Finite Group Databasebeta

← #37543841

Subgroups — #37543841

Order 23168.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041826C211yesmaximal, characteristic
2C4×Q8×D181C211yesmaximal, characteristic, complemented
3#26041594C211yesmaximal, characteristic, complemented
4#26041591C211yesmaximal, characteristic, complemented
5#26041573C211yesmaximal, characteristic, complemented
6#26041572C211yesmaximal, characteristic, complemented
7#26041478C211yesmaximal, characteristic, complemented
8#26041477C211yesmaximal, characteristic, complemented
9#26041419C211yesmaximal, characteristic, complemented
10#26041416C211yesmaximal, characteristic, complemented
11#26041381C211yesmaximal, characteristic, complemented
12#26041379C211yesmaximal, characteristic, complemented
13#26041377C211yesmaximal, characteristic, complemented
14#26041334C211yesmaximal, characteristic, complemented
15#440×C181C211yesmaximal, characteristic, complemented
16C2×#15668449—21no—
17C2×Q8×D181C2211yescharacteristic
18C2×#15668438C2211yescharacteristic
19C2×#15668437C2211yescharacteristic
20C2×#15668436C2211yescharacteristic
21#19153815C2211yescharacteristic
22#19153812C2211yescharacteristic
23#19153802—21no—
24#19153791C2211yescharacteristic
25#19153789—21no—
26#19153787C2211yescharacteristic
27Q8×Dic181C2211yescharacteristic
28C4×Dic362—21no—
29C4×Dic362C2211yescharacteristic
30#19153781—21no—
31#19153781C2211yescharacteristic
32C4×D724C2211yescharacteristic
33C4×C22⋊D181—21no—
34C4⋊C4×D181—21no—
35C4⋊C4×D181C2211yescharacteristic
36C42×D181—21no—
37C42×D181C2211yescharacteristic
38#19153750C2211yescharacteristic
39#19153749C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153745C2211yescharacteristic
42#19153745C2211yescharacteristic
43#19153741C2211yescharacteristic
44#19153740C2211yescharacteristic
45#19153739C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153727C2211yescharacteristic
52C2×SD16×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54C4×D4×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#102×C181C2211yescharacteristic
58#97×C181C2211yescharacteristic
59#15668449—44no—
60C2×Dic362—21no—
61C2×Dic362—21no—
62C2×Dic362C2311yescharacteristic
63Q8×D181—22no—
64Q8×D181—22no—
65C2×D724—21no—
66C2×D724C2311yescharacteristic
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C4×D181—21no—
71C2×C4×D181—21no—
72C2×C4×D181—21no—
73C2×C4×D181D411yescharacteristic
74C2×C4×D181D411yescharacteristic
75C2×C4×D181C2311yescharacteristic
76#15668438—22no—
77#15668437—22no—
78#15668436—22no—
79C2×#13089678C2311yescharacteristic
80C2×#13089678C2311yescharacteristic
81C4⋊Dic181—21no—
82C4⋊Dic181C2311yescharacteristic
83C4⋊Dic181C2311yescharacteristic
84#15668428—21no—
85#15668428—21no—
86#15668428—21no—
87#15668428—21no—
88C22⋊Dic181—21no—
89#15668426—21no—
90#15668426—21no—
91C4×Dic181—21no—
92C4×Dic181—21no—
93C4×Dic181—21no—
94C4×Dic181D411yescharacteristic
95C4×Dic181D411yescharacteristic
96C4×Dic181C2311yescharacteristic
97C2×Q8×C181C2311yescharacteristic
98C2×D4×C181C2311yescharacteristic
