Finite Group Databasebeta

← #37546010

Subgroups — #37546010

Order 23168.

386 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041993C211yesmaximal, characteristic
2#26041942C211yesmaximal, characteristic, complemented
3#26041628C211yesmaximal, characteristic, complemented
4#26041622C211yesmaximal, characteristic, complemented
5#26041617C211yesmaximal, characteristic, complemented
6#26041573C211yesmaximal, characteristic, complemented
7#26041524C211yesmaximal, characteristic, complemented
8#26041482C211yesmaximal, characteristic, complemented
9#26041417C211yesmaximal, characteristic, complemented
10#26041416C211yesmaximal, characteristic, complemented
11#26041370C211yesmaximal, characteristic, complemented
12#26041355C211yesmaximal, characteristic, complemented
13#26041289C211yesmaximal, characteristic, complemented
14#26041286C211yesmaximal, characteristic, complemented
15#468×C181C211yesmaximal, characteristic, complemented
16C22×Dic362C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18C2×#15668437C2211yescharacteristic
19C2×#15668437C2211yescharacteristic
20C2×#15668432C2211yescharacteristic
21#19153815—21no—
22#19153815C2211yescharacteristic
23#19153814C2211yescharacteristic
24#19153803—21no—
25#19153802—21no—
26#19153796C2211yescharacteristic
27#19153794—21no—
28#19153793—21no—
29#19153788—21no—
30#19153787C2211yescharacteristic
31#19153781C2211yescharacteristic
32D4×Dic181C2211yescharacteristic
33C4×C22⋊D181C2211yescharacteristic
34#19153776—21no—
35#19153775C2211yescharacteristic
36#19153774C2211yescharacteristic
37#19153772—21no—
38#19153749C2211yescharacteristic
39#19153747C2211yescharacteristic
40#19153745C2211yescharacteristic
41#19153741C2211yescharacteristic
42#19153741C2211yescharacteristic
43#19153741C2211yescharacteristic
44#19153741C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153740C2211yescharacteristic
47#19153739C2211yescharacteristic
48#19153739C2211yescharacteristic
49C22⋊#13089678C2211yescharacteristic
50#19153728C2211yescharacteristic
51C8×Dic181C2211yescharacteristic
52C2×SD16×C181C2211yescharacteristic
53C2×M16×C181C2211yescharacteristic
54C22⋊Q8×C181C2211yescharacteristic
55#121×C181C2211yescharacteristic
56#107×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic
59C2×Dic362—21no—
60C2×Dic362—21no—
61C2×Dic362—22no—
62C2×Dic362—21no—
63C2×Dic362—41no—
64C2×Dic362C2311yescharacteristic
65C2×Dic362C2311yescharacteristic
66Q8×D181—22no—
67Q8×D181—41no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181D411yescharacteristic
70C22×Dic181—21no—
71C22×Dic181—21no—
72C2×C4×D181—21no—
73C2×C4×D181C2311yescharacteristic
74#15668437—22no—
75#15668437—41no—
76#15668432—41no—
77C2×#13089678C2311yescharacteristic
78C2×#13089678C2311yescharacteristic
79C4⋊Dic181—21no—
80C4⋊Dic181C2311yescharacteristic
81C4⋊Dic181C2311yescharacteristic
82#15668428—21no—
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86#15668428D411yescharacteristic
87C22⋊Dic181—21no—
88C22⋊Dic181—21no—
89C22⋊Dic181—21no—
90C22⋊Dic181D411yescharacteristic
