Finite Group Databasebeta

← #37545915

Subgroups — #37545915

Order 23168.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041981C211yesmaximal, characteristic, complemented
2#26041917C211yesmaximal, characteristic, complemented
3#26041635C211yesmaximal, characteristic, complemented
4#26041619C211yesmaximal, characteristic, complemented
5#26041593C211yesmaximal, characteristic, complemented
6#26041577C211yesmaximal, characteristic, complemented
7#26041530C211yesmaximal, characteristic, complemented
8#26041522C211yesmaximal, characteristic, complemented
9#26041496C211yesmaximal, characteristic, complemented
10#26041411C211yesmaximal, characteristic, complemented
11D8×Dic181C211yesmaximal, characteristic, complemented
12#26041354C211yesmaximal, characteristic, complemented
13#26041282C211yesmaximal, characteristic, complemented
14#26041280C211yesmaximal, characteristic, complemented
15#449×C181C211yesmaximal, characteristic, complemented
16C2×D4×D181C2211yescharacteristic
17C2×#15668434C2211yescharacteristic
18C2×#15668433C2211yescharacteristic, complemented
19C2×#15668432C2211yescharacteristic
20#19153809—21no—
21#19153807C2211yescharacteristic, complemented
22#19153805C2211yescharacteristic
23#19153800—21no—
24#19153799—21no—
25#19153792C2211yescharacteristic
26#19153790C2211yescharacteristic
27#19153788—21no—
28#19153788—21no—
29#19153787C2211yescharacteristic
30#19153784—21no—
31D4×Dic181—21no—
32D4×Dic181C2211yescharacteristic
33D4×Dic181C2211yescharacteristic, complemented
34C4×C22⋊D181C2211yescharacteristic
35#19153774C2211yescharacteristic
36C2×C4⋊Dic181C2211yescharacteristic, complemented
37C2×#15668426—21no—
38#19153750C2211yescharacteristic, complemented
39#19153750C2211yescharacteristic, complemented
40#19153747C2211yescharacteristic, complemented
41#19153745C2211yescharacteristic, complemented
42#19153740C2211yescharacteristic, complemented
43#19153740C2211yescharacteristic, complemented
44#19153738C2211yescharacteristic
45#19153738C2211yescharacteristic
46#19153737C2211yescharacteristic
47#19153737C2211yescharacteristic
48#19153737C2211yescharacteristic, complemented
49#19153736C2211yescharacteristic
50#19153728C2211yescharacteristic
51C8×Dic181C2211yescharacteristic, complemented
52C2×#41×C181C2211yescharacteristic
53C2×SD16×C181C2211yescharacteristic, complemented
54C2×D8×C181C2211yescharacteristic
55#121×C181C2211yescharacteristic, complemented
56#103×C181C2211yescharacteristic, complemented
57#102×C181C2211yescharacteristic
58C22⋊C8×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60C2×D724C2311yescharacteristic
61C2×C22⋊D181—21no—
62C2×C22⋊D181—21no—
63C2×C22⋊D181—21no—
64C2×C22⋊D181D411yescharacteristic
65D4×D181—22no—
66D4×D181—41no—
67C22×Dic181—21no—
68C22×Dic181—21no—
69C22×Dic181—21no—
70C2×C4×D181C2311yescharacteristic
71#15668434—41no—
72#15668433—41no—
73#15668432—41no—
74C2×#13089678C2311yescharacteristic
75C2×#13089678C2311yescharacteristic
76C4⋊Dic181—21no—
77C4⋊Dic181—21no—
78C4⋊Dic181C2311yescharacteristic
79C4⋊Dic181C2311yescharacteristic
80C4⋊Dic181C2311yescharacteristic
81#15668428—21no—
82#15668428—21no—
83#15668428D411yescharacteristic
84C22⋊Dic181—21no—
85C22⋊Dic181—21no—
86C22⋊Dic181—21no—
87C22⋊Dic181D411yescharacteristic
88#15668426—21no—
89#15668426—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426D411yescharacteristic
94C4×Dic181—21no—
