Finite Group Databasebeta

← #26109430

Subgroups — #26109430

Order 12032.

434 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689201C211yesmaximal, characteristic
2#20243525C211yesmaximal, characteristic, complemented
3#20242134C211yesmaximal, characteristic, complemented
4#20241813C211yesmaximal, characteristic
5#20241284C211yesmaximal, characteristic, complemented
6#20241283C211yesmaximal, characteristic, complemented
7C47×#1486C211yesmaximal, characteristic
8C22×#10625329—21no—
9#15689201—21no—
10#15689201—21no—
11#15689201C412yes—
12#15689201C412yes—
13#15689176C2211yescharacteristic
14#15689147C2211yescharacteristic
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×C22⋊(C47⋊C8)—21no—
17C2×C4×C47⋊C8C2211yescharacteristic
18#15688887—21no—
19#15688887—21no—
20#15688875—21no—
21#15688873—21no—
22D4⋊(C47⋊C8)—22no—
23C2×C4×D4×C47C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C47×#423C2211yescharacteristic
26C47×D4⋊C8—22no—
27C2×#10625329—22no—
28C2×#10625329—22no—
29C2×#10625329—21no—
30C2×#10625329—42no—
31C2×#10625329—42no—
32C22×C47⋊C8—21no—
33C22×C47⋊C8D411yescharacteristic
34C22×C47⋊C8D411yescharacteristic
35C4⋊(C47⋊C8)—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)C2×C412yes—
40C22⋊(C47⋊C8)—22no—
41C22⋊(C47⋊C8)—22no—
42C22⋊(C47⋊C8)—21no—
43C4×C47⋊C8—21no—
44C4×C47⋊C8C2×C411yescharacteristic
45C4×C47⋊C8C2×C411yescharacteristic
46C4×C47⋊C8D411yescharacteristic
47C4×C47⋊C8D411yescharacteristic
48C22×D4×C47Q811yescharacteristic
49C23×C4×C47—21no—
50C2×M16×C47—21no—
51C4×D4×C47—21no—
52C4×D4×C47—21no—
53C4×D4×C47C2×C412yes—
54C4×D4×C47D412yes—
55C2×C4⋊C4×C47Q811yescharacteristic
56C2×C22⋊C4×C47—21no—
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C4⋊C8×C47—21no—
61C22⋊C8×C47—21no—
62C4×C8×C47—21no—
63C24×C47—21no—
64C2×D4×C47—21no—
65C2×D4×C47—42no—
66C2×D4×C47C4⋊C412yes—
67C22×C4×C47—21no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×Q811yescharacteristic
76C22×C4×C47#4111yescharacteristic
77M16×C47—41no—
78M16×C47—41no—
79C2×C8×C47—41no—
80C2×C8×C47—41no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47C4⋊C412yes—
83C22⋊C4×C47—22no—
84C22⋊C4×C47—22no—
85C42×C47C22×C411yescharacteristic
86C42×C47C2×D411yescharacteristic
87C42×C47C2×D411yescharacteristic
88C42×C47#4111yescharacteristic
89#10625329—44no—
90#10625329—44no—
91#10625329—41no—
92#10625329—41no—
93C2×C47⋊C8—21no—
94C2×C47⋊C8—21no—
95C2×C47⋊C8—21no—
96C2×C47⋊C8—21no—
97C2×C47⋊C8—22no—
98C2×C47⋊C8—41no—
99C2×C47⋊C8—41no—
100C2×C47⋊C8—41no—
101C2×C47⋊C8—41no—
102C2×C47⋊C8C22⋊C411yescharacteristic
103C2×C47⋊C8C22⋊C411yescharacteristic
104C2×C47⋊C8C22⋊C411yescharacteristic
105C2×C47⋊C8C22⋊C411yescharacteristic
106C23×C47—22no—
107C23×C47—42no—
108C23×C47—42no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47C22⋊Q811yescharacteristic
112D4×C47—24no—
113D4×C47—42no—
114C2×C4×C47—22no—
115C2×C4×C47—22no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47C2×C22⋊C411yescharacteristic
124C2×C4×C47C2×C4⋊C411yescharacteristic
125C2×C4×C47#11711yescharacteristic
126C2×C4×C47#12111yescharacteristic
127C2×C4×C47#12111yescharacteristic
128C2×C4×C47C22⋊Q811yescharacteristic
129C2×C4×C47#13511yescharacteristic
130C2×C4×C47#13511yescharacteristic
131C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
133C8×C47—81no—
134C8×C47—81no—
135C47⋊C8—21no—
136C47⋊C8—21no—
137C47⋊C8—21no—
138C47⋊C8—21no—
139C47⋊C8—42no—
140C47⋊C8—81no—
141C47⋊C8—81no—
142#8664—471nomaximal, Sylow
143C22×C47—21no—
144C22×C47—21no—
145C22×C47—42no—
146C22×C47—42no—
147C22×C47—42no—
148C22×C47—82no—
149C22×C47—82no—
150C22×C47#38011yescharacteristic
151C22×C47#47811yescharacteristic
152C22×C47#47811yescharacteristic
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—42no—
157C4×C47#41611yescharacteristic
158C4×C47#41811yescharacteristic
159C4×C47#43012yes—
160C4×C47#46311yescharacteristic
161C4×C47#46311yescharacteristic
