Finite Group Databasebeta

← #37545326

Subgroups — #37545326

Order 23168.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153840C211yesmaximal, characteristic, complemented
2#26041923C211yesmaximal, characteristic, complemented
3#26041917C211yesmaximal, characteristic, complemented
4#26041660C211yesmaximal, characteristic, complemented
5#26041525C211yesmaximal, characteristic, complemented
6#26041522C211yesmaximal, characteristic, complemented
7#26041496C211yesmaximal, characteristic, complemented
8#26041492C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041438C211yesmaximal, characteristic, complemented
11#26041413C211yesmaximal, characteristic, complemented
12#26041411C211yesmaximal, characteristic, complemented
13D8×Dic181C211yesmaximal, characteristic, complemented
14#26041355C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×D4×D181C2211yescharacteristic, complemented
18#19153840—24no—
19C2×#15668439C2211yescharacteristic, complemented
20C2×#15668438C2211yescharacteristic, complemented
21C2×#15668437C2211yescharacteristic, complemented
22C2×#15668434C2211yescharacteristic
23C2×#15668434C2211yescharacteristic
24C2×#15668432C2211yescharacteristic, complemented
25#19153805C2211yescharacteristic, complemented
26#19153803C2211yescharacteristic, complemented
27#19153799—21no—
28#19153797—21no—
29#19153795C2211yescharacteristic, complemented
30#19153792C2211yescharacteristic, complemented
31#19153790C2211yescharacteristic
32#19153788—21no—
33#19153788—21no—
34#19153787C2211yescharacteristic
35#19153785—21no—
36#19153784—21no—
37D4×Dic181C2211yescharacteristic, complemented
38D4×Dic181C2211yescharacteristic, complemented
39C4×C22⋊D181C2211yescharacteristic
40C2×C22⋊Dic181—21no—
41C2×#15668426—21no—
42#19153750C2211yescharacteristic, complemented
43#19153745C2211yescharacteristic, complemented
44#19153741C2211yescharacteristic, complemented
45#19153740C2211yescharacteristic, complemented
46#19153738C2211yescharacteristic, complemented
47#19153737C2211yescharacteristic, complemented
48#19153737C2211yescharacteristic, complemented
49#19153737C2211yescharacteristic, complemented
50C22⋊#13089678C2211yescharacteristic, complemented
51#19153728C2211yescharacteristic, complemented
52C8×Dic181C2211yescharacteristic, complemented
53C2×#41×C181C2211yescharacteristic
54C22×D4×C181C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C181—24no—
56C2×SD16×C181C2211yescharacteristic, complemented
57C2×SD16×C181C2211yescharacteristic, complemented
58C2×D8×C181C2211yescharacteristic
59C2×D8×C181C2211yescharacteristic
60C2×M16×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668448—22no—
63#15668448—41no—
64C2×Dic362C2311yescharacteristic
65C2×D724C2311yescharacteristic
66C2×C22⋊D181—21no—
67C2×C22⋊D181—21no—
68C2×C22⋊D181—21no—
69C2×C22⋊D181—21no—
70C2×C22⋊D181D411yescharacteristic
71D4×D181—22no—
72D4×D181—41no—
73C22×Dic181—21no—
74C22×Dic181—21no—
75C22×Dic181—21no—
76C2×C4×D181C2311yescharacteristic
77#15668439—41no—
78#15668438—22no—
79#15668437—22no—
80#15668434—22no—
81#15668434—41no—
82#15668432—22no—
83C2×#13089678C2311yescharacteristic
84C2×#13089678C2311yescharacteristic
85C4⋊Dic181C2311yescharacteristic
86C4⋊Dic181C2311yescharacteristic
87#15668428—21no—
88#15668428—21no—
89#15668428D411yescharacteristic
90C22⋊Dic181—21no—
91C22⋊Dic181—21no—
92C22⋊Dic181—21no—
93C22⋊Dic181—21no—
94C22⋊Dic181—21no—
95C22⋊Dic181—21no—
96C22⋊Dic181D411yescharacteristic
97#15668426—21no—
98#15668426—21no—
99#15668426—21no—
100#15668426—21no—
101#15668426D411yescharacteristic
102C4×Dic181C2311yescharacteristic
103C24×C181—21no—
104#41×C181—22no—
105#41×C181—41no—
106C2×Q8×C181C2311yescharacteristic
107C2×D4×C181—22no—
108C2×D4×C181—21no—
109C2×D4×C181—21no—
110C2×D4×C181—41no—
111C2×D4×C181C2311yescharacteristic
112C2×D4×C181C2311yescharacteristic
