Finite Group Databasebeta

← #37545332

Subgroups — #37545332

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×#19153842C211yesmaximal, characteristic, complemented
2C2×#19153840C211yesmaximal, characteristic, complemented
3#26041926C211yesmaximal, characteristic, complemented
4#26041923C211yesmaximal, characteristic, complemented
5#26041662C211yesmaximal, characteristic, complemented
6#26041524C211yesmaximal, characteristic, complemented
7#26041523C211yesmaximal, characteristic, complemented
8#26041493C211yesmaximal, characteristic, complemented
9#26041492C211yesmaximal, characteristic, complemented
10#26041443C211yesmaximal, characteristic, complemented
11#26041442C211yesmaximal, characteristic, complemented
12#26041411C211yesmaximal, characteristic, complemented
13#26041359C211yesmaximal, characteristic, complemented
14#26041354C211yesmaximal, characteristic, complemented
15#447×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C2×#15668448C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic, complemented
19#19153842—24no—
20#19153840—24no—
21C2×Dic724C2211yescharacteristic, complemented
22C2×#15668438C2211yescharacteristic, complemented
23C2×#15668437C2211yescharacteristic, complemented
24C2×#15668437C2211yescharacteristic, complemented
25C2×#15668437C2211yescharacteristic, complemented
26C2×#15668434C2211yescharacteristic
27C2×#15668434C2211yescharacteristic
28C2×#15668433C2211yescharacteristic, complemented
29C2×#15668432C2211yescharacteristic, complemented
30#19153803C2211yescharacteristic, complemented
31#19153803C2211yescharacteristic, complemented
32#19153798—21no—
33#19153797—21no—
34#19153796C2211yescharacteristic, complemented
35#19153795C2211yescharacteristic, complemented
36#19153788—21no—
37#19153787—21no—
38#19153787C2211yescharacteristic
39#19153785—21no—
40#19153784—21no—
41D4×Dic181C2211yescharacteristic, complemented
42#19153778C2211yescharacteristic, complemented
43C4×C22⋊D181C2211yescharacteristic
44C2×C22⋊Dic181—21no—
45C22⋊C4×D181—21no—
46#19153749C2211yescharacteristic, complemented
47#19153745C2211yescharacteristic, complemented
48#19153741C2211yescharacteristic, complemented
49#19153739C2211yescharacteristic, complemented
50#19153738C2211yescharacteristic, complemented
51#19153737C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic, complemented
53C8×Dic181C2211yescharacteristic, complemented
54C22×D4×C181C2211yescharacteristic, complemented
55C2×D8×C181C2211yescharacteristic
56C2×D8×C181C2211yescharacteristic
57#121×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic, complemented
59#102×C181C2211yescharacteristic, complemented
60C22⋊C8×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668449—22no—
63#15668449—41no—
64#15668448—22no—
65#15668448—41no—
66C2×Dic362C2311yescharacteristic
67C2×Dic362C2311yescharacteristic
68C2×D724C2311yescharacteristic
69C2×C22⋊D181—21no—
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181D411yescharacteristic
74D4×D181—22no—
75D4×D181—41no—
76C22×Dic181—21no—
77C22×Dic181—21no—
78C2×C4×D181—21no—
79C2×C4×D181—21no—
80C2×C4×D181C2311yescharacteristic
81Dic724—41no—
82#15668438—22no—
83#15668437—22no—
84#15668437—22no—
85#15668437—22no—
86#15668434—22no—
87#15668434D414yescomplemented
88#15668433—22no—
89#15668432—22no—
90C2×#13089678C2311yescharacteristic
91C2×#13089678C2311yescharacteristic
