Finite Group Databasebeta

← #37545737

Subgroups — #37545737

Order 23168.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041963C211yesmaximal, characteristic, complemented
2#26041913C211yesmaximal, characteristic
3#26041642C211yesmaximal, characteristic, complemented
4#26041600C211yesmaximal, characteristic, complemented
5#26041594C211yesmaximal, characteristic, complemented
6#26041582C211yesmaximal, characteristic, complemented
7#26041511C211yesmaximal, characteristic, complemented
8#26041477C211yesmaximal, characteristic, complemented
9#26041429C211yesmaximal, characteristic, complemented
10#26041428C211yesmaximal, characteristic, complemented
11#26041382C211yesmaximal, characteristic, complemented
12#26041348C211yesmaximal, characteristic, complemented
13#26041285C211yesmaximal, characteristic, complemented
14#102×D181C211yesmaximal, characteristic, complemented
15#468×C181C211yesmaximal, characteristic, complemented
16C22×D724C2211yescharacteristic
17C2×D4×D181—21no—
18C2×D4×D181C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668436C2211yescharacteristic, complemented
21C2×C8×D181C2211yescharacteristic, complemented
22#19153812C2211yescharacteristic
23#19153809—21no—
24#19153807C2211yescharacteristic, complemented
25#19153806—21no—
26#19153798—21no—
27#19153792C2211yescharacteristic, complemented
28#19153791C2211yescharacteristic
29#19153789—21no—
30#19153786—21no—
31#19153784—21no—
32Q8×Dic181C2211yescharacteristic, complemented
33#19153778C2211yescharacteristic
34C4×C22⋊D181C2211yescharacteristic
35#19153775C2211yescharacteristic, complemented
36#19153774—21no—
37C4⋊C4×D181C2211yescharacteristic, complemented
38#19153749C2211yescharacteristic, complemented
39#19153747C2211yescharacteristic, complemented
40#19153745C2211yescharacteristic, complemented
41#19153738C2211yescharacteristic
42#19153738C2211yescharacteristic
43#19153738C2211yescharacteristic, complemented
44#19153738C2211yescharacteristic, complemented
45#19153737C2211yescharacteristic, complemented
46#19153737C2211yescharacteristic, complemented
47#19153736C2211yescharacteristic
48#19153736C2211yescharacteristic, complemented
49C22⋊#13089678C2211yescharacteristic, complemented
50#19153728C2211yescharacteristic, complemented
51#19153726C2211yescharacteristic, complemented
52C2×SD16×C181C2211yescharacteristic, complemented
53C2×M16×C181C2211yescharacteristic, complemented
54C22⋊Q8×C181C2211yescharacteristic, complemented
55#121×C181C2211yescharacteristic
56#107×C181C2211yescharacteristic, complemented
57#103×C181C2211yescharacteristic, complemented
58#102×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60C23×D181—21no—
61C2×D724—22no—
62C2×D724—21no—
63C2×D724—21no—
64C2×D724—41no—
65C2×D724C2311yescharacteristic
66C2×D724C2311yescharacteristic
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C22⋊D181D411yescharacteristic
71D4×D181—22no—
72D4×D181—41no—
73D4×D181—44no—
74C2×C4×D181—21no—
75C2×C4×D181—21no—
76C2×C4×D181C2311yescharacteristic
77#15668436—22no—
78#15668436—41no—
79C2×#13089678C2311yescharacteristic
80C2×#13089678C2311yescharacteristic
81C8×D181—41no—
82C4⋊Dic181—21no—
83C4⋊Dic181—21no—
84C4⋊Dic181C2311yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86#15668428—21no—
87#15668428—21no—
88#15668428—21no—
89#15668428D411yescharacteristic
90C22⋊Dic181—21no—
91C22⋊Dic181D411yescharacteristic
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426D411yescharacteristic
96C4×Dic181—21no—
97C4×Dic181—21no—
98C4×Dic181C2311yescharacteristic
99C2×Q8×C181C2311yescharacteristic
100C2×D4×C181—21no—
101C2×D4×C181C2311yescharacteristic