113C2×D4×C181C2311yescharacteristic
114C22×C4×C181—21no—
115C22×C4×C181C2311yescharacteristic
116SD16×C181—22no—
117SD16×C181—22no—
118D8×C181—22no—
119D8×C181D414yescomplemented
120M16×C181—22no—
121C2×C8×C181C2311yescharacteristic
122C2×C8×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181—42no—
125C22×D181—42no—
126C22×D181—81no—
127C22×D181C2×D411yescharacteristic
128Dic362—21no—
129Dic362—21no—
130D724—21no—
131D724—21no—
132C22⋊D181—41no—
133C22⋊D181—42no—
134C22⋊D181—42no—
135C22⋊D181—81no—
136C22⋊D181—81no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C2×Dic181—21no—
140C2×Dic181—42no—
141C2×Dic181—81no—
142C2×Dic181—81no—
143C2×Dic181C2×D411yescharacteristic
144C2×Dic181C2×D411yescharacteristic
145C2×Dic181C2×D411yescharacteristic
146C4×D181—21no—
147C4×D181—21no—
148#13089678—21no—
149#13089678—21no—
150C23×C181—21no—
151C23×C181—21no—
152C23×C181—21no—
153C23×C181—42no—
154C23×C181—42no—
155C23×C181—81no—
156C23×C181C2×D411yescharacteristic
157Q8×C181—21no—
158Q8×C181—21no—
159D4×C181—21no—
160D4×C181—21no—
161D4×C181—21no—
162D4×C181—21no—
163D4×C181—42no—
164D4×C181—42no—
165D4×C181C2×D412yes—
166D4×C181C2×D412yes—
167C2×C4×C181—21no—
168C2×C4×C181—21no—
169C2×C4×C181—21no—
170C2×C4×C181—42no—
171C2×C4×C181C2×D411yescharacteristic
172C2×C4×C181C2411yescharacteristic
173C8×C181—21no—
174C8×C181C2×D412yes—
175C22×C181—21no—
176C22×C181—22no—
177C22×C181—22no—
178C22×C181—21no—
179C22×C181—21no—
180C22×C181—41no—
181C22×C181—41no—
182C22×C181—41no—
183C22×C181—41no—
184C22×C181—41no—
185C22×C181—41no—
186C22×C181—41no—
187C22×C181—84no—
188C22×C181C22×D411yescharacteristic
189C2×D181—21no—
190C2×D181—21no—
191C2×D181—21no—
192C2×D181—41no—
193C2×D181—41no—
194C2×D181—41no—
195C2×D181—84no—
196C4×C181—21no—
197C4×C181—41no—
198C4×C181C22×D411yescharacteristic
199C4×C181H(5,𝔽2)11yescharacteristic
200Dic181—21no—
201Dic181—21no—
202Dic181—21no—
203Dic181—41no—
204Dic181—41no—
205Dic181—41no—
206C2×C181—42no—
207C2×C181—41no—
208C2×C181—42no—
209C2×C181—81no—
210C2×C181D4211yescharacteristic
211C2×C181#57511yescharacteristic
212C2×C181#57611yescharacteristic
213D181—41no—
214D181—81no—
215C181#327311yescharacteristic, complemented, Sylow
216#3273—1811nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1811no—
218C2×AGL(1,ℤ/8)D18111yescharacteristic, complemented
219#546—1811no—
220#546—1811no—
221#495—1811no—
222#468—1811no—
223#468—1811no—
224#463—1811no—
225#463—1811no—
226#452—1811no—
227#452—1811no—
228#447—1811no—
229#447—1811no—
230C4×D8—1811no—
231#436—1811no—
232C2×#41—1811no—
233C2×#41C2×D18111yescharacteristic
234C22×D4—1811no—
235C22×D4C2×D18111yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—3624no—
238C2×Q16—1811no—
239C2×SD16—1811no—
240C2×SD16—1811no—
241C2×SD16C2×D18111yescharacteristic, complemented
242C2×SD16C2×D18111yescharacteristic, complemented
243C2×D8—1811no—
244C2×D8—1811no—
245C2×D8C2×D18111yescharacteristic
246C2×D8C2×D18111yescharacteristic
247C2×M16—1811no—
248C2×M16C2×D18111yescharacteristic, complemented
249#124—1811no—
250#124—1811no—
251#123—3621no—
252#123—3621no—
253C22⋊Q8—1811no—
254C22⋊Q8—1811no—
255#121—3621no—
256#121—3621no—
257#121—1811no—
258#121—1811no—
259#120—3621no—
260#120—3621no—
261C4×D4—1811no—
262C4×D4—1811no—
263C4×D4—1811no—
264C2×C22⋊C4—3621no—
265C2×C22⋊C4—3621no—
266#107—1811no—
267C4⋊C8—1811no—
268#103—1811no—
269#103—1811no—
270#102—1811no—
271#102—1811no—
272#102—1811no—
273#102—1811no—
274C22⋊C8—1811no—
275C22⋊C8—1811no—
276C4×C8—1811no—
277C24—21no—
278C24—3621no—
279#41—22no—
280#41—41no—
281#41—3622no—
282#41—7241no—
283C2×Q8—1811no—
284C2×Q8C22×D18111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—3622no—
290C2×D4—3621no—
291C2×D4—3621no—
292C2×D4—3621no—
293C2×D4—3621no—
294C2×D4—1811no—
295C2×D4—1811no—
296C2×D4—7241no—
297C2×D4C22×D18111yescharacteristic
298C2×D4C22×D18111yescharacteristic
