Finite Group Databasebeta

← #37546542

Subgroups — #37546542

Order 23168.

346 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041989C211yesmaximal, characteristic, complemented
2#26041647C211yesmaximal, characteristic, complemented
3#26041638C211yesmaximal, characteristic, complemented
4#26041633C211yesmaximal, characteristic, complemented
5#26041586C211yesmaximal, characteristic, complemented
6#26041579C211yesmaximal, characteristic, complemented
7#26041564C211yesmaximal, characteristic, complemented
8#26041491C211yesmaximal, characteristic
9#26041491C211yesmaximal, characteristic
10#26041481C211yesmaximal, characteristic, complemented
11#26041410C211yesmaximal, characteristic, complemented
12#26041389C211yesmaximal, characteristic
13#26041377C211yesmaximal, characteristic, complemented
14C4×#15668436C211yesmaximal, characteristic, complemented
15#556×C181C211yesmaximal, characteristic, complemented
16C2×#15668436C2211yescharacteristic
17C2×#15668436C2211yescharacteristic
18C2×#15668434C2211yescharacteristic
19#19153816C2211yescharacteristic
20#19153808C2211yescharacteristic
21#19153808C2211yescharacteristic
22#19153794—21no—
23#19153794—21no—
24#19153792—21no—
25C4×D724C2211yescharacteristic
26C4×D724C2211yescharacteristic
27C4×D724C2211yescharacteristic
28C4⋊C4×D181—21no—
29#19153749C2211yescharacteristic
30#19153747C2211yescharacteristic
31#19153747C2211yescharacteristic
32C4⋊#13089678C2211yescharacteristic
33C4⋊#13089678C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic
35#19153740C2211yescharacteristic
36#19153740C2211yescharacteristic
37#19153740C2211yescharacteristic
38#19153740C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153737C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44#19153736C2211yescharacteristic
45#19153727C2211yescharacteristic
46#19153727C2211yescharacteristic
47C4×#13089678C2211yescharacteristic
48C4⋊Q8×C181C2211yescharacteristic
49#126×C181—21no—
50#126×C181—21no—
51#124×C181C2211yescharacteristic
52#124×C181C2211yescharacteristic
53C22⋊Q8×C181—21no—
54C4×Q8×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic
56C4×Q8×C181C2211yescharacteristic
57C4×D4×C181C2211yescharacteristic
58#117×C181—21no—
59C2×Dic362—21no—
60C2×D724C2311yescharacteristic
61C2×C4×D181—21no—
62C2×C4×D181—21no—
63C2×C4×D181—21no—
64#15668436—22no—
65#15668436—22no—
66#15668434—22no—
67C2×#13089678C2311yescharacteristic
68C2×#13089678C2311yescharacteristic
69C2×#13089678C2311yescharacteristic
70C2×#13089678C2311yescharacteristic
71C4⋊Dic181—21no—
72C4⋊Dic181C2311yescharacteristic
73C4⋊Dic181C2311yescharacteristic
74C4⋊Dic181C2311yescharacteristic
75#15668428—21no—
76#15668428—21no—
77#15668428—21no—
78#15668428—21no—
79#15668426—21no—
80#15668426—21no—
81#15668426—21no—
82C2×Q8×C181D411yescharacteristic
83C2×Q8×C181C2311yescharacteristic
84C2×Q8×C181C2311yescharacteristic
85C2×D4×C181C2311yescharacteristic
86C22×C4×C181—21no—
87C4⋊C4×C181—21no—
88C4⋊C4×C181—21no—
89C4⋊C4×C181—21no—
90C4⋊C4×C181—21no—
91C4⋊C4×C181D411yescharacteristic
92C4⋊C4×C181D411yescharacteristic
93C4⋊C4×C181D411yescharacteristic
94C4⋊C4×C181C2311yescharacteristic
95C22⋊C4×C181—21no—
96C22⋊C4×C181—21no—
97C22⋊C4×C181—21no—
98C22⋊C4×C181—21no—
99C22⋊C4×C181—21no—
100C42×C181—21no—
101C42×C181—21no—
102C42×C181C2311yescharacteristic
103C42×C181C2311yescharacteristic
104C42×C181C2311yescharacteristic
105C22×D181—21no—
106Dic362—81no—
107D724—21no—
108D724#4112yes—
109C2×Dic181—21no—
110C2×Dic181—21no—
111C2×Dic181—21no—
112C4×D181—42no—
113C4×D181—81no—
114C4×D181—81no—
115#13089678—21no—
116#13089678—21no—
117#13089678—21no—
118#13089678—21no—
119C23×C181—21no—
120Q8×C181—21no—
121Q8×C181—21no—
122Q8×C181—21no—
123Q8×C181—21no—
124Q8×C181—41no—
125D4×C181—21no—
126D4×C181—21no—
127C2×C4×C181—21no—
128C2×C4×C181—21no—
129C2×C4×C181—21no—
130C2×C4×C181—42no—
131C2×C4×C181C2×D411yescharacteristic
132C2×C4×C181C2×D411yescharacteristic
133C2×C4×C181C2×D411yescharacteristic
134C2×C4×C181C2×D411yescharacteristic
135C2×C4×C181C2×D411yescharacteristic
136C2×C4×C181C2×D411yescharacteristic
137C2×C4×C181C2411yescharacteristic
138C22×C181—41no—
139C22×C181—41no—
140C22×C181—41no—
141C22×C181C22×D411yescharacteristic
142C2×D181—22no—
143C2×D181—41no—
144C2×D181—41no—
145C4×C181—21no—
146C4×C181—21no—