102C22×C4×C181C2311yescharacteristic
103SD16×C181—41no—
104M16×C181—41no—
105C2×C8×C181C2311yescharacteristic
106C2×C8×C181C2311yescharacteristic
107C4⋊C4×C181—21no—
108C4⋊C4×C181C2311yescharacteristic
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22⋊C4×C181—21no—
112C22×D181—21no—
113C22×D181—21no—
114C22×D181—42no—
115C22×D181—42no—
116C22×D181—42no—
117C22×D181—42no—
118C22×D181—81no—
119C22×D181—81no—
120C22×D181C2×D411yescharacteristic
121D724—21no—
122D724—21no—
123D724—21no—
124D724—42no—
125D724—42no—
126D724#4112yes—
127C22⋊D181—22no—
128C22⋊D181—42no—
129C22⋊D181—42no—
130C22⋊D181—81no—
131C2×Dic181—21no—
132C2×Dic181—21no—
133C2×Dic181C2×D411yescharacteristic
134C2×Dic181C2×D411yescharacteristic
135C2×Dic181C2×D411yescharacteristic
136C4×D181—21no—
137C4×D181—21no—
138C4×D181—42no—
139C4×D181—42no—
140#13089678—21no—
141#13089678—21no—
142C23×C181—21no—
143C23×C181C2×D411yescharacteristic
144Q8×C181—21no—
145Q8×C181—21no—
146D4×C181—21no—
147D4×C181—21no—
148D4×C181—42no—
149C2×C4×C181—21no—
150C2×C4×C181—21no—
151C2×C4×C181—21no—
152C2×C4×C181—21no—
153C2×C4×C181—21no—
154C2×C4×C181C2×D411yescharacteristic
155C2×C4×C181C2411yescharacteristic
156C8×C181—21no—
157C8×C181—21no—
158C22×C181—21no—
159C22×C181—21no—
160C22×C181—21no—
161C22×C181—41no—
162C22×C181—41no—
163C22×C181—41no—
164C22×C181C22×D411yescharacteristic
165C2×D181—21no—
166C2×D181—21no—
167C2×D181—22no—
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—41no—
172C2×D181—41no—
173C2×D181—84no—
174C2×D181—84no—
175C2×D181AGL(1,ℤ/8)12yes—
176C4×C181—21no—
177C4×C181—41no—
178C4×C181—41no—
179C4×C181—41no—
180C4×C181C2×#4111yescharacteristic
181C4×C181H(5,𝔽2)11yescharacteristic
182Dic181—21no—
183Dic181—21no—
184Dic181—21no—
185Dic181—41no—
186Dic181—41no—
187C2×C181—41no—
188C2×C181—81no—
189C2×C181#54611yescharacteristic
190C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C181#57711yescharacteristic
192D181—22no—
193D181—42no—
194D181—81no—
195C181#331011yescharacteristic, complemented, Sylow
196#3310—1811nomaximal, Sylow
197#552—1811no—
198D42—1811no—
199#490—1811no—
200#482—1811no—
201#480—1811no—
202#478—1811no—
203#468D18111yescharacteristic, complemented
204#465—1811no—
205#460—1811no—
206#450—1811no—
207#450—1811no—
208#440—1811no—
209#435—1811no—
210#418—1811no—
211C2×#102—1811no—
212C22×D4—3621no—
213C22×D4—1811no—
214C22×D4—1811no—
215C2×SD16—1811no—
216C2×SD16—1811no—
217C2×SD16C2×D18111yescharacteristic, complemented
218C2×M16C2×D18111yescharacteristic, complemented
219C22×C8—1811no—
220#127—1811no—
221#126—3621no—
222#125—3621no—
223#125—1811no—
224#123—3621no—
225C22⋊Q8—1811no—
226C22⋊Q8C2×D18111yescharacteristic, complemented
227#121—3621no—
228#121—1811no—
229#121C2×D18111yescharacteristic
230#120—3621no—
231#120—3621no—
232C4×Q8—1811no—
233C4×D4—1811no—
234C4×D4—1811no—
235#117—3621no—
236#117—1811no—
237C2×C4⋊C4—1811no—
238#107—1811no—
239#107C2×D18111yescharacteristic, complemented
240#106—1811no—
241C4⋊C8—1811no—
242#103C2×D18111yescharacteristic, complemented
243#102—1811no—
244#102—1811no—
245#102—1811no—
246#102—1811no—
247#102—1811no—
248#102—1811no—
249#102—1811no—
250#102—1811no—
251#102C2×D18111yescharacteristic, complemented
252C22⋊C8—1811no—
253C22⋊C8—1811no—
254#97—1811no—
255C24—3621no—
256C24—3621no—
257C2×Q8C22×D18111yescharacteristic
258C2×D4—21no—
259C2×D4—3622no—
260C2×D4—3621no—
261C2×D4—3622no—
262C2×D4—3621no—
263C2×D4—3621no—
264C2×D4—3621no—
265C2×D4—3621no—
266C2×D4—1811no—
267C2×D4—1811no—
268C2×D4—1811no—
269C2×D4—7241no—
270C2×D4—7241no—
271C2×D4—7244no—
272C2×D4C22×D18111yescharacteristic