91#15668426—21no—
92#15668426—21no—
93#15668426D411yescharacteristic
94C4×Dic181—21no—
95C4×Dic181—21no—
96C4×Dic181—21no—
97C4×Dic181C2311yescharacteristic
98C2×Q8×C181C2311yescharacteristic
99C2×D4×C181—21no—
100C2×D4×C181C2311yescharacteristic
101C22×C4×C181C2311yescharacteristic
102SD16×C181—41no—
103M16×C181—41no—
104C2×C8×C181C2311yescharacteristic
105C2×C8×C181C2311yescharacteristic
106C4⋊C4×C181—21no—
107C4⋊C4×C181C2311yescharacteristic
108C4⋊C4×C181C2311yescharacteristic
109C22⋊C4×C181—21no—
110C22⋊C4×C181—21no—
111C22×D181C2×D411yescharacteristic
112Dic362—21no—
113Dic362—21no—
114Dic362—21no—
115Dic362—42no—
116Dic362—42no—
117Dic362—81no—
118Dic362#4112yes—
119C22⋊D181—22no—
120C22⋊D181—81no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—21no—
125C2×Dic181—42no—
126C2×Dic181—42no—
127C2×Dic181C2×D411yescharacteristic
128C2×Dic181C2×D411yescharacteristic
129C2×Dic181C2×D411yescharacteristic
130C4×D181—21no—
131C4×D181—21no—
132C4×D181—42no—
133#13089678—21no—
134#13089678—21no—
135C23×C181—21no—
136C23×C181C2×D411yescharacteristic
137Q8×C181—21no—
138Q8×C181—21no—
139D4×C181—21no—
140D4×C181—21no—
141D4×C181—81no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2411yescharacteristic
149C8×C181—21no—
150C8×C181—21no—
151C22×C181—21no—
152C22×C181—21no—
153C22×C181—21no—
154C22×C181—41no—
155C22×C181—41no—
156C22×C181—41no—
157C22×C181C22×D411yescharacteristic
158C2×D181—21no—
159C2×D181—21no—
160C2×D181—21no—
161C4×C181—21no—
162C4×C181—41no—
163C4×C181—41no—
164C4×C181—41no—
165C4×C181C2×#4111yescharacteristic
166C4×C181H(5,𝔽2)11yescharacteristic
167Dic181—21no—
168Dic181—22no—
169Dic181—21no—
170Dic181—41no—
171Dic181—41no—
172Dic181—41no—
173Dic181#13712yes—
174C2×C181—41no—
175C2×C181—81no—
176C2×C181#54611yescharacteristic
177C2×C181C2×#13711yescharacteristic
178C2×C181#57711yescharacteristic
179D181—41no—
180C181#332911yescharacteristic, complemented, Sylow
181#3329—1811nomaximal, Sylow
182#555—1811no—
183D4×Q8—1811no—
184#487—1811no—
185#485—1811no—
186#485—1811no—
187#477—1811no—
188#468—1811no—
189#468D18111yescharacteristic, complemented
190#461—1811no—
191#448—1811no—
192#448—1811no—
193C4×SD16—1811no—
194#436—1811no—
195#419—1811no—
196#419—1811no—
197C22×Q8—1811no—
198C22×Q8—1811no—
199C2×SD16—1811no—
200C2×SD16—1811no—
201C2×SD16C2×D18111yescharacteristic
202C2×M16—1811no—
203C2×M16C2×D18111yescharacteristic
204C4⋊Q8—3621no—
205C4⋊Q8—1811no—
206C4⋊Q8—1811no—
207#124—3621no—
208#124—3621no—
209C22⋊Q8—3621no—
210C22⋊Q8—3621no—
211C22⋊Q8—1811no—
212C22⋊Q8C2×D18111yescharacteristic
213#121—3621no—
214#121—1811no—
215#121C2×D18111yescharacteristic
216C4×Q8—1811no—
217C4×D4—3621no—
218C4×D4—1811no—
219C4×D4—1811no—
220#117—3621no—
221#117—1811no—
222#117—1811no—
223#107—1811no—
224#107C2×D18111yescharacteristic
225#106—1811no—
226C4⋊C8—1811no—
227#103—1811no—
228#103—1811no—
229#103—1811no—
230#103—1811no—
231#103—1811no—
232#103—1811no—
233#103—1811no—
234#103—1811no—
235#103C2×D18111yescharacteristic
