Finite Group Databasebeta

← #37545573

Subgroups — #37545573

Order 23168.

420 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#26041975C211yesmaximal, characteristic, complemented
2#26041925C211yesmaximal, characteristic, complemented
3#26041634C211yesmaximal, characteristic, complemented
4#26041615C211yesmaximal, characteristic, complemented
5#26041604C211yesmaximal, characteristic, complemented
6#26041580C211yesmaximal, characteristic, complemented
7#26041515C211yesmaximal, characteristic, complemented
8#26041508C211yesmaximal, characteristic, complemented
9#26041495C211yesmaximal, characteristic, complemented
10#26041426C211yesmaximal, characteristic, complemented
11#26041415C211yesmaximal, characteristic, complemented
12C4×D1448C211yesmaximal, characteristic, complemented
13#26041276C211yesmaximal, characteristic, complemented
14C2×#19153738C211yesmaximal, characteristic, complemented
15C8×D4×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×D724C2211yescharacteristic
18C2×#15668438C2211yescharacteristic, complemented
19C2×D1448C2211yescharacteristic
20C2×D1448C2211yescharacteristic
21#19153811—21no—
22#19153808C2211yescharacteristic, complemented
23#19153804C2211yescharacteristic
24#19153801—21no—
25#19153798—21no—
26#19153795C2211yescharacteristic
27#19153790C2211yescharacteristic
28#19153790C2211yescharacteristic
29#19153790C2211yescharacteristic, complemented
30#19153789—21no—
31#19153788—21no—
32#19153784—21no—
33C4×D724C2211yescharacteristic
34C4×D724C2211yescharacteristic, complemented
35#19153776—21no—
36#19153775C2211yescharacteristic, complemented
37C2×C4⋊Dic181C2211yescharacteristic
38C22⋊C4×D181—21no—
39#19153750C2211yescharacteristic, complemented
40#19153750C2211yescharacteristic, complemented
41#19153748C2211yescharacteristic, complemented
42#19153741C2211yescharacteristic, complemented
43#19153741C2211yescharacteristic, complemented
44#19153741C2211yescharacteristic, complemented
45#19153738C2211yescharacteristic
46#19153738C2211yescharacteristic
47#19153738C2211yescharacteristic
48#19153738C2211yescharacteristic
49#19153738C2211yescharacteristic
50#19153738C2211yescharacteristic
51#19153738C2211yescharacteristic, complemented
52C22×C8×C181C2211yescharacteristic
53C22×C8×C181C2211yescharacteristic, complemented
54C4×D4×C181C2211yescharacteristic
55C4⋊C8×C181C2211yescharacteristic, complemented
56C22⋊C8×C181C2211yescharacteristic
57C22⋊C8×C181C2211yescharacteristic, complemented
58C4×C8×C181C2211yescharacteristic, complemented
59C23×D181—21no—
60#15668449—22no—
61#15668449—41no—
62C2×Dic362C2311yescharacteristic
63C2×D724—22no—
64C2×D724—21no—
65C2×D724—41no—
66C2×D724C2311yescharacteristic
67C2×D724C2311yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C22×Dic181—21no—
74C2×C4×D181—21no—
75C2×C4×D181—21no—
76C2×C4×D181—21no—
77#15668438—41no—
78D1448—22no—
79D1448—41no—
80C4⋊Dic181—21no—
81C4⋊Dic181C2311yescharacteristic
82C4⋊Dic181C2311yescharacteristic
83C4⋊Dic181C2311yescharacteristic
84C4⋊Dic181C2311yescharacteristic
85#15668428—21no—
86#15668428—21no—
87#15668428—21no—
88C22⋊Dic181—21no—
89C22⋊Dic181—21no—
90#15668426—21no—
91#15668426—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#15668426—21no—
97C4×Dic181—21no—
98C2×D4×C181D411yescharacteristic
99C22×C4×C181C2311yescharacteristic