99C22×C4×C181—21no—
100SD16×C181—22no—
101C2×C8×C181C2311yescharacteristic
102C2×C8×C181C2311yescharacteristic
103C4⋊C4×C181—21no—
104C4⋊C4×C181C2311yescharacteristic
105C4⋊C4×C181C2311yescharacteristic
106C22⋊C4×C181—21no—
107C42×C181—21no—
108C42×C181C2311yescharacteristic
109C22×D181—21no—
110C22×D181C2×D411yescharacteristic
111Dic362—24no—
112Dic362—21no—
113Dic362—42no—
114Dic362#4112yescomplemented
115D724—21no—
116D724—21no—
117D724—42no—
118C22⋊D181—42no—
119C22⋊D181—42no—
120C22⋊D181—42no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181C2×D411yescharacteristic
125C2×Dic181C2×D411yescharacteristic
126C2×Dic181C2×D411yescharacteristic
127C4×D181—24no—
128C4×D181—22no—
129C4×D181—21no—
130C4×D181—21no—
131C4×D181—22no—
132C4×D181—42no—
133C4×D181—42no—
134#13089678—21no—
135#13089678—21no—
136C23×C181—21no—
137Q8×C181—21no—
138Q8×C181#4112yescomplemented
139D4×C181—21no—
140D4×C181—21no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—42no—
145C2×C4×C181C2×D411yescharacteristic
146C2×C4×C181C2×D411yescharacteristic
147C2×C4×C181C2411yescharacteristic
148C8×C181—21no—
149C8×C181—21no—
150C22×C181—41no—
151C22×C181—41no—
152C22×C181—41no—
153C22×C181C22×D411yescharacteristic
154C2×D181—21no—
155C2×D181—21no—
156C2×D181—41no—
157C2×D181—41no—
158C2×D181—41no—
159C2×D181AGL(1,ℤ/8)12yes—
160C4×C181—22no—
161C4×C181—21no—
162C4×C181—21no—
163C4×C181—41no—
164C4×C181—41no—
165C4×C181C2×#4111yescharacteristic
166C4×C181C2×#4111yescharacteristic
167Dic181—22no—
168Dic181—21no—
169Dic181—21no—
170Dic181—41no—
171Dic181—41no—
172Dic181#13712yes—
173C2×C181—81no—
174C2×C181#52511yescharacteristic
175C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C181C2×#13711yescharacteristic
177D181—22no—
178D181—81no—
179C181#311211yescharacteristic, complemented, Sylow
180#3112—1811nomaximal, Sylow
181#532—1811no—
182C2×C4×Q8—1811no—
183#480—1811no—
184#480—1811no—
185#477—1811no—
186#477—1811no—
187#460—1811no—
188#460—1811no—
189#448—1811no—
190#448—1811no—
191#440—1811no—
192#440—1811no—
193#440—1811no—
194#440D18111yescharacteristic, complemented
195#432—1811no—
196C2×#41—3621no—
197C22×Q8—1811no—
198C2×SD16—1811no—
199C2×SD16—1811no—
200C2×SD16—1811no—
201C2×SD16C2×D18111yescharacteristic
202C4⋊Q8—1811no—
203#127—1811no—
204#124—3621no—
205#121—3621no—
206#121—1811no—
207#121—1811no—
208C4×Q8—3621no—
209C4×Q8—3621no—
210C4×Q8—1811no—
211C4×Q8—1811no—
212C4×Q8—1811no—
213C4×Q8C2×D18111yescharacteristic
214C4×D4—3621no—
215C4×D4—1811no—
216C4×D4C2×D18111yescharacteristic
217C2×C4⋊C4—3621no—
218C2×C4⋊C4—1811no—
219C2×C42—3621no—
220C2×C42—1811no—
221#107—1811no—
222#107—1811no—
223#107—1811no—
224#107C2×D18111yescharacteristic
225C4⋊C8—1811no—
226C4⋊C8—1811no—
227#103—1811no—
228#103—1811no—
229#103—1811no—
230#103C2×D18111yescharacteristic
231#102—1811no—
232#102—1811no—
233#102—1811no—
234#102C2×D18111yescharacteristic
235C22⋊C8—1811no—
236C22⋊C8—1811no—
237#97—1811no—
238#97C2×D18111yescharacteristic
239#41—7244no—
240C2×Q8—3621no—
241C2×Q8—3621no—
242C2×Q8—3622no—