299C2×D4C22×D18111yescharacteristic
300C22×C4—21no—
301C22×C4—3621no—
302C22×C4—3621no—
303C22×C4—3621no—
304C22×C4—1811no—
305C22×C4C22×D18111yescharacteristic
306Q16—7241no—
307SD16—22no—
308SD16—22no—
309SD16—3622no—
310SD16—3622no—
311D8—22no—
312D8—3622no—
313D8—7241no—
314D8C22⋊D18114yescomplemented
315M16—22no—
316M16—3622no—
317C2×C8—1811no—
318C2×C8—1811no—
319C2×C8C22×D18111yescharacteristic
320C2×C8C22×D18111yescharacteristic
321C4⋊C4—3621no—
322C4⋊C4—3621no—
323C4⋊C4—1811no—
324C4⋊C4—1811no—
325C4⋊C4—1811no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4—3621no—
329C22⋊C4—3621no—
330C22⋊C4—3621no—
331C22⋊C4—3621no—
332C22⋊C4—3621no—
333C22⋊C4—3621no—
334C22⋊C4—3621no—
335C22⋊C4—3621no—
336C22⋊C4—1811no—
337C22⋊C4—1811no—
338C42—1811no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—3621no—
346C23—14481no—
347C23—1811no—
348C23—7242no—
349C23—7242no—
350C23D4×D18111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3621no—
354Q8—3621no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—3621no—
362D4—3621no—
363D4—14481no—
364D4—14481no—
365D4—7241no—
366D4—7242no—
367D4—7242no—
368D4C2×C22⋊D18112yes—
369D4C2×C22⋊D18112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—3621no—
375C2×C4—3621no—
376C2×C4—3621no—
377C2×C4—3621no—
378C2×C4—3621no—
379C2×C4—14481no—
380C2×C4—14481no—
381C2×C4—1811no—
382C2×C4—1811no—
383C2×C4—1811no—
384C2×C4—7242no—
385C2×C4D4×D18111yescharacteristic
386C2×C4C23×D18111yescharacteristic
387C8—21no—
388C8—3621no—
389C8—3621no—
390C8C2×C22⋊D18112yes—
391C22—21no—
392C22—22no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—3621no—
405C22—3621no—
406C22—3621no—
407C22—14484no—
408C22—7241no—
409C22—7241no—
410C22—7241no—
411C22C2×D4×D18111yescharacteristic
412C4—21no—
413C4—41no—
414C4—3621no—
415C4—3621no—
416C4—3621no—
417C4—7241no—
418C4—7241no—
419C4—7241no—
420C4C22×C22⋊D18111yescharacteristic
421C4#1915386111yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—14481no—
427C2—7241no—
428C2D4×C22⋊D18111yescharacteristic
429C2#2604207511yescharacteristic
430C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×AGL(1,ℤ/8)×C181C2
socleC22×C181C22×D4
hypercenterC2×AGL(1,ℤ/8)D181
hypocenterC181#3273
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19153840C2semidirect
2. #26041923C2semidirect
3. #26041917C2semidirect
4. #26041660C2semidirect
5. #26041525C2semidirect
6. #26041522C2semidirect
7. #26041496C2semidirect
8. #26041492C2semidirect
9. #26041441C2semidirect
10. #26041438C2semidirect
11. #26041413C2semidirect
12. #26041411C2semidirect
13. D8×Dic181C2semidirect
14. #26041355C2semidirect
15. C2×AGL(1,ℤ/8)×C181C2semidirect
17. C2×D4×D181C22semidirect
19. C2×#15668439C22semidirect
20. C2×#15668438C22semidirect
21. C2×#15668437C22semidirect
24. C2×#15668432C22semidirect
25. #19153805C22semidirect
26. #19153803C22semidirect
29. #19153795C22semidirect
30. #19153792C22semidirect
37. D4×Dic181C22semidirect
38. D4×Dic181C22semidirect
42. #19153750C22semidirect
43. #19153745C22semidirect
44. #19153741C22semidirect
45. #19153740C22semidirect
46. #19153738C22semidirect
47. #19153737C22semidirect
48. #19153737C22semidirect
49. #19153737C22semidirect
50. C22⋊#13089678C22semidirect
51. #19153728C22semidirect
52. C8×Dic181C22semidirect
54. C22×D4×C181C22semidirect
56. C2×SD16×C181C22semidirect
57. C2×SD16×C181C22semidirect
60. C2×M16×C181C22semidirect
119. D8×C181D4semidirect
215. C181#3273semidirect
218. C2×AGL(1,ℤ/8)D181semidirect
235. C22×D4C2×D181semidirect
241. C2×SD16C2×D181semidirect
242. C2×SD16C2×D181semidirect
248. C2×M16C2×D181semidirect
314. D8C22⋊D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538402
#260419232
#260419172
#260416602
#260415252
#260415222
#260414962
#260414922
#260414412
#260414382
#260414132
#260414112
D8×Dic1812
#260413552
C2×AGL(1,ℤ/8)×C1812
#3273181