Dic47—41no—
48C2×C4⋊Dic47—22no—
49#13093970—22no—
50#13093970—42no—
51C2×C47⋊C16—42no—
52C2×C47⋊C16C2311yescharacteristic
53C2×C47⋊C16C2312yes—
54C2×C47⋊C16C2311yescharacteristic
55#13093958—21no—
56#13093958—22no—
57#13093958C2311yescharacteristic
58#13093958C2312yes—
59#13093949—21no—
60#13093949—22no—
61C22×Q8×C47—22no—
62C2×Q16×C47—24no—
63C2×Q16×C47—22no—
64C2×Q16×C47—21no—
65C2×Q16×C47—42no—
66C2×Q16×C47C2312yes—
67C2×Q16×C47C2311yescharacteristic
68C22×C8×C47C2312yes—
69C4⋊Q8×C47—21no—
70C4⋊Q8×C47—41no—
71C22⋊Q8×C47—22no—
72C22⋊Q8×C47—42no—
73C4×Q8×C47—21no—
74C4×D4×C47—41no—
75C4×D4×C47D411yescharacteristic
76#107×C47C2311yescharacteristic
77C4⋊C8×C47D411yescharacteristic
78#103×C47—21no—
79#103×C47—22no—
80C22⋊C8×C47D412yes—
81C4×C8×C47C2311yescharacteristic
82C2×Q8×C47—21no—
83C2×Q8×C47—22no—
84C2×Q8×C47—44no—
85C2×Q8×C47—42no—
86C2×Q8×C47—82no—
87C2×D4×C47D811yescharacteristic
88C22×C4×C47—42no—
89C22×C4×C47C2×D412yes—
90C22×Dic47—42no—
91C2×Dic94—21no—
92C2×Dic94—41no—
93Q16×C47—22no—
94Q16×C47—21no—
95Q16×C47—22no—
96Q16×C47—44no—
97Q16×C47—81no—
98Q16×C47#4112yes—
99C2×C8×C47—22no—
100C2×C8×C47—22no—
101C2×C8×C47C2×D412yes—
102C2×C8×C47C2×D411yescharacteristic
103C2×C8×C47C2411yescharacteristic
104C4⋊C4×C47—21no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—42no—
108C4⋊C4×C47—41no—
109C4⋊C4×C47D811yescharacteristic
110C22⋊C4×C47—42no—
111C22⋊C4×C47D812yes—
112C42×C47—41no—
113C42×C47C2×D411yescharacteristic
114C22⋊Dic47—42no—
115C4⋊Dic47—21no—
116C4⋊Dic47—22no—
117C4⋊Dic47—42no—
118C4⋊Dic47—41no—
119Dic47⋊C4—41no—
120Dic47⋊C4—42no—
121C4×Dic47—41no—
122Dic188—21no—
123Dic188—21no—
124C47⋊C16—21no—
125C47⋊C16—21no—
126C47⋊C16—22no—
127C23×C47C2×D812yes—
128Q8×C47—22no—
129Q8×C47—41no—
130Q8×C47—42no—
131Q8×C47—42no—
132Q8×C47—84no—
133D4×C47Q3214yescomplemented
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—42no—
139C2×C4×C47—84no—
140C2×C4×C47C2×D812yes—
141C2×C4×C47C2×D811yescharacteristic
142C2×C4×C47C2×D811yescharacteristic
143C2×C4×C47C22×D411yescharacteristic
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—42no—
147C2×Dic47—162no—
148Dic94—41no—
149Dic94—41no—
150Dic94—161no—
151C8×C47—22no—
152C8×C47—21no—
153C8×C47C2×#4111yescharacteristic
154C8×C47H(5,𝔽2)11yescharacteristic
155#33866—471nomaximal, Sylow
156C22×C47—22no—
157C22×C47—22no—
158C22×C47C2×Q3214yes—
159C22×C47C22×D811yescharacteristic
160C4×C47—21no—
161C4×C47—22no—
162C4×C47—42no—
163C4×C47—81no—
164C4×C47—82no—
165C4×C47C2×Q3212yes—
166C4×C47#54611yescharacteristic
167C4×C47#57711yescharacteristic
168Dic47—82no—
169Dic47—81no—
170Dic47—81no—
171#3338—471no—
172D4×Q16D4711yescharacteristic
173#2241—471no—
174#2236—472no—
175#2227—471no—
176#2216—472no—
177#2209—471no—
178#2188—472no—
179C4×Q32—471no—
180D4×C16—471no—
181C2×#358—472no—
182C2×C47—24no—
183C2×C47#329411yescharacteristic
184C2×C47C22×Q3211yescharacteristic
185C2×C47#341611yescharacteristic
186C22×Q16C2×D4712yes—
187#554—941no—
188D4×Q8—21no—
189C2×Q32—471no—
190C2×Q32—472no—
191C22×C16—472no—
192#500—471no—
193#494C2×D4711yescharacteristic
194#484—942no—
195#477—941no—
196#467—472no—
197#467C2×D4712yes—
198#462—21no—
199#451—22no—
200C4×Q16—471no—
201C4×Q16C2×D4711yescharacteristic
202C8×D4C2×D4711yescharacteristic
203C2×#107—472no—
204#366—471no—
205#366—472no—
206C4⋊C16—471no—
207#358—471no—
208#358—472no—
209#358—472no—