92C4⋊Dic181—21no—
93C4⋊Dic181C2311yescharacteristic
94#15668428—21no—
95#15668428D411yescharacteristic
96C22⋊Dic181—21no—
97C22⋊Dic181—21no—
98C22⋊Dic181—21no—
99C22⋊Dic181—21no—
100C22⋊Dic181—21no—
101C22⋊Dic181—21no—
102C22⋊Dic181—21no—
103C22⋊Dic181D411yescharacteristic
104#15668426—21no—
105#15668426—21no—
106#15668426D411yescharacteristic
107C4×Dic181C2311yescharacteristic
108C24×C181—21no—
109C2×D4×C181—22no—
110C2×D4×C181—21no—
111C2×D4×C181—21no—
112C2×D4×C181—41no—
113C2×D4×C181C2311yescharacteristic
114C2×D4×C181C2311yescharacteristic
115C2×D4×C181C2311yescharacteristic
116C22×C4×C181C2311yescharacteristic
117D8×C181—22no—
118D8×C181—41no—
119C2×C8×C181C2311yescharacteristic
120C2×C8×C181C2311yescharacteristic
121C4⋊C4×C181C2311yescharacteristic
122C22⋊C4×C181—21no—
123C22×D181—21no—
124C22×D181—42no—
125C22×D181—42no—
126C22×D181—81no—
127C22×D181C2×D411yescharacteristic
128Dic362—21no—
129Dic362—21no—
130Dic362—21no—
131Dic362—21no—
132D724—21no—
133D724C2×D412yes—
134C22⋊D181—41no—
135C22⋊D181—42no—
136C22⋊D181—42no—
137C22⋊D181—42no—
138C22⋊D181—81no—
139C2×Dic181—21no—
140C2×Dic181—21no—
141C2×Dic181—21no—
142C2×Dic181—42no—
143C2×Dic181—81no—
144C2×Dic181C2×D411yescharacteristic
145C2×Dic181C2×D411yescharacteristic
146C2×Dic181C2×D411yescharacteristic
147C4×D181—21no—
148C4×D181—21no—
149C4×D181—42no—
150C4×D181—81no—
151#13089678—21no—
152#13089678C2×D412yes—
153C23×C181—21no—
154C23×C181—21no—
155C23×C181—21no—
156C23×C181—42no—
157C23×C181—42no—
158C23×C181—81no—
159C23×C181C2×D411yescharacteristic
160D4×C181—21no—
161D4×C181—21no—
162D4×C181—21no—
163D4×C181—21no—
164D4×C181—21no—
165D4×C181—42no—
166D4×C181—81no—
167D4×C181C2×D412yes—
168C2×C4×C181—21no—
169C2×C4×C181—21no—
170C2×C4×C181—21no—
171C2×C4×C181C2×D411yescharacteristic
172C2×C4×C181C2411yescharacteristic
173C8×C181—21no—
174C8×C181—21no—
175C22×C181—21no—
176C22×C181—22no—
177C22×C181—21no—
178C22×C181—21no—
179C22×C181—41no—
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—22no—
191C2×D181—21no—
192C2×D181—21no—
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—41no—
207C2×C181—42no—
208C2×C181—81no—
209C2×C181—81no—
210C2×C181D4211yescharacteristic
211C2×C181#57511yescharacteristic
212C2×C181#57611yescharacteristic
213D181—41no—
214D181—42no—
215C181#327311yescharacteristic, complemented, Sylow
216#3273—1811nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1811no—
218C2×AGL(1,ℤ/8)—1811no—
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#447D18111yescharacteristic, complemented
230C4×D8—1811no—
231#436—1811no—
232C2×#41—1811no—
233C2×#41—1811no—
234C22×D4—1811no—
235C22×D4C2×D18111yescharacteristic, complemented
236AGL(1,ℤ/8)—3624no—
237AGL(1,ℤ/8)—3624no—
238C2×Q16—1811no—
239C2×SD16—1811no—
240C2×SD16—1811no—
241C2×SD16—1811no—
242C2×SD16—1811no—
243C2×D8—1811no—
244C2×D8—1811no—
245C2×D8C2×D18111yescharacteristic
246C2×D8C2×D18111yescharacteristic
247C2×M16—1811no—
248C2×M16—1811no—
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#121C2×D18111yescharacteristic
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#102C2×D18111yescharacteristic, complemented
273#102C2×D18111yescharacteristic, complemented
274C22⋊C8—1811no—
275C22⋊C8C2×D18111yescharacteristic, complemented
276C4×C8—1811no—
277C24—21no—
278C24—3621no—
279#41—3622no—
280#41—3622no—
281#41—7241no—
