Finite Group Databasebeta

← #37545709

Subgroups — #37545709

Order 23168.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041941C211yesmaximal, characteristic, complemented
2#26041932C211yesmaximal, characteristic, complemented
3#26041641C211yesmaximal, characteristic, complemented
4#26041616C211yesmaximal, characteristic, complemented
5#26041594C211yesmaximal, characteristic, complemented
6#26041526C211yesmaximal, characteristic, complemented
7#26041516C211yesmaximal, characteristic, complemented
8#26041495C211yesmaximal, characteristic, complemented
9#26041427C211yesmaximal, characteristic, complemented
10#26041419C211yesmaximal, characteristic, complemented
11#26041377C211yesmaximal, characteristic, complemented
12#26041349C211yesmaximal, characteristic, complemented
13#26041279C211yesmaximal, characteristic, complemented
14#103×D181C211yesmaximal, characteristic, complemented
15#477×C181C211yesmaximal, characteristic, complemented
16C2×#15668450—21no—
17C2×#15668450C2211yescharacteristic
18C2×Q8×D181C2211yescharacteristic
19C2×#15668438C2211yescharacteristic
20C2×#15668436C2211yescharacteristic
21C2×D1448C2211yescharacteristic
22C2×#15668432C2211yescharacteristic
23C2×C8×D181C2211yescharacteristic
24#19153804—21no—
25#19153804C2211yescharacteristic
26#19153801—21no—
27#19153801—21no—
28#19153796—21no—
29#19153796C2211yescharacteristic
30#19153792—21no—
31#19153791—21no—
32#19153791C2211yescharacteristic
33#19153791C2211yescharacteristic
34#19153791C2211yescharacteristic
35C4×D724—21no—
36C4×D724C2211yescharacteristic
37C4×D724C2211yescharacteristic
38C4⋊C4×D181C2211yescharacteristic
39C4⋊C4×D181C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153741C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153739C2211yescharacteristic
44#19153739C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153736C2211yescharacteristic
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic
49#19153728C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153727C2211yescharacteristic
52C4⋊Q8×C181C2211yescharacteristic
53C4×Q8×C181C2211yescharacteristic
54#107×C181C2211yescharacteristic
55#107×C181C2211yescharacteristic
56C4⋊C8×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58#103×C181C2211yescharacteristic
59#15668450—22no—
60#15668450—41no—
61#15668450—44no—
62C2×Dic362—21no—
63C2×Dic362C2311yescharacteristic
64Q8×D181—22no—
65Q8×D181—41no—
66C2×D724—21no—
67C2×D724—21no—
68C2×D724—21no—
69C2×D724—21no—
70C2×D724D411yescharacteristic
71C2×D724C2311yescharacteristic
72C2×D724C2311yescharacteristic
73C2×C4×D181—21no—
74C2×C4×D181—21no—
75C2×C4×D181—21no—
76C2×C4×D181—21no—
77C2×C4×D181—21no—
78C2×C4×D181C2311yescharacteristic
79C2×C4×D181C2311yescharacteristic
80#15668438—41no—
81#15668436—22no—
82D1448—41no—
83#15668432—41no—
84C2×#13089678C2311yescharacteristic
85C2×#13089678C2311yescharacteristic
86C8×D181—41no—
87C4⋊Dic181—21no—
88C4⋊Dic181D411yescharacteristic
89C4⋊Dic181C2311yescharacteristic
90#15668428—21no—
91#15668428—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97#15668426—21no—
98#15668426—21no—
99#15668426D411yescharacteristic
100#15668426D411yescharacteristic