236#102C2×D18111yescharacteristic
237C22⋊C8—1811no—
238C22⋊C8—1811no—
239C4×C8—1811no—
240C2×Q8—3621no—
241C2×Q8—3621no—
242C2×Q8—3622no—
243C2×Q8—3621no—
244C2×Q8—3622no—
245C2×Q8—1811no—
246C2×Q8—1811no—
247C2×Q8—7241no—
248C2×Q8—7241no—
249C2×Q8C22×D18111yescharacteristic
250C2×D4—21no—
251C2×D4—3621no—
252C2×D4—1811no—
253C2×D4C22×D18111yescharacteristic
254C22×C4—3621no—
255C22×C4—3621no—
256C22×C4—3621no—
257C22×C4—1811no—
258C22×C4C22×D18111yescharacteristic
259SD16—41no—
260SD16—3622no—
261SD16—7241no—
262M16—41no—
263M16—7241no—
264C2×C8—1811no—
265C2×C8—1811no—
266C2×C8C22×D18111yescharacteristic
267C2×C8C22×D18111yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—3621no—
270C4⋊C4—3621no—
271C4⋊C4—3621no—
272C4⋊C4—3621no—
273C4⋊C4—3621no—
274C4⋊C4—1811no—
275C4⋊C4—1811no—
276C4⋊C4—1811no—
277C4⋊C4C22×D18111yescharacteristic
278C4⋊C4C22×D18111yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—3621no—
282C22⋊C4—3621no—
283C22⋊C4—3621no—
284C22⋊C4—3621no—
285C22⋊C4—3621no—
286C22⋊C4—1811no—
287C22⋊C4—1811no—
288C42—3621no—
289C42—3621no—
290C42—3621no—
291C42—1811no—
292C23—21no—
293C23—1811no—
294C23D4×D18111yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—3621no—
298Q8—3621no—
299Q8—3621no—
300Q8—14481no—
301Q8—1812no—
302Q8—7242no—
303Q8—7242no—
304D4—21no—
305D4—21no—
306D4—81no—
307D4—3622no—
308D4—14481no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—3621no—
315C2×C4—3621no—
316C2×C4—3621no—
317C2×C4—3621no—
318C2×C4—3621no—
319C2×C4—3621no—
320C2×C4—1811no—
321C2×C4—1811no—
322C2×C4—1811no—
323C2×C4—7242no—
324C2×C4—7242no—
325C2×C4—7242no—
326C2×C4D4×D18111yescharacteristic
327C2×C4C23×D18111yescharacteristic
328C8—21no—
329C8—21no—
330C8—3621no—
331C8—3621no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—3621no—
339C22—3621no—
340C22—3621no—
341C22C2×D4×D18111yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3621no—
347C4—3622no—
348C4—3621no—
349C4—1812no—
350C4—7241no—
351C4—7241no—
352C4—7241no—
353C4#41×D18111yescharacteristic
354C4#1915386111yescharacteristic
355C2—41no—
356C2—81no—
357C2—7241no—
358C2#2604191811yescharacteristic
359C2#137×D18111yescharacteristic
360C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#468×C181C2
socleC22×C181C22×D4
prime radical#26041993C2
hypercenter#468D181
hypocenterC181#3329
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #26041942C2semidirect
3. #26041628C2semidirect
4. #26041622C2semidirect
5. #26041617C2semidirect
6. #26041573C2semidirect
7. #26041524C2semidirect
8. #26041482C2semidirect
9. #26041417C2semidirect
10. #26041416C2semidirect
11. #26041370C2semidirect
12. #26041355C2semidirect
13. #26041289C2semidirect
14. #26041286C2semidirect
15. #468×C181C2semidirect
180. C181#3329semidirect
189. #468D181semidirect

Maximal subgroups

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

subgroupindex
#260419932
#260419422
#260416282
#260416222
#260416172
#260415732
#260415242
#260414822
#260414172
#260414162
#260413702
#260413552
#260412892
#260412862
#468×C1812
#3329181