Finite Group Databasebeta

← #37545716

Subgroups — #37545716

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#26041940C211yesmaximal, characteristic, complemented
2#26041929C211yesmaximal, characteristic, complemented
3#26041640C211yesmaximal, characteristic, complemented
4#26041613C211yesmaximal, characteristic, complemented
5#26041593C211yesmaximal, characteristic, complemented
6#26041572C211yesmaximal, characteristic, complemented
7#26041522C211yesmaximal, characteristic, complemented
8#26041513C211yesmaximal, characteristic, complemented
9#26041496C211yesmaximal, characteristic, complemented
10#26041423C211yesmaximal, characteristic, complemented
11#26041380C211yesmaximal, characteristic, complemented
12#26041347C211yesmaximal, characteristic, complemented
13#26041279C211yesmaximal, characteristic, complemented
14C2×#19153740C211yesmaximal, characteristic, complemented
15#448×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668448—21no—
18C2×#15668434C2211yescharacteristic
19C2×#15668432C2211yescharacteristic
20C22×#13089678C2211yescharacteristic
21#19153815C2211yescharacteristic
22#19153805—21no—
23#19153805C2211yescharacteristic
24#19153798—21no—
25#19153797—21no—
26#19153793—21no—
27#19153792—21no—
28#19153792C2211yescharacteristic
29#19153790—21no—
30#19153790C2211yescharacteristic
31#19153787C2211yescharacteristic
32Q8×Dic181C2211yescharacteristic
33D4×Dic181C2211yescharacteristic
34C4×C22⋊D181—21no—
35C4×C22⋊D181C2211yescharacteristic
36C2×C4⋊Dic181C2211yescharacteristic
37C4⋊C4×D181C2211yescharacteristic
38#19153750C2211yescharacteristic
39#19153750C2211yescharacteristic
40#19153749C2211yescharacteristic
41#19153745C2211yescharacteristic
42#19153740C2211yescharacteristic
43#19153740C2211yescharacteristic
44#19153740C2211yescharacteristic
45#19153740C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153737C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153728C2211yescharacteristic
51#19153726C2211yescharacteristic
52C22×Q8×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic
54C2×SD16×C181C2211yescharacteristic
55#121×C181C2211yescharacteristic
56#103×C181C2211yescharacteristic
57#103×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic
59#15668450—22no—
60#15668450—41no—
61#15668448—44no—
62C2×Dic362—21no—
63C2×D724—21no—
64C2×D724C2311yescharacteristic
65C2×C22⋊D181—21no—
66C2×C22⋊D181—21no—
67C2×C22⋊D181D411yescharacteristic
68C22×Dic181—21no—
69C22×Dic181—21no—
70C2×C4×D181—21no—
71C2×C4×D181—21no—
72C2×C4×D181C2311yescharacteristic
73#15668434—41no—
74#15668432—41no—
75C2×#13089678—41no—
76C2×#13089678C2311yescharacteristic
77C2×#13089678C2311yescharacteristic
78C4⋊Dic181—21no—
79C4⋊Dic181—21no—
80C4⋊Dic181C2311yescharacteristic
81C4⋊Dic181C2311yescharacteristic
82C4⋊Dic181C2311yescharacteristic
83#15668428—21no—
84#15668428—21no—
85#15668428—21no—
86#15668428D411yescharacteristic
87C22⋊Dic181—21no—
88C22⋊Dic181—21no—
89C22⋊Dic181D411yescharacteristic
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426D411yescharacteristic
95C4×Dic181—21no—
96C4×Dic181C2311yescharacteristic
97C2×Q8×C181—21no—
98C2×Q8×C181—22no—
99C2×Q8×C181—41no—
100C2×Q8×C181C2311yescharacteristic
101C2×Q8×C181C2311yescharacteristic
102C2×D4×C181—21no—
103C2×D4×C181C2311yescharacteristic
104C22×C4×C181—21no—
105C22×C4×C181C2311yescharacteristic
106SD16×C181—22no—
107SD16×C181—41no—
108C2×C8×C181C2311yescharacteristic
109C2×C8×C181C2311yescharacteristic
110C4⋊C4×C181C2311yescharacteristic
111C22⋊C4×C181—21no—
112C22×D181—21no—
113C22×D181C2×D411yescharacteristic
114Dic362—42no—
115D724—21no—
116D724—21no—
117D724—42no—
118C22⋊D181—22no—
119C22⋊D181—42no—
120C22⋊D181—42no—
121C2×Dic181—21no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—42no—
125C2×Dic181—42no—
126C2×Dic181C2×D411yescharacteristic
127C2×Dic181C2×D411yescharacteristic
128C2×Dic181C2×D411yescharacteristic
129C4×D181—21no—
130C4×D181—21no—
131C4×D181—42no—
132C4×D181—42no—
133#13089678—21no—
134#13089678—21no—
135C23×C181—21no—
136C23×C181C2×D411yescharacteristic
137Q8×C181—21no—
138Q8×C181—21no—
139Q8×C181—21no—
140Q8×C181—42no—
141Q8×C181#4112yescomplemented
142D4×C181—21no—
143D4×C181—21no—
144D4×C181—42no—
145C2×C4×C181—21no—
146C2×C4×C181—21no—
147C2×C4×C181—21no—
148C2×C4×C181—21no—
149C2×C4×C181—21no—
150C2×C4×C181—42no—
151C2×C4×C181C2×D411yescharacteristic
152C2×C4×C181C2411yescharacteristic
153C8×C181—21no—
154C8×C181—21no—