282#41—7241no—
283C2×Q8—1811no—
284C2×Q8—1811no—
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—3621no—
301C22×C4—3621no—
302C22×C4—3621no—
303C22×C4—3621no—
304C22×C4—1811no—
305C22×C4C22×D18111yescharacteristic
306Q16—7241no—
307SD16—3622no—
308SD16—3622no—
309SD16—3622no—
310SD16—3622no—
311D8—22no—
312D8—41no—
313D8—3622no—
314D8—1814no—
315M16—3622no—
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⋊C4C22×D18111yescharacteristic
326C22⋊C4—21no—
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—3621no—
352Q8—3621no—
353Q8—3621no—
354Q8—3621no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—21no—
360D4—42no—
361D4—81no—
362D4—3621no—
363D4—14481no—
364D4—1812no—
365D4—7241no—
366D4—7242no—
367D4—7242no—
368D4—7242no—
369D4D4×D18112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—3621no—
374C2×C4—3621no—
375C2×C4—3621no—
376C2×C4—3621no—
377C2×C4—3621no—
378C2×C4—14481no—
379C2×C4—14481no—
380C2×C4—1811no—
381C2×C4—1811no—
382C2×C4—1811no—
383C2×C4—7242no—
384C2×C4—7242no—
385C2×C4D4×D18111yescharacteristic
386C2×C4C23×D18111yescharacteristic
387C8—21no—
388C8—21no—
389C8—3621no—
390C8—1812no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—3621no—
405C22—3622no—
406C22—3621no—
407C22—3621no—
408C22—14484no—
409C22—7241no—
410C22—7241no—
411C22C2×D4×D18111yescharacteristic
412C4—21no—
413C4—41no—
414C4—3621no—
415C4—3621no—
416C4—3621no—
417C4—7241no—
418C4—7241no—
419C4—7241no—
420C4C2×D4×D18111yescharacteristic
421C4#1915386111yescharacteristic
422C2—41no—
423C2—42no—
424C2—81no—
425C2—81no—
426C2—7241no—
427C2—7242no—
428C2#2604191111yescharacteristic
429C2#2604207511yescharacteristic
430C2#2604208011yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#447×C181C2
socleC22×C181C22×D4
hypercenter#447D181
hypocenterC181#3273
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#19153842C2semidirect
2. C2×#19153840C2semidirect
3. #26041926C2semidirect
4. #26041923C2semidirect
5. #26041662C2semidirect
6. #26041524C2semidirect
7. #26041523C2semidirect
8. #26041493C2semidirect
9. #26041492C2semidirect
10. #26041443C2semidirect
11. #26041442C2semidirect
12. #26041411C2semidirect
13. #26041359C2semidirect
14. #26041354C2semidirect
15. #447×C181C2semidirect
18. C2×D4×D181C22semidirect
21. C2×Dic724C22semidirect
22. C2×#15668438C22semidirect
23. C2×#15668437C22semidirect
24. C2×#15668437C22semidirect
25. C2×#15668437C22semidirect
28. C2×#15668433C22semidirect
29. C2×#15668432C22semidirect
30. #19153803C22semidirect
31. #19153803C22semidirect
34. #19153796C22semidirect
35. #19153795C22semidirect
41. D4×Dic181C22semidirect
42. #19153778C22semidirect
46. #19153749C22semidirect
47. #19153745C22semidirect
48. #19153741C22semidirect
49. #19153739C22semidirect
50. #19153738C22semidirect
51. #19153737C22semidirect
52. #19153728C22semidirect
53. C8×Dic181C22semidirect
54. C22×D4×C181C22semidirect
58. #102×C181C22semidirect
59. #102×C181C22semidirect
60. C22⋊C8×C181C22semidirect
87. #15668434D4semidirect
215. C181#3273semidirect
229. #447D181semidirect
235. C22×D4C2×D181semidirect
272. #102C2×D181semidirect
273. #102C2×D181semidirect
275. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538422
C2×#191538402
#260419262
#260419232
#260416622
#260415242
#260415232
#260414932
#260414922
#260414432
#260414422
#260414112
#260413592
#260413542
#447×C1812
#3273181