Finite Group Databasebeta

← #37545712

Subgroups — #37545712

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#26041933C211yesmaximal, characteristic, complemented
2#26041638C211yesmaximal, characteristic, complemented
3#26041612C211yesmaximal, characteristic, complemented
4#26041590C211yesmaximal, characteristic, complemented
5#26041574C211yesmaximal, characteristic, complemented
6#26041527C211yesmaximal, characteristic, complemented
7#26041513C211yesmaximal, characteristic, complemented
8#26041491C211yesmaximal, characteristic, complemented
9#26041424C211yesmaximal, characteristic, complemented
10#26041418C211yesmaximal, characteristic, complemented
11#26041377C211yesmaximal, characteristic, complemented
12#26041351C211yesmaximal, characteristic, complemented
13#26041280C211yesmaximal, characteristic, complemented
14C2×#19153740C211yesmaximal, characteristic, complemented
15#548×C181C211yesmaximal, characteristic, complemented
16C2×#15668448—21no—
17C2×#15668436C2211yescharacteristic
18C2×#15668436C2211yescharacteristic
19C2×#15668434C2211yescharacteristic
20C2×#15668433C2211yescharacteristic
21C22×#13089678C2211yescharacteristic
22#19153816C2211yescharacteristic
23#19153800—21no—
24#19153800—21no—
25#19153796—21no—
26#19153790C2211yescharacteristic
27#19153790C2211yescharacteristic
28C4×D724C2211yescharacteristic
29C2×C4⋊Dic181C2211yescharacteristic
30C2×C4⋊Dic181C2211yescharacteristic
31#19153749C2211yescharacteristic
32#19153749C2211yescharacteristic
33#19153749C2211yescharacteristic
34C4⋊#13089678C2211yescharacteristic
35#19153740C2211yescharacteristic
36#19153740C2211yescharacteristic
37#19153740C2211yescharacteristic
38#19153740C2211yescharacteristic
39#19153740C2211yescharacteristic
40#19153740C2211yescharacteristic
41#19153737C2211yescharacteristic
42#19153737C2211yescharacteristic
43#19153737C2211yescharacteristic
44#19153736C2211yescharacteristic
45C22⋊#13089678C2211yescharacteristic
46C22⋊#13089678C2211yescharacteristic
47#19153727C2211yescharacteristic
48C2×#41×C181C2211yescharacteristic
49C22×Q8×C181C2211yescharacteristic
50#124×C181—21no—
51#124×C181C2211yescharacteristic
52C22⋊Q8×C181—21no—
53C22⋊Q8×C181C2211yescharacteristic
54#121×C181—21no—
55#121×C181C2211yescharacteristic
56C4×Q8×C181C2211yescharacteristic
57C4×D4×C181—21no—
58C4×D4×C181C2211yescharacteristic
59#15668448—44no—
60C2×Dic362—21no—
61C2×D724C2311yescharacteristic
62C2×C22⋊D181—21no—
63C2×C22⋊D181—21no—
64C22×Dic181—21no—
65C22×Dic181—21no—
66C2×C4×D181—21no—
67#15668436—22no—
68#15668436—41no—
69#15668434—41no—
70#15668433—41no—
71C2×#13089678—41no—
72C2×#13089678C2311yescharacteristic
73C2×#13089678C2311yescharacteristic
74C2×#13089678C2311yescharacteristic
75C2×#13089678C2311yescharacteristic
76C4⋊Dic181—21no—
77C4⋊Dic181—21no—
78C4⋊Dic181—21no—
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81C4⋊Dic181C2311yescharacteristic
82#15668426—21no—
83#15668426—21no—
84#15668426—21no—
85#41×C181—22no—
86#41×C181—41no—
87C2×Q8×C181—21no—
88C2×Q8×C181—22no—
89C2×Q8×C181—41no—
90C2×Q8×C181C2311yescharacteristic
91C2×Q8×C181C2311yescharacteristic
92C2×D4×C181—21no—
