Finite Group Databasebeta

← #37546456

Subgroups — #37546456

Order 23168.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic362C211yesmaximal, characteristic, complemented
2#26041998C211yesmaximal, characteristic
3#26041638C211yesmaximal, characteristic
4#26041637C211yesmaximal, characteristic
5#26041628C211yesmaximal, characteristic, complemented
6#26041567C211yesmaximal, characteristic
7#26041565C211yesmaximal, characteristic, complemented
8#26041562C211yesmaximal, characteristic, complemented
9#26041499C211yesmaximal, characteristic
10#26041487C211yesmaximal, characteristic, complemented
11#26041408C211yesmaximal, characteristic, complemented
12#26041386C211yesmaximal, characteristic, complemented
13#26041384C211yesmaximal, characteristic, complemented
14SD16×Dic181C211yesmaximal, characteristic
15#464×C181C211yesmaximal, characteristic
16C2×#15668439C2211yescharacteristic
17#19153816—21no—
18#19153816C2211yescharacteristic
19#19153815—21no—
20#19153815—21no—
21#19153815C2211yescharacteristic
22#19153815C2211yescharacteristic
23#19153814C2211yescharacteristic
24#19153809—21no—
25#19153809—21no—
26#19153803C2211yescharacteristic
27#19153793—21no—
28Q8×Dic181C2211yescharacteristic
29Q8×Dic181C2211yescharacteristic
30Q8×Dic181C2211yescharacteristic
31C4×Dic362—21no—
32C4×Dic362C2211yescharacteristic
33C4×Dic362C2211yescharacteristic
34D4×Dic181C2211yescharacteristic
35#19153772—21no—
36#19153748C2211yescharacteristic
37#19153748C2211yescharacteristic
38#19153747C2211yescharacteristic
39#19153745C2211yescharacteristic
40#19153745C2211yescharacteristic
41#19153741C2211yescharacteristic
42#19153741C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153740C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153739C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153727C2211yescharacteristic
50#19153726C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×Q16×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic
54#124×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362—21no—
61C2×Dic362—21no—
62C2×Dic362D411yescharacteristic
63C2×Dic362C2311yescharacteristic
64C22×Dic181—21no—
65#15668439—22no—
66C2×#13089678C2311yescharacteristic
67C2×#13089678C2311yescharacteristic
68C4⋊Dic181—21no—
69C4⋊Dic181—21no—
70C4⋊Dic181—21no—
71C4⋊Dic181—21no—
72C4⋊Dic181D411yescharacteristic
73C4⋊Dic181C2311yescharacteristic
74C4⋊Dic181C2311yescharacteristic
75C4⋊Dic181C2311yescharacteristic
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79#15668428—21no—
80#15668428—21no—
81#15668428D411yescharacteristic
82#15668428D411yescharacteristic
83C22⋊Dic181—21no—
84C22⋊Dic181—21no—
85C22⋊Dic181—21no—
86C4×Dic181—21no—
87C4×Dic181—21no—
88C4×Dic181—21no—
89C4×Dic181—21no—
90C4×Dic181C2311yescharacteristic
91C4×Dic181C2311yescharacteristic
92C2×Q8×C181C2311yescharacteristic
93C2×Q8×C181C2311yescharacteristic
94C2×D4×C181C2311yescharacteristic
95Q16×C181—22no—
96SD16×C181—22no—
97C2×C8×C181C2311yescharacteristic
98C2×C8×C181C2311yescharacteristic
99C4⋊C4×C181—21no—
100C4⋊C4×C181C2311yescharacteristic
101C22⋊C4×C181—21no—
102C22⋊C4×C181—21no—
103C42×C181—21no—
104C42×C181C2311yescharacteristic
105Dic362—22no—
106Dic362—21no—
107Dic362—21no—
108Dic362—81no—
109Dic362—81no—
110Dic362—81no—
111C2×Dic181—21no—
112C2×Dic181—21no—
113C2×Dic181—21no—
114C2×Dic181—21no—
115C2×Dic181—42no—
116C2×Dic181C2×D411yescharacteristic
117C2×Dic181C2×D411yescharacteristic
118C2×Dic181C2×D411yescharacteristic
119C2×Dic181C2×D411yescharacteristic
120#13089678—21no—
121#13089678—21no—
122C23×C181—21no—
123Q8×C181—21no—
124Q8×C181—21no—
125Q8×C181—21no—
126Q8×C181#4112yes—
127D4×C181—21no—
128D4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181—21no—
131C2×C4×C181—21no—
132C2×C4×C181C2×D411yescharacteristic
133C2×C4×C181C2×D411yescharacteristic
134C2×C4×C181C2411yescharacteristic
135C8×C181—21no—
136C8×C181—21no—
137C22×C181—41no—
138C22×C181—41no—
139C22×C181—41no—
140C22×C181C22×D411yescharacteristic
141C4×C181—22no—
142C4×C181—21no—
143C4×C181—21no—
