Finite Group Databasebeta

← #37545399

Subgroups — #37545399

Order 23168.

526 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×#19153837C211yesmaximal, characteristic, complemented
2C22×#15668434C211yesmaximal, characteristic, complemented
3#26041923C211yesmaximal, characteristic, complemented
4#26041920C211yesmaximal, characteristic, complemented
5#26041660C211yesmaximal, characteristic, complemented
6#26041518C211yesmaximal, characteristic, complemented
7#26041514C211yesmaximal, characteristic, complemented
8#26041496C211yesmaximal, characteristic, complemented
9#26041493C211yesmaximal, characteristic, complemented
10#26041432C211yesmaximal, characteristic, complemented
11#26041425C211yesmaximal, characteristic, complemented
12#26041423C211yesmaximal, characteristic, complemented
13#26041359C211yesmaximal, characteristic, complemented
14#26041342C211yesmaximal, characteristic, complemented
15#450×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic, complemented
17C2×#15668448C2211yescharacteristic, complemented
18C22×D724C2211yescharacteristic
19#19153837—24no—
20C2×#15668439C2211yescharacteristic, complemented
21C2×#15668438C2211yescharacteristic, complemented
22C2×#15668438C2211yescharacteristic, complemented
23C2×#15668434—24no—
24C2×#15668434C2211yescharacteristic
25C2×#15668434C2211yescharacteristic
26C2×#15668434C2211yescharacteristic
27C2×#15668434C2211yescharacteristic
28C22×#13089678C2211yescharacteristic
29C2×C8×D181C2211yescharacteristic, complemented
30#19153805C2211yescharacteristic, complemented
31#19153803C2211yescharacteristic, complemented
32#19153801—21no—
33#19153797—21no—
34#19153795C2211yescharacteristic
35#19153791—21no—
36#19153791C2211yescharacteristic, complemented
37#19153788—21no—
38#19153787C2211yescharacteristic, complemented
39#19153786—21no—
40#19153785—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#19153740C2211yescharacteristic, complemented
49#19153739C2211yescharacteristic, complemented
50#19153738C2211yescharacteristic, complemented
51#19153738C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic, complemented
53C8×Dic181C2211yescharacteristic, complemented
54C22×D4×C181C2211yescharacteristic
55C2×SD16×C181C2211yescharacteristic, complemented
56C2×SD16×C181C2211yescharacteristic, complemented
57C22⋊Q8×C181C2211yescharacteristic
58#102×C181C2211yescharacteristic, complemented
59#102×C181C2211yescharacteristic, complemented
60C22⋊C8×C181C2211yescharacteristic
61C23×D181—21no—
62#15668450—22no—
63#15668450—41no—
64#15668448—22no—
65#15668448—41no—
66C2×Dic362C2311yescharacteristic
67C2×D724—22no—
68C2×D724—21no—
69C2×D724—21no—
70C2×D724—21no—
71C2×D724—41no—
72C2×D724C2311yescharacteristic
73C2×D724C2311yescharacteristic
74C2×C22⋊D181—21no—
75C2×C22⋊D181—21no—
76C2×C22⋊D181D411yescharacteristic
77C22×Dic181—21no—
78C22×Dic181—21no—
79C2×C4×D181—21no—
80C2×C4×D181—21no—
81C2×C4×D181C2311yescharacteristic
82#15668439—22no—
83#15668438—22no—
84#15668438—41no—
85#15668434—22no—
86#15668434—22no—
87#15668434—22no—
88#15668434D414yescomplemented
89C2×#13089678—22no—
90C2×#13089678C2311yescharacteristic
91C2×#13089678C2311yescharacteristic
92C8×D181—22no—