147C4×C181—21no—
148C4×C181—21no—
149C4×C181—21no—
150C4×C181—21no—
151C4×C181—41no—
152C4×C181—41no—
153C4×C181—41no—
154C4×C181C2×#4111yescharacteristic
155C4×C1812−1+411yescharacteristic
156Dic181—41no—
157Dic181—41no—
158Dic181—41no—
159C2×C181—81no—
160C2×C181#54811yescharacteristic
161C2×C181#57511yescharacteristic
162C2×C181#57711yescharacteristic
163D181—42no—
164C181#337611yescharacteristic, complemented, Sylow
165#3376—1811nomaximal, Sylow
166#556D18111yescharacteristic, complemented
167#554—1811no—
168#491—1811no—
169#489—1811no—
170#488—1811no—
171#479—1811no—
172#478—1811no—
173#475—1811no—
174#463—1811no—
175#463—1811no—
176#461—1811no—
177#446—1811no—
178#442—1811no—
179#440—1811no—
180C4×SD16—1811no—
181C2×SD16—1811no—
182C2×SD16—1811no—
183C2×D8—1811no—
184C4⋊Q8—1811no—
185C4⋊Q8C2×D18111yescharacteristic
186#126—21no—
187#126—21no—
188#125—1811no—
189#125—1811no—
190#124C2×D18111yescharacteristic
191#124C2×D18111yescharacteristic
192C22⋊Q8—21no—
193C22⋊Q8—3621no—
194C22⋊Q8—3621no—
195C22⋊Q8—3621no—
196C4×Q8C2×D18111yescharacteristic
197C4×Q8C2×D18111yescharacteristic
198C4×Q8C2×D18111yescharacteristic
199C4×D4—1811no—
200C4×D4—1811no—
201C4×D4—1811no—
202C4×D4C2×D18111yescharacteristic
203#117—21no—
204C2×C4⋊C4—3621no—
205#107—1811no—
206#106—1811no—
207#106—1811no—
208C4⋊C8—1811no—
209C4⋊C8—1811no—
210C4⋊C8—1811no—
211#103—1811no—
212#103—1811no—
213#103—1811no—
214#103—1811no—
215#103—1811no—
216#103—1811no—
217#102—1811no—
218#102—1811no—
219#102—1811no—
220#102—1811no—
221#97—1811no—
222#97—1811no—
223C4×C8—1811no—
224C2×Q8—3621no—
225C2×Q8C22⋊D18111yescharacteristic
226C2×Q8C22×D18111yescharacteristic
227C2×Q8C22×D18111yescharacteristic
228C2×D4—1811no—
229C2×D4C22×D18111yescharacteristic
230C22×C4—21no—
231C22×C4—3621no—
232C22×C4—3621no—
233C22×C4—3621no—
234SD16—3622no—
235SD16—3622no—
236D8—3622no—
237C2×C8—1811no—
238C2×C8—1811no—
239C2×C8—1811no—
240C2×C8—1811no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—3621no—
246C4⋊C4—3621no—
247C4⋊C4—3621no—
248C4⋊C4—3621no—
249C4⋊C4—3621no—
250C4⋊C4—1811no—
251C4⋊C4—1811no—
252C4⋊C4—1811no—
253C4⋊C4C22⋊D18111yescharacteristic
254C4⋊C4C22⋊D18111yescharacteristic
255C4⋊C4C22⋊D18111yescharacteristic
256C4⋊C4C22×D18111yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—3621no—
263C22⋊C4—3621no—
264C22⋊C4—3621no—
265C42—21no—
266C42—21no—
267C42C22×D18111yescharacteristic
268C42C22×D18111yescharacteristic
269C42C22×D18111yescharacteristic
270C23—21no—
271C23—3621no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—41no—
277Q8—14481no—
278D4—21no—
279D4—21no—
280D4—3621no—
281D4—1812no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—3621no—
287C2×C4—3621no—
288C2×C4—3621no—
289C2×C4—14481no—
290C2×C4—14481no—
291C2×C4—7242no—
292C2×C4C2×C22⋊D18111yescharacteristic
293C2×C4C2×C22⋊D18111yescharacteristic
294C2×C4C2×C22⋊D18111yescharacteristic
295C2×C4C2×C22⋊D18111yescharacteristic
296C2×C4C2×C22⋊D18111yescharacteristic
297C2×C4C2×C22⋊D18111yescharacteristic
298C2×C4C23×D18111yescharacteristic
299C8—3621no—
300C8—3621no—
301C8—3621no—
302C8—3621no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—3622no—
307C22—7241no—
308C22—7241no—
309C22C22×C22⋊D18111yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—7241no—
320C4—7241no—
321C4—7241no—
322C4#41×D18111yescharacteristic
323C4#1915386311yescharacteristic
324C2—81no—
325C2—7242no—
326C2#2604193611yescharacteristic
327C2#2604207711yescharacteristic
328C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#556×C181C2
socleC22×C181C22×D4
prime radicalC2×#15668434C22
hypercenter#556D181
hypocenterC181#3376
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #26041989C2semidirect
2. #26041647C2semidirect
3. #26041638C2semidirect
4. #26041633C2semidirect
5. #26041586C2semidirect
6. #26041579C2semidirect
7. #26041564C2semidirect
10. #26041481C2semidirect
11. #26041410C2semidirect
13. #26041377C2semidirect
14. C4×#15668436C2semidirect
15. #556×C181C2semidirect
164. C181#3376semidirect
166. #556D181semidirect

Maximal subgroups

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

subgroupindex
#260419892
#260416472
#260416382
#260416332
#260415862
#260415792
#260415642
#260414912
#260414812
#260414102
#260413892
#260413772
C4×#156684362
#556×C1812
#3376181