144C4×C181—41no—
145C4×C181—41no—
146C4×C181C2×#4111yescharacteristic
147C4×C1812−1+411yescharacteristic
148Dic181—21no—
149Dic181—21no—
150Dic181—21no—
151Dic181—41no—
152Dic181—41no—
153Dic181—41no—
154Dic181—41no—
155Dic181#13712yes—
156C2×C181—81no—
157C2×C181#54811yescharacteristic
158C2×C181C2×#13711yescharacteristic
159C2×C181#57711yescharacteristic
160C181#336911yescharacteristic, complemented, Sylow
161#3369—1811nomaximal, Sylow
162Q82—1811no—
163#556—1811no—
164#489—1811no—
165#489—1811no—
166#487—1811no—
167#476—1811no—
168#475—1811no—
169#475—1811no—
170#464—1811no—
171#464D18111yescharacteristic
172#462—1811no—
173#446—1811no—
174#441—1811no—
175#441—1811no—
176C4×SD16—1811no—
177C2×Q16—1811no—
178C2×Q16C2×D18111yescharacteristic
179C2×SD16C2×D18111yescharacteristic
180C4⋊Q8—3621no—
181C4⋊Q8—3621no—
182C4⋊Q8—3621no—
183C4⋊Q8—1811no—
184C4⋊Q8—1811no—
185C4⋊Q8—1811no—
186C4⋊Q8—1811no—
187#126—3621no—
188#126—3621no—
189#124—1811no—
190#124C2×D18111yescharacteristic
191C22⋊Q8—3621no—
192C4×Q8—3621no—
193C4×Q8—1811no—
194C4×Q8—1811no—
195C4×Q8—1811no—
196C4×Q8—1811no—
197C4×Q8—1811no—
198C4×Q8C2×D18111yescharacteristic
199C4×D4—1811no—
200#117—3621no—
201#106—1811no—
202#106—1811no—
203#106—1811no—
204C4⋊C8—1811no—
205C4⋊C8—1811no—
206C4⋊C8C2×D18111yescharacteristic
207#103—1811no—
208#103—1811no—
209#103—1811no—
210#103—1811no—
211#103—1811no—
212#103—1811no—
213#103C2×D18111yescharacteristic
214#102—1811no—
215#102—1811no—
216#102C2×D18111yescharacteristic
217#97—1811no—
218#97—1811no—
219C4×C8—1811no—
220C2×Q8—3621no—
221C2×Q8—3621no—
222C2×Q8—3621no—
223C2×Q8—1811no—
224C2×Q8—1811no—
225C2×Q8C22×D18111yescharacteristic
226C2×Q8C22×D18111yescharacteristic
227C2×D4C22×D18111yescharacteristic
228C22×C4—3621no—
229Q16—22no—
230Q16—3622no—
231SD16—22no—
232C2×C8—1811no—
233C2×C8—1811no—
234C2×C8C22×D18111yescharacteristic
235C2×C8C22×D18111yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—3621no—
238C4⋊C4—3621no—
239C4⋊C4—3621no—
240C4⋊C4—3621no—
241C4⋊C4—3621no—
242C4⋊C4—3621no—
243C4⋊C4—3621no—
244C4⋊C4—3621no—
245C4⋊C4—3621no—
246C4⋊C4—1811no—
247C4⋊C4—1811no—
248C4⋊C4—1811no—
249C4⋊C4—1811no—
250C4⋊C4—1811no—
251C4⋊C4—1811no—
252C4⋊C4C22×D18111yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—3621no—
256C22⋊C4—3621no—
257C22⋊C4—3621no—
258C42—21no—
259C42—3621no—
260C42—3621no—
261C42—3621no—
262C42—3621no—
263C42—1811no—
264C42—1811no—
265C42C22×D18111yescharacteristic
266C23—21no—
267Q8—21no—
268Q8—21no—
269Q8—21no—
270Q8—3622no—
271Q8—3621no—
272Q8—3621no—
273Q8—14481no—
274Q8—14481no—
275Q8—14481no—
276Q8#1566844812yes—
277D4—21no—
278D4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—3621no—
283C2×C4—3621no—
284C2×C4—3621no—
285C2×C4—3621no—
286C2×C4—1811no—
287C2×C4—1811no—
288C2×C4—1811no—
289C2×C4—1811no—
290C2×C4—7242no—
291C2×C4D4×D18111yescharacteristic
292C2×C4D4×D18111yescharacteristic
293C2×C4C23×D18111yescharacteristic
294C8—21no—
295C8—21no—
296C8—3621no—
297C8—3621no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22C2×D4×D18111yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—3621no—
308C4—3621no—
309C4—3621no—
310C4—1812no—
311C4—7241no—
312C4—7241no—
313C4—7241no—
314C4—7241no—
315C4C2×#1566844811yescharacteristic
316C4#1915386311yescharacteristic
317C2—81no—
318C2#2604193511yescharacteristic
319C2#137×D18111yescharacteristic
320C2#2604208611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#464×C181C2
socleC22×C181C22×D4
prime radicalC2×D4×C181C23
hypercenter#464D181
hypocenterC181#3369
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. Q8×Dic362C2semidirect
5. #26041628C2semidirect
7. #26041565C2semidirect
8. #26041562C2semidirect
10. #26041487C2semidirect
11. #26041408C2semidirect
12. #26041386C2semidirect
13. #26041384C2semidirect
160. C181#3369semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3622
#260419982
#260416382
#260416372
#260416282
#260415672
#260415652
#260415622
#260414992
#260414872
#260414082
#260413862
#260413842
SD16×Dic1812
#464×C1812
#3369181