210#358—472no—
211#358—472no—
212#358—471no—
213C22⋊C16—472no—
214C4×C16—471no—
215C47#3386611yescharacteristic, complemented, Sylow
216C22×Q8—22no—
217C2×Q16—24no—
218C2×Q16—22no—
219C2×Q16—21no—
220C2×Q16—42no—
221C2×Q16—471no—
222C2×Q16C22×D4712yes—
223C2×Q16C22×D4711yescharacteristic
224C22×C8C22×D4712yes—
225C4⋊Q8—21no—
226C4⋊Q8—41no—
227C4⋊Q8—941no—
228#125—1881no—
229C22⋊Q8—22no—
230C22⋊Q8—42no—
231C22⋊Q8—942no—
232C22⋊Q8—1882no—
233C4×Q8—21no—
234C4×Q8—941no—
235C4×D4—41no—
236C4×D4—1881no—
237C4×D4C22⋊D4711yescharacteristic
238C2×C4⋊C4—942no—
239Q32—942no—
240Q32—1882no—
241C2×C16—471no—
242C2×C16—471no—
243C2×C16—472no—
244C2×C16—1882no—
245#107—471no—
246#107—472no—
247#107—942no—
248#107—941no—
249#107C22×D4711yescharacteristic
250C4⋊C8C22⋊D4711yescharacteristic
251#103—21no—
252#103—22no—
253#103—941no—
254#103—942no—
255C22⋊C8C22⋊D4712yes—
256C4×C8C22×D4711yescharacteristic
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—44no—
260C2×Q8—42no—
261C2×Q8—82no—
262C2×Q8—941no—
263C2×Q8—1881no—
264C2×D4#1062533411yescharacteristic
265C22×C4—42no—
266C22×C4—1882no—
267C22×C4C2×C22⋊D4712yes—
268Q16—22no—
269Q16—21no—
270Q16—22no—
271Q16—44no—
272Q16—81no—
273Q16—941no—
274Q16—941no—
275Q16#1062535212yes—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8C2×C22⋊D4711yescharacteristic
279C2×C8C2×C22⋊D4712yes—
280C2×C8C23×D4711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—941no—
287C4⋊C4—942no—
288C4⋊C4—1881no—
289C4⋊C4—1882no—
290C4⋊C4—1882no—
291C4⋊C4—1881no—
292C4⋊C4#1062533411yescharacteristic
293C22⋊C4—42no—
294C22⋊C4—1882no—
295C22⋊C4#1062533412yes—
296C42—41no—
297C42—1881no—
298C42C2×C22⋊D4711yescharacteristic
299C16—941no—
300C16—941no—
301C16—942no—
302C23C2×#1062533412yes—
303Q8—22no—
304Q8—41no—
305Q8—42no—
306Q8—42no—
307Q8—84no—
308Q8—1881no—
309Q8—1881no—
310Q8—7521no—
311D4#1309397014yescomplemented
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—84no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4—1882no—
321C2×C4—7522no—
322C2×C4C2×#1062533411yescharacteristic
323C2×C4C2×#1062533412yes—
324C2×C4C2×#1062533411yescharacteristic
325C2×C4C22×C22⋊D4711yescharacteristic
326C8—22no—
327C8—21no—
328C8C2×#1062535211yescharacteristic
329C8#1309406911yescharacteristic
330C22—22no—
331C22—22no—
332C22C2×#1309397014yes—
333C22C22×#1062533411yescharacteristic
334C4—21no—
335C4—22no—
336C4—42no—
337C4—81no—
338C4—82no—
339C4—3761no—
340C4—3762no—
341C4—3761no—
342C4C2×#1309397012yes—
343C4#1568976711yescharacteristic
344C4#1568993411yescharacteristic
345C2—24no—
346C2#2025520111yescharacteristic
347C2C22×#1309397011yescharacteristic
348C2#2025653211yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingD4×Q16×C47C2
socleC22×C47C22×D8
prime radicalC2×D4×C47D8
hypercenterD4×Q16D47
hypocenterC47#33866
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
2. #20245615C2semidirect
3. #20245588C2semidirect
4. #20245540C2semidirect
5. #20245477C2semidirect
6. #20245435C2semidirect
7. #20245322C2semidirect
8. C4×#13093970C2semidirect
10. C2×#15688927C2semidirect
133. D4×C47Q32semidirect
215. C47#33866semidirect
311. D4#13093970semidirect

Maximal subgroups

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

subgroupindex
#202557812
#202456152
#202455882
#202455402
#202454772
#202454352
#202453222
C4×#130939702
D4×C47⋊C162
C2×#156889272
D4×Q16×C472
#3386647