243C2×Q8—3622no—
244C2×Q8—1811no—
245C2×Q8C22×D18111yescharacteristic
246C2×D4—3621no—
247C2×D4—3621no—
248C2×D4—3621no—
249C2×D4—3621no—
250C2×D4—1811no—
251C2×D4C22×D18111yescharacteristic
252C22×C4—21no—
253C22×C4—3621no—
254C22×C4—3621no—
255C22×C4—3621no—
256C22×C4—1811no—
257C22×C4—1811no—
258C22×C4—1811no—
259SD16—22no—
260SD16—3622no—
261SD16—3622no—
262SD16—3622no—
263C2×C8—1811no—
264C2×C8—1811no—
265C2×C8C22×D18111yescharacteristic
266C2×C8C22×D18111yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—3621no—
269C4⋊C4—3621no—
270C4⋊C4—3621no—
271C4⋊C4—3621no—
272C4⋊C4—3621no—
273C4⋊C4—1811no—
274C4⋊C4—1811no—
275C4⋊C4C22×D18111yescharacteristic
276C4⋊C4C22×D18111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3621no—
279C22⋊C4—3621no—
280C22⋊C4—3621no—
281C42—21no—
282C42—3621no—
283C42—3621no—
284C42—3621no—
285C42—1811no—
286C42—1811no—
287C42—1811no—
288C42C22×D18111yescharacteristic
289C23—21no—
290C23—3621no—
291C23—1811no—
292Q8—21no—
293Q8—3624no—
294Q8—3621no—
295Q8—1812no—
296Q8—7242no—
297Q8#1566844912yescomplemented
298D4—21no—
299D4—21no—
300D4—3621no—
301D4—3621no—
302D4—7242no—
303D4—7242no—
304D4—7242no—
305D4—7242no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—3624no—
311C2×C4—3622no—
312C2×C4—3622no—
313C2×C4—3621no—
314C2×C4—3621no—
315C2×C4—3621no—
316C2×C4—3621no—
317C2×C4—3621no—
318C2×C4—1811no—
319C2×C4—1811no—
320C2×C4—1811no—
321C2×C4—7242no—
322C2×C4—7242no—
323C2×C4D4×D18111yescharacteristic
324C2×C4D4×D18111yescharacteristic
325C2×C4C23×D18111yescharacteristic
326C8—21no—
327C8—21no—
328C8—3621no—
329C8—3621no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—3621no—
334C22—3621no—
335C22—1812no—
336C22—7241no—
337C22—7241no—
338C22—7241no—
339C22C2×D4×D18111yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—3622no—
346C4—3621no—
347C4—3621no—
348C4—1812no—
349C4—7241no—
350C4—7241no—
351C4#41×D18111yescharacteristic
352C4C2×#1566844911yescharacteristic
353C2—81no—
354C2—3622no—
355C2—14481no—
356C2#2604178811yescharacteristic
357C2AGL(1,ℤ/8)×D18111yescharacteristic
358C2#137×D18111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#440×C181C2
socleC22×C181C22×D4
prime radical#26041826C2
hypercenter#440D181
hypocenterC181#3112
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D181C2semidirect
3. #26041594C2semidirect
4. #26041591C2semidirect
5. #26041573C2semidirect
6. #26041572C2semidirect
7. #26041478C2semidirect
8. #26041477C2semidirect
9. #26041419C2semidirect
10. #26041416C2semidirect
11. #26041381C2semidirect
12. #26041379C2semidirect
13. #26041377C2semidirect
14. #26041334C2semidirect
15. #440×C181C2semidirect
114. Dic362#41semidirect
138. Q8×C181#41semidirect
179. C181#3112semidirect
194. #440D181semidirect
297. Q8#15668449semidirect

Maximal subgroups

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

subgroupindex
#260418262
C4×Q8×D1812
#260415942
#260415912
#260415732
#260415722
#260414782
#260414772
#260414192
#260414162
#260413812
#260413792
#260413772
#260413342
#440×C1812
#3112181