101C2×Q8×C181—21no—
102C2×Q8×C181C2311yescharacteristic
103C2×C8×C181C2311yescharacteristic
104C2×C8×C181C2311yescharacteristic
105C4⋊C4×C181—21no—
106C4⋊C4×C181—21no—
107C4⋊C4×C181C2311yescharacteristic
108C4⋊C4×C181C2311yescharacteristic
109C4⋊C4×C181C2311yescharacteristic
110C42×C181—21no—
111C42×C181C2311yescharacteristic
112C22×D181—21no—
113C22×D181—21no—
114C22×D181C2×D411yescharacteristic
115C22×D181C2×D411yescharacteristic
116Dic362—21no—
117Dic362—21no—
118Dic362—42no—
119D724—22no—
120D724—21no—
121D724—21no—
122D724—21no—
123D724—21no—
124D724—42no—
125D724—42no—
126D724—42no—
127D724—42no—
128C2×Dic181—21no—
129C2×Dic181—21no—
130C2×Dic181C2×D411yescharacteristic
131C2×Dic181C2×D411yescharacteristic
132C4×D181—21no—
133C4×D181—21no—
134C4×D181—21no—
135C4×D181—21no—
136C4×D181—42no—
137C4×D181—42no—
138C4×D181—42no—
139C4×D181—42no—
140C4×D181—42no—
141#13089678—21no—
142#13089678—21no—
143Q8×C181—21no—
144Q8×C181—42no—
145Q8×C181#4112yescomplemented
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181C2×D411yescharacteristic
151C2×C4×C181C2×D411yescharacteristic
152C2×C4×C181C2411yescharacteristic
153C8×C181—21no—
154C8×C181—21no—
155C22×C181C22×D411yescharacteristic
156C2×D181—21no—
157C2×D181—21no—
158C2×D181—21no—
159C2×D181—21no—
160C2×D181—21no—
161C2×D181—41no—
162C2×D181—41no—
163C2×D181—41no—
164C2×D181—41no—
165C2×D181—41no—
166C2×D181—41no—
167C2×D181AGL(1,ℤ/8)12yes—
168C4×C181—22no—
169C4×C181—21no—
170C4×C181—21no—
171C4×C181—41no—
172C4×C181—41no—
173C4×C181—41no—
174C4×C181C2×#4111yescharacteristic
175C4×C181H(5,𝔽2)11yescharacteristic
176Dic181—21no—
177Dic181—21no—
178Dic181—41no—
179Dic181—41no—
180C2×C181#54611yescharacteristic
181C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
182C2×C181#57811yescharacteristic
183D181—22no—
184D181—41no—
185D181—81no—
186D181—81no—
187C181#330811yescharacteristic, complemented, Sylow
188#3308—1811nomaximal, Sylow
189#548—1811no—
190#547—1811no—
191#489—1811no—
192#484—1811no—
193#480—1811no—
194#477D18111yescharacteristic, complemented
195#468—1811no—
196#466—1811no—
197#463—1811no—
198#449—1811no—
199#448—1811no—
200#440—1811no—
201#435—1811no—
202#417—1811no—
203C2×#103—1811no—
204C2×#41—3621no—
205C2×#41—1811no—
206C22×Q8—1811no—
207C2×SD16—1811no—
208C2×SD16—1811no—
209C2×D8—1811no—
210C2×M16—1811no—
211C22×C8—1811no—
212C4⋊Q8C2×D18111yescharacteristic
213#124—3621no—
214#124—1811no—
215#123—3621no—
216#123—3621no—
217C22⋊Q8—3621no—
218C22⋊Q8—3621no—
219C22⋊Q8—1811no—
220#121—3621no—
221#121—1811no—
222#121—1811no—
223#121—1811no—
224C4×Q8C2×D18111yescharacteristic
225C4×D4—3621no—
226C4×D4—1811no—
227C4×D4—1811no—
228C2×C4⋊C4—1811no—
229C2×C4⋊C4—1811no—
230#107—1811no—
231#107C2×D18111yescharacteristic
232#107C2×D18111yescharacteristic
233C4⋊C8C2×D18111yescharacteristic
234#103—1811no—
235#103—1811no—
236#103—1811no—
237#103—1811no—
238#103C2×D18111yescharacteristic
239#103C2×D18111yescharacteristic
240#102—1811no—
241#102—1811no—
242#102—1811no—
243#102—1811no—
244C22⋊C8—1811no—
245C22⋊C8—1811no—
246#97—1811no—
247#41—3622no—
248#41—7241no—
249#41—7244no—
250C2×Q8—21no—
251C2×Q8—3621no—