100C22×C4×C181C2311yescharacteristic
101C2×C8×C181—41no—
102C2×C8×C181—41no—
103C2×C8×C181C2311yescharacteristic
104C2×C8×C181C2311yescharacteristic
105C2×C8×C181C2311yescharacteristic
106C2×C8×C181C2311yescharacteristic
107C4⋊C4×C181D411yescharacteristic
108C22⋊C4×C181D411yescharacteristic
109C22⋊C4×C181D411yescharacteristic
110C42×C181C2311yescharacteristic
111C22×D181—21no—
112C22×D181—21no—
113C22×D181—21no—
114C22×D181—42no—
115C22×D181—42no—
116C22×D181—81no—
117Dic362—21no—
118Dic362—21no—
119D724—21no—
120D724—21no—
121D724—21no—
122D724—21no—
123D724—21no—
124D724—42no—
125D724#4112yes—
126C22⋊D181—42no—
127C22⋊D181—42no—
128C22⋊D181—81no—
129C22⋊D181—81no—
130C2×Dic181—21no—
131C2×Dic181—21no—
132C2×Dic181—21no—
133C2×Dic181—21no—
134C2×Dic181—21no—
135C2×Dic181—81no—
136C4×D181—42no—
137C4×D181—81no—
138C4×D181—81no—
139C23×C181C2×D411yescharacteristic
140C23×C181C2×D411yescharacteristic
141D4×C181—22no—
142C2×C4×C181—21no—
143C2×C4×C181—21no—
144C2×C4×C181—21no—
145C2×C4×C181—21no—
146C2×C4×C181C2×D411yescharacteristic
147C2×C4×C181C2×D411yescharacteristic
148C2×C4×C181C2×D411yescharacteristic
149C2×C4×C181C2×D411yescharacteristic
150C2×C4×C181C2411yescharacteristic
151C8×C181—21no—
152C8×C181—21no—
153C8×C181—21no—
154C8×C181—21no—
155C22×C181—21no—
156C22×C181—21no—
157C22×C181—21no—
158C22×C181—21no—
159C22×C181—21no—
160C22×C181#13512yes—
161C22×C181C22×D411yescharacteristic
162C2×D181—22no—
163C2×D181—41no—
164C2×D181—41no—
165C2×D181—41no—
166C2×D181—41no—
167C2×D181—41no—
168C2×D181—41no—
169C2×D181—41no—
170C2×D181—41no—
171C2×D181—84no—
172C4×C181—21no—
173C4×C181—21no—
174C4×C181—21no—
175C4×C181—21no—
176C4×C181C2×#4111yescharacteristic
177C4×C181H(5,𝔽2)11yescharacteristic
178Dic181—41no—
179Dic181—41no—
180Dic181—41no—
181Dic181—41no—
182Dic181—41no—
183C2×C181—22no—
184C2×C181—41no—
185C2×C181#54611yescharacteristic
186C2×C181C2×#13511yescharacteristic
187C2×C181#57611yescharacteristic
188D181—42no—
189D181—81no—
190D181—81no—
191C181#329711yescharacteristic, complemented, Sylow
192#3297—1811nomaximal, Sylow
193#553—1811no—
194#546—1811no—
195#488—1811no—
196#484—1811no—
197#482—1811no—
198#478—1811no—
199#466—1811no—
200#465—1811no—
201#463—1811no—
202#449—1811no—
203#447—1811no—
204C4×D8—1811no—
205C8×D4D18111yescharacteristic, complemented
206#416—1811no—
207C2×#102—1811no—
208C2×#41—1811no—
209C22×D4—1811no—
210C2×SD16—1811no—
211C2×D8—1811no—
212C2×D8—1811no—
213C22×C8C2×D18111yescharacteristic
214C22×C8C2×D18111yescharacteristic, complemented
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#121—1811no—
226#120—3621no—
227C4×D4—3621no—
228C4×D4—1811no—
229C4×D4—1811no—
230C4×D4C2×D18111yescharacteristic
231#117—1811no—
232C2×C4⋊C4—1811no—
233C2×C22⋊C4—3621no—
234#107—1811no—
235#107—1811no—
236#106—1811no—
237C4⋊C8C2×D18111yescharacteristic, complemented
238#103—1811no—
239#103—1811no—
240#103—1811no—
241#102—1811no—
242#102—1811no—
243#102—1811no—
244#102—1811no—
245#102—1811no—
246#102—1811no—
247#102—1811no—
248C22⋊C8C2×D18111yescharacteristic
249C22⋊C8C2×D18111yescharacteristic, complemented
250C4×C8C2×D18111yescharacteristic, complemented
251C24—3621no—
252#41—3622no—
253#41—7241no—
254C2×Q8—1811no—
255C2×D4—3622no—
256C2×D4—3621no—
257C2×D4—3621no—
258C2×D4—3621no—
259C2×D4—3621no—
260C2×D4—3621no—
261C2×D4—1811no—