95C4×Dic181C2311yescharacteristic
96#41×C181—22no—
97#41×C181—41no—
98C2×Q8×C181C2311yescharacteristic
99C2×D4×C181—21no—
100C2×D4×C181—21no—
101C2×D4×C181C2311yescharacteristic
102C2×D4×C181C2311yescharacteristic
103C22×C4×C181—21no—
104C22×C4×C181C2311yescharacteristic
105SD16×C181—41no—
106D8×C181—22no—
107C2×C8×C181C2311yescharacteristic
108C2×C8×C181C2311yescharacteristic
109C4⋊C4×C181C2311yescharacteristic
110C22⋊C4×C181—21no—
111C22×D181—21no—
112C22×D181—42no—
113C22×D181—42no—
114C22×D181—81no—
115C22×D181C2×D411yescharacteristic
116D724—21no—
117D724—21no—
118C22⋊D181—41no—
119C22⋊D181—42no—
120C22⋊D181—81no—
121C22⋊D181—81no—
122C2×Dic181—21no—
123C2×Dic181—21no—
124C2×Dic181—21no—
125C2×Dic181—81no—
126C2×Dic181—81no—
127C2×Dic181—81no—
128C2×Dic181C2×D411yescharacteristic
129C2×Dic181C2×D411yescharacteristic
130C2×Dic181C2×D411yescharacteristic
131C4×D181—21no—
132C4×D181—21no—
133#13089678—21no—
134#13089678—21no—
135C23×C181—21no—
136C23×C181—21no—
137C23×C181C2×D411yescharacteristic
138Q8×C181—21no—
139Q8×C181—21no—
140D4×C181—21no—
141D4×C181—21no—
142D4×C181—21no—
143D4×C181—42no—
144D4×C181—42no—
145D4×C181#4112yes—
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—22no—
157C22×C181—21no—
158C22×C181—21no—
159C22×C181—41no—
160C22×C181—41no—
161C22×C181—41no—
162C22×C181—41no—
163C22×C181—41no—
164C22×C181C22×D411yescharacteristic
165C2×D181—21no—
166C2×D181—21no—
167C2×D181—21no—
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—84no—
172C4×C181—21no—
173C4×C181—41no—
174C4×C181—41no—
175C4×C181C2×#4111yescharacteristic
176C4×C181H(5,𝔽2)11yescharacteristic
177Dic181—21no—
178Dic181—21no—
179Dic181—21no—
180Dic181—41no—
181Dic181—41no—
182Dic181—41no—
183C2×C181—42no—
184C2×C181—41no—
185C2×C181—81no—
186C2×C181#54611yescharacteristic
187C2×C181#57511yescharacteristic
188C2×C181#57611yescharacteristic
189D181—41no—
190D181—81no—
191C181#332211yescharacteristic, complemented, Sylow
192#3322—1811nomaximal, Sylow
193#553—1811no—
194#546—1811no—
195#488—1811no—
196#485—1811no—
197#480—1811no—
198#478—1811no—
199#469—1811no—
200#468—1811no—
201#463—1811no—
202#449D18111yescharacteristic, complemented
203#447—1811no—
204C4×D8—1811no—
205#436—1811no—
206#418—1811no—
207#417—1811no—
208C2×#41C2×D18111yescharacteristic
209C22×D4—1811no—
210C2×SD16C2×D18111yescharacteristic, complemented
211C2×D8—1811no—
212C2×D8C2×D18111yescharacteristic
213C2×M16—1811no—
214C2×M16—1811no—
215#126—3621no—
216#125—1811no—
217#124—1811no—
218#123—3621no—
219#123—3621no—
220C22⋊Q8—1811no—
221#121—3621no—
222#121—3621no—
223#121—1811no—
224#121—1811no—
225#121C2×D18111yescharacteristic, complemented
226#120—3621no—
227C4×D4—3621no—
228C4×D4—1811no—
229C4×D4—1811no—
230C4×D4—1811no—
231#117—1811no—
232C2×C4⋊C4—1811no—
233C2×C22⋊C4—3621no—
234#107—1811no—
235#107—1811no—
236#106—1811no—
237C4⋊C8—1811no—
238#103—1811no—
239#103—1811no—
240#103C2×D18111yescharacteristic, complemented
241#102—1811no—
242#102—1811no—
243#102—1811no—
244#102—1811no—
245#102—1811no—
246#102—1811no—
247#102C2×D18111yescharacteristic
248C22⋊C8—1811no—
249C22⋊C8C2×D18111yescharacteristic, complemented
250C4×C8—1811no—
251C24—3621no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D18111yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—3622no—
258C2×D4—3621no—
259C2×D4—3621no—
260C2×D4—3621no—
261C2×D4—1811no—