252C2×Q8—3622no—
253C2×Q8—1811no—
254C2×Q8—7241no—
255C2×Q8C22×D18111yescharacteristic
256C2×D4—3621no—
257C2×D4—3621no—
258C2×D4—3621no—
259C2×D4—3621no—
260C2×D4—1811no—
261C2×D4—1811no—
262C2×D4—1811no—
263C22×C4—3621no—
264C22×C4—3621no—
265C22×C4—3621no—
266C22×C4—3621no—
267C22×C4—3621no—
268C22×C4—1811no—
269C22×C4—1811no—
270SD16—3622no—
271SD16—7241no—
272D8—7241no—
273M16—7241no—
274C2×C8—1811no—
275C2×C8—1811no—
276C2×C8—7241no—
277C2×C8C22×D18111yescharacteristic
278C2×C8C22×D18111yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—3621no—
282C4⋊C4—3621no—
283C4⋊C4—3621no—
284C4⋊C4—1811no—
285C4⋊C4—1811no—
286C4⋊C4C22×D18111yescharacteristic
287C4⋊C4C22×D18111yescharacteristic
288C4⋊C4C22×D18111yescharacteristic
289C22⋊C4—3621no—
290C22⋊C4—3621no—
291C22⋊C4—3621no—
292C22⋊C4—3621no—
293C22⋊C4—3621no—
294C22⋊C4—3621no—
295C22⋊C4—3621no—
296C22⋊C4—1811no—
297C22⋊C4—1811no—
298C42—21no—
299C42C22×D18111yescharacteristic
300C23—3621no—
301C23—3621no—
302C23—1811no—
303C23—1811no—
304Q8—21no—
305Q8—42no—
306Q8—3621no—
307Q8—3621no—
308Q8—7242no—
309Q8#1566845012yescomplemented
310D4—3622no—
311D4—3621no—
312D4—3621no—
313D4—3621no—
314D4—3621no—
315D4—7242no—
316D4—7242no—
317D4—7242no—
318D4—7242no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—3621no—
324C2×C4—3621no—
325C2×C4—3621no—
326C2×C4—3621no—
327C2×C4—3621no—
328C2×C4—3621no—
329C2×C4—1811no—
330C2×C4—1811no—
331C2×C4—7242no—
332C2×C4—7242no—
333C2×C4—7242no—
334C2×C4—7242no—
335C2×C4—7242no—
336C2×C4D4×D18111yescharacteristic
337C2×C4D4×D18111yescharacteristic
338C2×C4C23×D18111yescharacteristic
339C8—21no—
340C8—21no—
341C8—3621no—
342C8—3621no—
343C22—3621no—
344C22—3621no—
345C22—3621no—
346C22—3621no—
347C22—3621no—
348C22—1812no—
349C22—7241no—
350C22—7241no—
351C22—7241no—
352C22—7241no—
353C22—7241no—
354C22—7241no—
355C22C2×D4×D18111yescharacteristic
356C4—22no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—3621no—
363C4—3621no—
364C4—7241no—
365C4—7241no—
366C4C2×#1566845011yescharacteristic
367C4#1915386211yescharacteristic
368C2—3622no—
369C2—14481no—
370C2—14481no—
371C2—7241no—
372C2#2604192011yescharacteristic
373C2AGL(1,ℤ/8)×D18111yescharacteristic
374C2#2604209211yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#477×C181C2
socleC22×C181C22×D4
hypercenter#477D181
hypocenterC181#3308
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #26041941C2semidirect
2. #26041932C2semidirect
3. #26041641C2semidirect
4. #26041616C2semidirect
5. #26041594C2semidirect
6. #26041526C2semidirect
7. #26041516C2semidirect
8. #26041495C2semidirect
9. #26041427C2semidirect
10. #26041419C2semidirect
11. #26041377C2semidirect
12. #26041349C2semidirect
13. #26041279C2semidirect
14. #103×D181C2semidirect
15. #477×C181C2semidirect
145. Q8×C181#41semidirect
187. C181#3308semidirect
194. #477D181semidirect
309. Q8#15668450semidirect

Maximal subgroups

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

subgroupindex
#260419412
#260419322
#260416412
#260416162
#260415942
#260415262
#260415162
#260414952
#260414272
#260414192
#260413772
#260413492
#260412792
#103×D1812
#477×C1812
#3308181