162C2×#436—471no—
163#1837—471no—
164#1564—471no—
165#1486—471no—
166#1486D4711yescharacteristic
167#1301—471no—
168#1301—471no—
169C2×C47—21no—
170C2×C47—21no—
171C2×C47—82no—
172C2×C47#180411yescharacteristic
173C2×C47#181111yescharacteristic
174C2×C47#181411yescharacteristic
175C22×M16—941no—
176C2×C4×D4C2×D4711yescharacteristic
177#436—472no—
178#436—472no—
179#436—941no—
180#436—941no—
181#431—471no—
182#431C2×D4711yescharacteristic
183#423—471no—
184#423C2×D4711yescharacteristic
185C2×C4⋊C8—471no—
186C2×C22⋊C8—941no—
187C2×C4×C8—471no—
188#341—941no—
189#341—941no—
190#335—941no—
191#334—941no—
192D4⋊C8—22no—
193D4⋊C8—942no—
194C47#866411yescharacteristic, complemented, Sylow
195C22×D4Dic9411yescharacteristic
196C23×C4—21no—
197C2×M16—21no—
198C2×M16—942no—
199C2×M16—942no—
200C2×M16—941no—
201C2×M16—1882no—
202C2×M16—1882no—
203C22×C8—471no—
204C22×C8—471no—
205C22×C8—941no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4C4×D4712yes—
209C4×D4C22⋊D4712yes—
210C2×C4⋊C4Dic9411yescharacteristic
211C2×C22⋊C4—21no—
212C2×C42C22×D4711yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—472no—
217C4⋊C8—941no—
218C4⋊C8—941no—
219C4⋊C8—941no—
220C4⋊C8—941no—
221C22⋊C8—21no—
222C22⋊C8—942no—
223C22⋊C8—942no—
224C22⋊C8—941no—
225C4×C8—21no—
226C4×C8—471no—
227C4×C8—471no—
228C4×C8—471no—
229C4×C8—471no—
230C4×C8—941no—
231C24—21no—
232C2×D4—21no—
233C2×D4—42no—
234C2×D4Dic47⋊C412yes—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4C2×Dic9411yescharacteristic
243C22×C4D4×D4711yescharacteristic
244C22×C4#1062535211yescharacteristic
245M16—41no—
246M16—41no—
247M16—1884no—
248M16—1884no—
249M16—1881no—
250M16—1881no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—471no—
254C2×C8—471no—
255C2×C8—471no—
256C2×C8—471no—
257C2×C8—941no—
258C2×C8—941no—
259C2×C8—941no—
260C2×C8—941no—
261C2×C8—942no—
262C2×C8—1881no—
263C2×C8—1881no—
264C2×C8—1881no—
265C2×C8—1881no—
266C4⋊C4—21no—
267C4⋊C4Dic47⋊C412yes—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C42C2×C4×D4711yescharacteristic
271C42D4×D4711yescharacteristic
272C42#1062535211yescharacteristic
273C42C2×C22⋊D4711yescharacteristic
274C23—22no—
275C23—42no—
276C23—42no—
277C23—41no—
278C23—41no—
279C23#1309400111yescharacteristic
280D4—24no—
281D4—42no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4C22⋊C4×D4711yescharacteristic
292C2×C4C2×Dic47⋊C411yescharacteristic
293C2×C4#1309398011yescharacteristic
294C2×C4#1309399611yescharacteristic
295C2×C4#1309399611yescharacteristic
296C2×C4#1309400111yescharacteristic
297C2×C4#1309404311yescharacteristic
298C2×C4#1309404511yescharacteristic
299C2×C4#1309404811yescharacteristic
300C2×C4#1309405111yescharacteristic
301C8—81no—
302C8—81no—
303C8—941no—
304C8—941no—
305C8—941no—
306C8—941no—
307C8—3761no—
308C8—3761no—
309C8—1882no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—82no—
316C22—82no—
317C22#1568898111yescharacteristic
318C22#1568942111yescharacteristic
319C22#1568942511yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—42no—
324C4#1568911611yescharacteristic
325C4#1568912611yescharacteristic
326C4#1568917112yes—
327C4#1568933611yescharacteristic
328C4#1568934011yescharacteristic
329C2—21no—
330C2—21no—
331C2—82no—
332C2#2024339611yescharacteristic
333C2#2024342811yescharacteristic
334C2#2024344011yescharacteristic
subgroupG/H
centerC2×C4#13093996
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1486C2
socleC22×C47#380
prime radicalC22×D4×C47Q8
cosocleC2×C42×C47C23
hypercenter#1486D47
hypocenterC47#8664
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #20243525C2semidirect
3. #20242134C2semidirect
5. #20241284C2semidirect
6. #20241283C2semidirect
194. C47#8664semidirect

Maximal subgroups

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

subgroupindex
C2×#156892012
#202435252
#202421342
#202418132
#202412842
#202412832
C47×#14862
#866447