262C2×D4—1811no—
263C2×D4—7241no—
264C2×D4C22×D18111yescharacteristic
265C2×D4C22×D18111yescharacteristic
266C22×C4—21no—
267C22×C4—3621no—
268C22×C4—3621no—
269C22×C4—3621no—
270C22×C4—1811no—
271C22×C4C22×D18111yescharacteristic
272SD16—41no—
273D8—22no—
274D8—7241no—
275M16—7241no—
276M16—7241no—
277C2×C8—1811no—
278C2×C8—1811no—
279C2×C8C22×D18111yescharacteristic
280C2×C8C22×D18111yescharacteristic
281C4⋊C4—3621no—
282C4⋊C4—3621no—
283C4⋊C4—3621no—
284C4⋊C4—3621no—
285C4⋊C4—1811no—
286C4⋊C4—1811no—
287C4⋊C4—1811no—
288C4⋊C4—1811no—
289C4⋊C4C22×D18111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—3621no—
292C22⋊C4—3621no—
293C22⋊C4—3621no—
294C22⋊C4—3621no—
295C22⋊C4—3621no—
296C22⋊C4—3621no—
297C22⋊C4—3621no—
298C22⋊C4—3621no—
299C22⋊C4—1811no—
300C22⋊C4—1811no—
301C42—3621no—
302C42—1811no—
303C23—21no—
304C23—21no—
305C23—3621no—
306C23—14481no—
307C23—1811no—
308C23—7242no—
309C23—7242no—
310C23D4×D18111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—42no—
318D4—3621no—
319D4—3621no—
320D4—14481no—
321D4—14481no—
322D4—7241no—
323D4—7242no—
324D4#1566844812yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—3621no—
331C2×C4—3621no—
332C2×C4—3621no—
333C2×C4—3621no—
334C2×C4—3621no—
335C2×C4—14481no—
336C2×C4—14481no—
337C2×C4—14481no—
338C2×C4—1811no—
339C2×C4—1811no—
340C2×C4—1811no—
341C2×C4D4×D18111yescharacteristic
342C2×C4C23×D18111yescharacteristic
343C8—21no—
344C8—21no—
345C8—3621no—
346C8—3621no—
347C22—21no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—3621no—
357C22—3621no—
358C22—3621no—
359C22—14484no—
360C22—7241no—
361C22—7241no—
362C22—7241no—
363C22C2×D4×D18111yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—3621no—
368C4—3621no—
369C4—3621no—
370C4—7241no—
371C4—7241no—
372C4—7241no—
373C4C2×#1566844811yescharacteristic
374C4#1915386111yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—14481no—
379C2—7241no—
380C2#2604191611yescharacteristic
381C2#2604207511yescharacteristic
382C2#2604208111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#449×C181C2
socleC22×C181C22×D4
hypercenter#449D181
hypocenterC181#3322
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #26041981C2semidirect
2. #26041917C2semidirect
3. #26041635C2semidirect
4. #26041619C2semidirect
5. #26041593C2semidirect
6. #26041577C2semidirect
7. #26041530C2semidirect
8. #26041522C2semidirect
9. #26041496C2semidirect
10. #26041411C2semidirect
11. D8×Dic181C2semidirect
12. #26041354C2semidirect
13. #26041282C2semidirect
14. #26041280C2semidirect
15. #449×C181C2semidirect
18. C2×#15668433C22semidirect
21. #19153807C22semidirect
33. D4×Dic181C22semidirect
36. C2×C4⋊Dic181C22semidirect
38. #19153750C22semidirect
39. #19153750C22semidirect
40. #19153747C22semidirect
41. #19153745C22semidirect
42. #19153740C22semidirect
43. #19153740C22semidirect
48. #19153737C22semidirect
51. C8×Dic181C22semidirect
53. C2×SD16×C181C22semidirect
55. #121×C181C22semidirect
56. #103×C181C22semidirect
58. C22⋊C8×C181C22semidirect
191. C181#3322semidirect
202. #449D181semidirect
210. C2×SD16C2×D181semidirect
225. #121C2×D181semidirect
240. #103C2×D181semidirect
249. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419812
#260419172
#260416352
#260416192
#260415932
#260415772
#260415302
#260415222
#260414962
#260414112
D8×Dic1812
#260413542
#260412822
#260412802
#449×C1812
#3322181