155C22×C181—21no—
156C22×C181—21no—
157C22×C181—41no—
158C22×C181—41no—
159C22×C181—41no—
160C22×C181AGL(1,ℤ/8)12yes—
161C22×C181C22×D411yescharacteristic
162C2×D181—21no—
163C2×D181—21no—
164C2×D181—21no—
165C2×D181—41no—
166C2×D181—41no—
167C2×D181—41no—
168C4×C181—21no—
169C4×C181—22no—
170C4×C181—41no—
171C4×C181—41no—
172C4×C181C2×#4111yescharacteristic
173C4×C181H(5,𝔽2)11yescharacteristic
174Dic181—21no—
175Dic181—21no—
176Dic181—21no—
177Dic181—41no—
178Dic181—41no—
179Dic181—41no—
180C2×C181—22no—
181C2×C181—81no—
182C2×C181#54611yescharacteristic
183C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C181#57811yescharacteristic
185D181—41no—
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#477—1811no—
195#468—1811no—
196#466—1811no—
197#463—1811no—
198#449—1811no—
199#448D18111yescharacteristic, complemented
200#440—1811no—
201#435—1811no—
202#417—1811no—
203C2×#103—1811no—
204C2×#41—3621no—
205C2×#41—1811no—
206C22×Q8C2×D18111yescharacteristic
207C2×SD16C2×D18111yescharacteristic
208C2×SD16C2×D18111yescharacteristic
209C2×D8—1811no—
210C2×M16—1811no—
211C22×C8—1811no—
212C4⋊Q8—1811no—
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#121C2×D18111yescharacteristic
224C4×Q8—1811no—
225C4×D4—3621no—
226C4×D4—1811no—
227C4×D4—1811no—
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#103C2×D18111yescharacteristic
239#103C2×D18111yescharacteristic
240#102—1811no—
241#102—1811no—
242#102—1811no—
243#102—1811no—
244C22⋊C8—1811no—
245C22⋊C8C2×D18111yescharacteristic
246#97—1811no—
247#41—3622no—
248#41—7241no—
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—3621no—
258C2×D4—3621no—
259C2×D4—3621no—
260C2×D4—1811no—
261C2×D4—1811no—
262C2×D4C22×D18111yescharacteristic
263C22×C4—21no—
264C22×C4—3621no—
265C22×C4—3621no—
266C22×C4—3621no—
267C22×C4—3621no—
268C22×C4—1811no—
269C22×C4C22×D18111yescharacteristic
270SD16—22no—
271SD16—41no—
272D8—7241no—
273M16—7241no—
274C2×C8—1811no—
275C2×C8—1811no—
276C2×C8—7241no—
277C2×C8C22×D18111yescharacteristic
278C2×C8C22×D18111yescharacteristic
279C4⋊C4—3621no—
280C4⋊C4—3621no—
281C4⋊C4—3621no—
282C4⋊C4—3621no—
283C4⋊C4—3621no—
284C4⋊C4—1811no—
285C4⋊C4—1811no—
286C4⋊C4—1811no—
287C4⋊C4—1811no—
288C4⋊C4C22×D18111yescharacteristic
289C22⋊C4—21no—
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—3621no—
299C42—1811no—
300C23—21no—
301C23—3621no—
302C23—1811no—
303C23D4×D18111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—7242no—
309Q8#1566844812yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—3622no—
314D4—3621no—
315D4—3621no—
316D4—7242no—
317D4—7242no—
318D4—7242no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—3621no—
326C2×C4—3621no—
327C2×C4—3621no—
328C2×C4—3621no—
329C2×C4—3621no—
330C2×C4—1811no—
331C2×C4—1811no—
332C2×C4—1811no—
333C2×C4—7242no—
334C2×C4—7242no—
335C2×C4—7242no—
336C2×C4—7242no—
337C2×C4D4×D18111yescharacteristic
338C2×C4C23×D18111yescharacteristic
339C8—21no—
340C8—21no—
341C8—3621no—
342C8—3621no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—3621no—
349C22—3621no—
350C22—3621no—
351C22—7241no—
352C22—7241no—
353C22—7241no—
354C22#1915384312yes—
355C22C2×D4×D18111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—3621no—
361C4—3621no—
362C4—3621no—
363C4—7241no—
364C4—7241no—
365C4—7241no—
366C4C2×#1566844811yescharacteristic
367C4#1915386111yescharacteristic
368C2—22no—
369C2—81no—
370C2—14481no—
371C2—7241no—
372C2#2604191611yescharacteristic
373C2C2×#1915384311yescharacteristic
374C2#2604209511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#448×C181C2
socleC22×C181C22×D4
hypercenter#448D181
hypocenterC181#3308
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #26041940C2semidirect
2. #26041929C2semidirect
3. #26041640C2semidirect
4. #26041613C2semidirect
5. #26041593C2semidirect
6. #26041572C2semidirect
7. #26041522C2semidirect
8. #26041513C2semidirect
9. #26041496C2semidirect
10. #26041423C2semidirect
11. #26041380C2semidirect
12. #26041347C2semidirect
13. #26041279C2semidirect
14. C2×#19153740C2semidirect
15. #448×C181C2semidirect
141. Q8×C181#41semidirect
187. C181#3308semidirect
199. #448D181semidirect
309. Q8#15668448semidirect

Maximal subgroups

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

subgroupindex
#260419402
#260419292
#260416402
#260416132
#260415932
#260415722
#260415222
#260415132
#260414962
#260414232
#260413802
#260413472
#260412792
C2×#191537402
#448×C1812
#3308181