273C22×C4—3621no—
274C22×C4—3621no—
275C22×C4—1811no—
276C22×C4C22×D18111yescharacteristic
277SD16—41no—
278SD16—3622no—
279SD16—7241no—
280M16—41no—
281C2×C8—1811no—
282C2×C8—1811no—
283C2×C8—7241no—
284C2×C8C22×D18111yescharacteristic
285C2×C8C22×D18111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—3621no—
288C4⋊C4—3621no—
289C4⋊C4—3621no—
290C4⋊C4—3621no—
291C4⋊C4—3621no—
292C4⋊C4—1811no—
293C4⋊C4—1811no—
294C4⋊C4—1811no—
295C4⋊C4C22×D18111yescharacteristic
296C4⋊C4C22×D18111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—3621no—
300C22⋊C4—3621no—
301C22⋊C4—3621no—
302C22⋊C4—3621no—
303C22⋊C4—1811no—
304C22⋊C4—1811no—
305C42—3621no—
306C42—3621no—
307C42—1811no—
308C23—21no—
309C23—3621no—
310C23—3621no—
311C23—14481no—
312C23—14481no—
313C23—1811no—
314C23—7242no—
315C23—7242no—
316C23—7242no—
317C23—7242no—
318C23D4×D18111yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—3622no—
325D4—3621no—
326D4—3621no—
327D4—3621no—
328D4—14481no—
329D4—1812no—
330D4—7242no—
331D4—7242no—
332D4—7242no—
333D4—7242no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—3621no—
340C2×C4—3621no—
341C2×C4—3621no—
342C2×C4—3621no—
343C2×C4—1811no—
344C2×C4—1811no—
345C2×C4—1811no—
346C2×C4—7242no—
347C2×C4—7242no—
348C2×C4D4×D18111yescharacteristic
349C2×C4C23×D18111yescharacteristic
350C8—21no—
351C8—21no—
352C8—3621no—
353C8—3621no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—3621no—
361C22—3621no—
362C22—3622no—
363C22—14484no—
364C22—14484no—
365C22—1812no—
366C22—7241no—
367C22—7241no—
368C22—7241no—
369C22—7241no—
370C22—7241no—
371C22C2×D4×D18111yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—3621no—
377C4—3621no—
378C4—3621no—
379C4—7241no—
380C4—7241no—
381C4#41×D18111yescharacteristic
382C4#1915386111yescharacteristic
383C2—41no—
384C2—81no—
385C2—3622no—
386C2—14481no—
387C2—7242no—
388C2#2604191811yescharacteristic
389C2AGL(1,ℤ/8)×D18111yescharacteristic
390C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#468×C181C2
socleC22×C181C22×D4
prime radical#26041913C2
hypercenter#468D181
hypocenterC181#3310
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
1. #26041963C2semidirect
3. #26041642C2semidirect
4. #26041600C2semidirect
5. #26041594C2semidirect
6. #26041582C2semidirect
7. #26041511C2semidirect
8. #26041477C2semidirect
9. #26041429C2semidirect
10. #26041428C2semidirect
11. #26041382C2semidirect
12. #26041348C2semidirect
13. #26041285C2semidirect
14. #102×D181C2semidirect
15. #468×C181C2semidirect
20. C2×#15668436C22semidirect
21. C2×C8×D181C22semidirect
24. #19153807C22semidirect
27. #19153792C22semidirect
32. Q8×Dic181C22semidirect
35. #19153775C22semidirect
37. C4⋊C4×D181C22semidirect
38. #19153749C22semidirect
39. #19153747C22semidirect
40. #19153745C22semidirect
43. #19153738C22semidirect
44. #19153738C22semidirect
45. #19153737C22semidirect
46. #19153737C22semidirect
48. #19153736C22semidirect
49. C22⋊#13089678C22semidirect
50. #19153728C22semidirect
51. #19153726C22semidirect
52. C2×SD16×C181C22semidirect
53. C2×M16×C181C22semidirect
54. C22⋊Q8×C181C22semidirect
56. #107×C181C22semidirect
57. #103×C181C22semidirect
58. #102×C181C22semidirect
195. C181#3310semidirect
203. #468D181semidirect
217. C2×SD16C2×D181semidirect
218. C2×M16C2×D181semidirect
226. C22⋊Q8C2×D181semidirect
239. #107C2×D181semidirect
242. #103C2×D181semidirect
251. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419632
#260419132
#260416422
#260416002
#260415942
#260415822
#260415112
#260414772
#260414292
#260414282
#260413822
#260413482
#260412852
#102×D1812
#468×C1812
#3310181