93C4⋊Dic181C2311yescharacteristic
94#15668428—21no—
95#15668428D411yescharacteristic
96C22⋊Dic181—21no—
97C22⋊Dic181—21no—
98C22⋊Dic181—21no—
99C22⋊Dic181—21no—
100C22⋊Dic181—21no—
101C22⋊Dic181D411yescharacteristic
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106#15668426D411yescharacteristic
107C4×Dic181C2311yescharacteristic
108C24×C181—21no—
109C2×Q8×C181C2311yescharacteristic
110C2×D4×C181—22no—
111C2×D4×C181—21no—
112C2×D4×C181—41no—
113C2×D4×C181C2311yescharacteristic
114C2×D4×C181C2311yescharacteristic
115C22×C4×C181C2311yescharacteristic
116SD16×C181—22no—
117SD16×C181—41no—
118C2×C8×C181C2311yescharacteristic
119C2×C8×C181C2311yescharacteristic
120C4⋊C4×C181—21no—
121C4⋊C4×C181C2311yescharacteristic
122C22⋊C4×C181—21no—
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—42no—
126C22×D181—42no—
127C22×D181—81no—
128C22×D181C2×D411yescharacteristic
129Dic362—21no—
130Dic362—21no—
131D724—21no—
132D724—21no—
133D724—21no—
134D724—42no—
135D724—42no—
136D724—81no—
137D724C2×D412yes—
138C22⋊D181—22no—
139C22⋊D181—42no—
140C22⋊D181—81no—
141C2×Dic181—21no—
142C2×Dic181—21no—
143C2×Dic181—42no—
144C2×Dic181—81no—
145C2×Dic181C2×D411yescharacteristic
146C2×Dic181C2×D411yescharacteristic
147C2×Dic181C2×D411yescharacteristic
148C4×D181—21no—
149C4×D181—21no—
150C4×D181—42no—
151C4×D181—81no—
152#13089678—21no—
153#13089678C2×D412yes—
154C23×C181—21no—
155C23×C181—21no—
156C23×C181—42no—
157C23×C181—42no—
158C23×C181—81no—
159C23×C181C2×D411yescharacteristic
160Q8×C181—21no—
161Q8×C181—21no—
162D4×C181—21no—
163D4×C181—21no—
164D4×C181—21no—
165D4×C181—42no—
166D4×C181C2×D412yes—
167C2×C4×C181—21no—
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—21no—
177C22×C181—22no—
178C22×C181—41no—
179C22×C181—41no—
180C22×C181—41no—
181C22×C181—41no—
182C22×C181—41no—
183C22×C181—84no—
184C22×C181#13512yescomplemented
185C22×C181C22×D411yescharacteristic
186C2×D181—21no—
187C2×D181—22no—
188C2×D181—21no—
189C2×D181—21no—
190C2×D181—41no—
191C2×D181—41no—
192C2×D181—41no—
193C2×D181—41no—
194C2×D181—41no—
195C2×D181—84no—
196C4×C181—21no—
197C4×C181—41no—
198C4×C181—41no—
199C4×C181C22×D411yescharacteristic
200C4×C181H(5,𝔽2)11yescharacteristic
201Dic181—21no—
202Dic181—21no—
203Dic181—21no—
204Dic181—41no—
205Dic181—41no—
206C2×C181—22no—
207C2×C181—42no—
208C2×C181—81no—
209C2×C181D4211yescharacteristic
210C2×C181C2×#13511yescharacteristic
211C2×C181#57611yescharacteristic
212D181—41no—
213D181—42no—
214D181—81no—
215C181#328011yescharacteristic, complemented, Sylow
216#3280—1811nomaximal, Sylow
217C2×#135—1811no—
218C22×D8—1811no—
219#546—1811no—
220#546—1811no—
221#495—1811no—
222#467—1811no—
223#466—1811no—
224#463—1811no—
225#463—1811no—
226#450—1811no—
227#450D18111yescharacteristic, complemented
228#449—1811no—
229#449—1811no—
230C4×D8—1811no—
231C8×D4—1811no—
232C2×#41—1811no—
233C2×#41—1811no—
234C22×D4—1811no—
235C22×D4C2×D18111yescharacteristic
236#135—3624no—
237C2×Q16—1811no—
238C2×SD16—1811no—
239C2×SD16—1811no—
240C2×SD16C2×D18111yescharacteristic, complemented
241C2×SD16C2×D18111yescharacteristic, complemented
242C2×D8—3624no—
243C2×D8—1811no—
244C2×D8—1811no—
245C2×D8—1811no—
246C2×D8—1811no—
247C22×C8—1811no—