262C2×D4—1811no—
263C2×D4—1811no—
264C2×D4—7241no—
265C2×D4D72411yescharacteristic
266C22×C4—3621no—
267C22×C4—3621no—
268C22×C4—3621no—
269C22×C4—3621no—
270C22×C4C22×D18111yescharacteristic
271C22×C4C22×D18111yescharacteristic
272SD16—7241no—
273D8—3622no—
274D8—7241no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D18111yescharacteristic
278C2×C8C22×D18111yescharacteristic
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⋊C4D72411yescharacteristic
290C22⋊C4—3621no—
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⋊C4D72411yescharacteristic
300C22⋊C4D72411yescharacteristic
301C42—3621no—
302C42C22×D18111yescharacteristic
303C23—3621no—
304C23—3621no—
305C23—3621no—
306C23—14481no—
307C23—7242no—
308C23—7242no—
309C23C2×D72411yescharacteristic
310C23C2×D72411yescharacteristic
311Q8—3621no—
312Q8—3621no—
313D4—22no—
314D4—3621no—
315D4—3621no—
316D4—3621no—
317D4—3621no—
318D4—3621no—
319D4—14481no—
320D4—14481no—
321D4—1812no—
322D4—7242no—
323D4—7242no—
324D4—7242no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—3621no—
330C2×C4—3621no—
331C2×C4—3621no—
332C2×C4—3621no—
333C2×C4—3621no—
334C2×C4—14481no—
335C2×C4—14481no—
336C2×C4—14481no—
337C2×C4—7242no—
338C2×C4C2×D72411yescharacteristic
339C2×C4C2×D72411yescharacteristic
340C2×C4C2×D72411yescharacteristic
341C2×C4C2×D72411yescharacteristic
342C2×C4C23×D18111yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—3622no—
353C22—14484no—
354C22—7241no—
355C22—7241no—
356C22—7241no—
357C22—7241no—
358C22—7241no—
359C22—7241no—
360C22—7241no—
361C22—7241no—
362C22#1915383812yes—
363C22C22×D72411yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—7241no—
369C4—7241no—
370C4—7241no—
371C4—7241no—
372C4—7241no—
373C4#41×D18111yescharacteristic
374C4#1915386111yescharacteristic
375C2—22no—
376C2—41no—
377C2—14481no—
378C2—14481no—
379C2—7242no—
380C2#2604192411yescharacteristic
381C2C2×#1915383811yescharacteristic
382C2#2604208411yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC8×D4×C181C2
socleC22×C181C22×D4
hypercenterC8×D4D181
hypocenterC181#3297
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #26041975C2semidirect
2. #26041925C2semidirect
3. #26041634C2semidirect
4. #26041615C2semidirect
5. #26041604C2semidirect
6. #26041580C2semidirect
7. #26041515C2semidirect
8. #26041508C2semidirect
9. #26041495C2semidirect
10. #26041426C2semidirect
11. #26041415C2semidirect
12. C4×D1448C2semidirect
13. #26041276C2semidirect
14. C2×#19153738C2semidirect
15. C8×D4×C181C2semidirect
18. C2×#15668438C22semidirect
22. #19153808C22semidirect
29. #19153790C22semidirect
34. C4×D724C22semidirect
36. #19153775C22semidirect
39. #19153750C22semidirect
40. #19153750C22semidirect
41. #19153748C22semidirect
42. #19153741C22semidirect
43. #19153741C22semidirect
44. #19153741C22semidirect
51. #19153738C22semidirect
53. C22×C8×C181C22semidirect
55. C4⋊C8×C181C22semidirect
57. C22⋊C8×C181C22semidirect
58. C4×C8×C181C22semidirect
191. C181#3297semidirect
205. C8×D4D181semidirect
214. C22×C8C2×D181semidirect
237. C4⋊C8C2×D181semidirect
249. C22⋊C8C2×D181semidirect
250. C4×C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260419752
#260419252
#260416342
#260416152
#260416042
#260415802
#260415152
#260415082
#260414952
#260414262
#260414152
C4×D14482
#260412762
C2×#191537382
C8×D4×C1812
#3297181