Finite Group Databasebeta

← #37545947

Subgroups — #37545947

Order 23168.

448 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#26042000C211yesmaximal, characteristic, complemented
2#26041913C211yesmaximal, characteristic
3#26041624C211yesmaximal, characteristic, complemented
4#26041606C211yesmaximal, characteristic, complemented
5#26041600C211yesmaximal, characteristic, complemented
6#26041558C211yesmaximal, characteristic, complemented
7#26041528C211yesmaximal, characteristic, complemented
8#26041477C211yesmaximal, characteristic, complemented
9#26041429C211yesmaximal, characteristic, complemented
10#26041428C211yesmaximal, characteristic, complemented
11#26041364C211yesmaximal, characteristic, complemented
12#26041355C211yesmaximal, characteristic, complemented
13#26041285C211yesmaximal, characteristic, complemented
14#26041282C211yesmaximal, characteristic, complemented
15#468×C181C211yesmaximal, characteristic, complemented
16C22×D724C2211yescharacteristic
17C2×D4×D181—21no—
18C2×D4×D181C2211yescharacteristic
19C2×#15668436C2211yescharacteristic
20C2×#15668436C2211yescharacteristic, complemented
21C2×#15668432C2211yescharacteristic, complemented
22#19153815C2211yescharacteristic, complemented
23#19153812C2211yescharacteristic
24#19153811—21no—
25#19153809—21no—
26#19153802—21no—
27#19153792C2211yescharacteristic, complemented
28#19153791C2211yescharacteristic
29#19153789—21no—
30#19153786—21no—
31#19153784—21no—
32Q8×Dic181C2211yescharacteristic, complemented
33#19153781—21no—
34#19153778C2211yescharacteristic
35C4×C22⋊D181C2211yescharacteristic
36#19153775C2211yescharacteristic, complemented
37#19153774C2211yescharacteristic, 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
51C8×Dic181C2211yescharacteristic, 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×Dic362—21no—
62C2×D724—22no—
63C2×D724—21no—
64C2×D724—21no—
65C2×D724—41no—
66C2×D724C2311yescharacteristic
67C2×D724C2311yescharacteristic
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181D411yescharacteristic
72D4×D181—22no—
73D4×D181—41no—
74D4×D181—44no—
75C2×C4×D181—21no—
76C2×C4×D181C2311yescharacteristic
77#15668436—22no—
78#15668436—41no—
79#15668432—41no—
80C2×#13089678C2311yescharacteristic
81C2×#13089678C2311yescharacteristic
82C4⋊Dic181—21no—
83C4⋊Dic181C2311yescharacteristic
84C4⋊Dic181C2311yescharacteristic
85#15668428—21no—
86#15668428—21no—
87#15668428D411yescharacteristic
88C22⋊Dic181—21no—
89C22⋊Dic181D411yescharacteristic
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426D411yescharacteristic
95C4×Dic181—21no—
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
121Dic362—42no—
122D724—21no—
123D724—21no—
124D724—21no—
125D724—42no—
126D724—42no—
127D724#4112yes—
128C22⋊D181—22no—
129C22⋊D181—42no—
130C22⋊D181—42no—
131C22⋊D181—81no—
132C2×Dic181—21no—
133C2×Dic181—21no—
134C2×Dic181C2×D411yescharacteristic
135C2×Dic181C2×D411yescharacteristic
136C2×Dic181C2×D411yescharacteristic
137C4×D181—21no—
138C4×D181—21no—
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—22no—
167C2×D181—21no—
168C2×D181—21no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—41no—
172C2×D181—41no—
173C2×D181—41no—
174C2×D181—84no—
175C2×D181—84no—
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—41no—
185Dic181—41no—
186Dic181AGL(1,ℤ/8)12yes—
187C2×C181—41no—
188C2×C181—81no—
189C2×C181#54611yescharacteristic
190C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C181#57711yescharacteristic
192D181—41no—
193D181—42no—
194D181—81no—
195C181#332411yescharacteristic, complemented, Sylow
196#3324—1811nomaximal, Sylow
197#556—1811no—
198D42—1811no—
199#486—1811no—
200#482—1811no—
201#482—1811no—
202#474—1811no—
203#468—1811no—
204#468D18111yescharacteristic, complemented
205#460—1811no—
206#450—1811no—