93C2×D4×C181—21no—
94C2×D4×C181D411yescharacteristic
95C2×D4×C181C2311yescharacteristic
96C22×C4×C181—21no—
97C22×C4×C181—21no—
98C22×C4×C181C2311yescharacteristic
99C22×C4×C181C2311yescharacteristic
100C4⋊C4×C181—21no—
101C4⋊C4×C181—21no—
102C4⋊C4×C181D411yescharacteristic
103C4⋊C4×C181C2311yescharacteristic
104C22⋊C4×C181—21no—
105C22⋊C4×C181—21no—
106C22⋊C4×C181—21no—
107C22⋊C4×C181—21no—
108C22⋊C4×C181D411yescharacteristic
109C22⋊C4×C181D411yescharacteristic
110C42×C181—21no—
111C42×C181C2311yescharacteristic
112C22×D181—21no—
113Dic362—42no—
114D724—21no—
115D724—21no—
116C22⋊D181—42no—
117C22⋊D181—42no—
118C2×Dic181—21no—
119C2×Dic181—21no—
120C2×Dic181—21no—
121C2×Dic181—42no—
122C2×Dic181—42no—
123C4×D181—42no—
124#13089678—21no—
125#13089678—21no—
126#13089678—21no—
127#13089678—21no—
128C23×C181—21no—
129C23×C181C2×D411yescharacteristic
130C23×C181C2×D411yescharacteristic
131Q8×C181—21no—
132Q8×C181—21no—
133Q8×C181—21no—
134Q8×C181—42no—
135Q8×C181#4112yescomplemented
136D4×C181—22no—
137D4×C181—21no—
138D4×C181—21no—
139D4×C181—42no—
140D4×C181—42no—
141C2×C4×C181—21no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—42no—
149C2×C4×C181—42no—
150C2×C4×C181C2×D411yescharacteristic
151C2×C4×C181C2×D411yescharacteristic
152C2×C4×C181C2×D411yescharacteristic
153C2×C4×C181C2×D411yescharacteristic
154C2×C4×C181C2411yescharacteristic
155C22×C181—21no—
156C22×C181—21no—
157C22×C181—21no—
158C22×C181—21no—
159C22×C181—21no—
160C22×C181—41no—
161C22×C181—41no—
162C22×C181—41no—
163C22×C181AGL(1,ℤ/8)12yes—
164C22×C181C22×D411yescharacteristic
165C2×D181—41no—
166C2×D181—41no—
167C2×D181—41no—
168C4×C181—21no—
169C4×C181—21no—
170C4×C181—22no—
171C4×C181—21no—
172C4×C181—21no—
173C4×C181—41no—
174C4×C181—41no—
175C4×C181C2×#4111yescharacteristic
176C4×C181H(5,𝔽2)11yescharacteristic
177Dic181—41no—
178Dic181—41no—
179Dic181—41no—
180C2×C181—22no—
181C2×C181—41no—
182C2×C181—81no—
183C2×C181#54611yescharacteristic
184C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C181#57811yescharacteristic
186D181—81no—
187C181#330811yescharacteristic, complemented, Sylow
188#3308—1811nomaximal, Sylow
189#548D18111yescharacteristic, complemented
190#547—1811no—
191#489—1811no—
192#484—1811no—
193#480—1811no—
194#477—1811no—
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×#41C2×D18111yescharacteristic
206C22×Q8C2×D18111yescharacteristic
207C2×SD16—1811no—
208C2×SD16—1811no—
209C2×D8—1811no—
210C2×M16—1811no—
211C22×C8—1811no—
212C4⋊Q8—1811no—
213#124—21no—
214#124C2×D18111yescharacteristic
215#123—3621no—
216#123—3621no—
217C22⋊Q8—21no—
218C22⋊Q8—3621no—
219C22⋊Q8C2×D18111yescharacteristic
220#121—21no—
221#121—1811no—
222#121—1811no—
223#121C2×D18111yescharacteristic
224C4×Q8C2×D18111yescharacteristic
225C4×D4—21no—
226C4×D4—1811no—
227C4×D4C2×D18111yescharacteristic
228C2×C4⋊C4—1811no—
229C2×C4⋊C4—1811no—
230#107—1811no—
231#107—1811no—
232#107—1811no—
233C4⋊C8—1811no—
234#103—1811no—
235#103—1811no—
236#103—1811no—
237#103—1811no—
238#103—1811no—
239#103—1811no—
240#102—1811no—
241#102—1811no—
242#102—1811no—
243#102—1811no—
244C22⋊C8—1811no—
245C22⋊C8—1811no—
246#97—1811no—
247#41—22no—
248#41—41no—
249#41—7244no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—3621no—
254C2×Q8C22×D18111yescharacteristic
255C2×Q8C22×D18111yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3621no—
259C2×D4—3621no—
260C2×D4—1811no—
261C2×D4C22⋊D18111yescharacteristic
262C2×D4C22×D18111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3621no—
266C22×C4—3621no—
267C22×C4—3621no—
268C22×C4C22×D18111yescharacteristic
269C22×C4C22×D18111yescharacteristic
270SD16—3622no—
271SD16—7241no—
272D8—7241no—
273M16—7241no—
274C2×C8—1811no—
275C2×C8—1811no—
276C2×C8—1811no—
277C2×C8—1811no—
278C2×C8—7241no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—3621no—
282C4⋊C4—3621no—
283C4⋊C4—3621no—
284C4⋊C4—1811no—
285C4⋊C4—1811no—
286C4⋊C4—1811no—
287C4⋊C4C22⋊D18111yescharacteristic
288C4⋊C4C22×D18111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—3621no—
294C22⋊C4—3621no—
295C22⋊C4—3621no—
296C22⋊C4C22⋊D18111yescharacteristic
297C22⋊C4C22⋊D18111yescharacteristic
298C42—21no—
299C42C22×D18111yescharacteristic
300C23—21no—
301C23—3621no—
302C23C2×C22⋊D18111yescharacteristic
303C23C2×C22⋊D18111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—7242no—
309Q8#1566844812yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—3621no—
316D4—3621no—
317D4—7242no—
318D4—7242no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—3621no—
329C2×C4—3621no—
330C2×C4—3621no—
331C2×C4—7242no—
332C2×C4—7242no—
333C2×C4—7242no—
334C2×C4C2×C22⋊D18111yescharacteristic
335C2×C4C2×C22⋊D18111yescharacteristic
336C2×C4C2×C22⋊D18111yescharacteristic
337C2×C4C2×C22⋊D18111yescharacteristic
338C2×C4C23×D18111yescharacteristic
339C8—3621no—
340C8—3621no—
341C8—3621no—
342C8—3621no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—7241no—
352C22—7241no—
353C22—7241no—
354C22#1915384112yes—
355C22C22×C22⋊D18111yescharacteristic
356C4—21no—
357C4—21no—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—7241no—
364C4—7241no—
365C4—7241no—
366C4C2×#1566844811yescharacteristic
367C4#1915386111yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—14481no—
372C2#2604192311yescharacteristic
373C2C2×#1915384111yescharacteristic
374C2#2604209411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#548×C181C2
socleC22×C181C22×D4
hypercenter#548D181
hypocenterC181#3308
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #26041933C2semidirect
2. #26041638C2semidirect
3. #26041612C2semidirect
4. #26041590C2semidirect
5. #26041574C2semidirect
6. #26041527C2semidirect
7. #26041513C2semidirect
8. #26041491C2semidirect
9. #26041424C2semidirect
10. #26041418C2semidirect
11. #26041377C2semidirect
12. #26041351C2semidirect
13. #26041280C2semidirect
14. C2×#19153740C2semidirect
15. #548×C181C2semidirect
135. Q8×C181#41semidirect
187. C181#3308semidirect
189. #548D181semidirect
309. Q8#15668448semidirect

Maximal subgroups

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

subgroupindex
#260419332
#260416382
#260416122
#260415902
#260415742
#260415272
#260415132
#260414912
#260414242
#260414182
#260413772
#260413512
#260412802
C2×#191537402
#548×C1812
#3308181