207#450—1811no—
208C4×SD16—1811no—
209#436—1811no—
210#418—1811no—
211#418—1811no—
212C22×D4—3621no—
213C22×D4—1811no—
214C22×D4—1811no—
215C2×SD16—1811no—
216C2×SD16—1811no—
217C2×SD16C2×D18111yescharacteristic, complemented
218C2×M16—1811no—
219C2×M16C2×D18111yescharacteristic, complemented
220C4⋊Q8—1811no—
221#127—1811no—
222#126—3621no—
223#126—3621no—
224#124—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—3621no—
233C4×Q8—1811no—
234C4×D4—1811no—
235C4×D4—1811no—
236#117—1811no—
237#117—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—
254C4×C8—1811no—
255C24—3621no—
256C24—3621no—
257C2×Q8—3621no—
258C2×Q8C22×D18111yescharacteristic
259C2×D4—21no—
260C2×D4—3622no—
261C2×D4—3621no—
262C2×D4—3622no—
263C2×D4—3621no—
264C2×D4—3621no—
265C2×D4—3621no—
266C2×D4—3621no—
267C2×D4—1811no—
268C2×D4—1811no—
269C2×D4—1811no—
270C2×D4—7241no—
271C2×D4—7241no—
272C2×D4—7244no—
273C2×D4C22×D18111yescharacteristic
274C22×C4—3621no—
275C22×C4—1811no—
276C22×C4C22×D18111yescharacteristic
277SD16—41no—
278SD16—3622no—
279SD16—7241no—
280M16—41no—
281M16—7241no—
282C2×C8—1811no—
283C2×C8—1811no—
284C2×C8C22×D18111yescharacteristic
285C2×C8C22×D18111yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—3621no—
288C4⋊C4—3621no—
289C4⋊C4—3621no—
290C4⋊C4—1811no—
291C4⋊C4—1811no—
292C4⋊C4—1811no—
293C4⋊C4C22×D18111yescharacteristic
294C4⋊C4C22×D18111yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—3621no—
298C22⋊C4—3621no—
299C22⋊C4—3621no—
300C22⋊C4—3621no—
301C22⋊C4—3621no—
302C22⋊C4—1811no—
303C22⋊C4—1811no—
304C42—3621no—
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—
321Q8—7242no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—3622no—
326D4—3621no—
327D4—3621no—
328D4—3621no—
329D4—14481no—
330D4—1812no—
331D4—7242no—
332D4—7242no—
333D4—7242no—
334D4—7242no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—3621no—
341C2×C4—3621no—
342C2×C4—3621no—
343C2×C4—3621no—
344C2×C4—1811no—
345C2×C4—1811no—
346C2×C4—1811no—
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—3622no—
362C22—3621no—
363C22—3621no—
364C22—14484no—
365C22—14484no—
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—1812no—
379C4—7241no—
380C4—7241no—
381C4#41×D18111yescharacteristic
382C4#1915386111yescharacteristic
383C2—41no—
384C2—81no—
385C2—14481no—
386C2—7241no—
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#3324
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. #26042000C2semidirect
3. #26041624C2semidirect
4. #26041606C2semidirect
5. #26041600C2semidirect
6. #26041558C2semidirect
7. #26041528C2semidirect
8. #26041477C2semidirect
9. #26041429C2semidirect
10. #26041428C2semidirect
11. #26041364C2semidirect
12. #26041355C2semidirect
13. #26041285C2semidirect
14. #26041282C2semidirect
15. #468×C181C2semidirect
20. C2×#15668436C22semidirect
21. C2×#15668432C22semidirect
22. #19153815C22semidirect
27. #19153792C22semidirect
32. Q8×Dic181C22semidirect
36. #19153775C22semidirect
37. #19153774C22semidirect
38. #19153749C22semidirect
39. #19153747C22semidirect
40. #19153745C22semidirect
43. #19153738C22semidirect
44. #19153738C22semidirect
45. #19153737C22semidirect
46. #19153737C22semidirect
48. #19153736C22semidirect
49. C22⋊#13089678C22semidirect
50. #19153728C22semidirect
51. C8×Dic181C22semidirect
52. C2×SD16×C181C22semidirect
53. C2×M16×C181C22semidirect
54. C22⋊Q8×C181C22semidirect
56. #107×C181C22semidirect
57. #103×C181C22semidirect
58. #102×C181C22semidirect
195. C181#3324semidirect
204. #468D181semidirect
217. C2×SD16C2×D181semidirect
219. 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
#260420002
#260419132
#260416242
#260416062
#260416002
#260415582
#260415282
#260414772
#260414292
#260414282
#260413642
#260413552
#260412852
#260412822
#468×C1812
#3324181