248C22×C8—1811no—
249#124—1811no—
250#124—1811no—
251#123—3621no—
252#123—3621no—
253C22⋊Q8—1811no—
254C22⋊Q8C2×D18111yescharacteristic
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#102C2×D18111yescharacteristic, complemented
273#102C2×D18111yescharacteristic, complemented
274C22⋊C8—1811no—
275C22⋊C8C2×D18111yescharacteristic
276C4×C8—1811no—
277C24—21no—
278C24—3621no—
279#41—3622no—
280#41—3622no—
281#41—7241no—
282#41—7241no—
283C2×Q8—1811no—
284C2×Q8C22×D18111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—41no—
288C2×D4—3622no—
289C2×D4—3621no—
290C2×D4—3621no—
291C2×D4—3621no—
292C2×D4—3621no—
293C2×D4—3621no—
294C2×D4—1811no—
295C2×D4—1811no—
296C2×D4—1811no—
297C2×D4—7241no—
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—3622no—
307SD16—22no—
308SD16—41no—
309SD16—3622no—
310SD16—7241no—
311D8—3622no—
312D8—3622no—
313D8—3622no—
314D8—1814no—
315C2×C8—3622no—
316C2×C8—3622no—
317C2×C8—1811no—
318C2×C8—1811no—
319C2×C8C22×D18111yescharacteristic
320C2×C8C22×D18111yescharacteristic
321C4⋊C4—21no—
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—42no—
342C23—42no—
343C23—81no—
344C23—3621no—
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—42no—
359D4—3622no—
360D4—3621no—
361D4—3621no—
362D4—3621no—
363D4—14481no—
364D4—14481no—
365D4—1812no—
366D4—7242no—
367D4—7242no—
368D4—7242no—
369D4D4×D18112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
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—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—84no—
400C22—3621no—
401C22—3622no—
402C22—3621no—
403C22—3621no—
404C22—14484no—
405C22—7241no—
406C22—7241no—
407C22—7241no—
408C22—7241no—
409C22—7241no—
410C22#1915383712yescomplemented
411C22C2×D4×D18111yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—3621no—
416C4—3621no—
417C4—3621no—
418C4—7241no—
419C4—7241no—
420C4C2×D4×D18111yescharacteristic
421C4#1915386111yescharacteristic
422C2—22no—
423C2—42no—
424C2—81no—
425C2—14481no—
426C2—7241no—
427C2—7242no—
428C2#2604191111yescharacteristic
429C2C2×#1915383711yescharacteristic
430C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#450×C181C2
socleC22×C181C22×D4
hypercenter#450D181
hypocenterC181#3280
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#19153837C2semidirect
2. C22×#15668434C2semidirect
3. #26041923C2semidirect
4. #26041920C2semidirect
5. #26041660C2semidirect
6. #26041518C2semidirect
7. #26041514C2semidirect
8. #26041496C2semidirect
9. #26041493C2semidirect
10. #26041432C2semidirect
11. #26041425C2semidirect
12. #26041423C2semidirect
13. #26041359C2semidirect
14. #26041342C2semidirect
15. #450×C181C2semidirect
16. C2×#15668450C22semidirect
17. C2×#15668448C22semidirect
20. C2×#15668439C22semidirect
21. C2×#15668438C22semidirect
22. C2×#15668438C22semidirect
29. C2×C8×D181C22semidirect
30. #19153805C22semidirect
31. #19153803C22semidirect
36. #19153791C22semidirect
38. #19153787C22semidirect
41. D4×Dic181C22semidirect
42. #19153778C22semidirect
46. #19153749C22semidirect
47. #19153745C22semidirect
48. #19153740C22semidirect
49. #19153739C22semidirect
50. #19153738C22semidirect
51. #19153738C22semidirect
52. #19153728C22semidirect
53. C8×Dic181C22semidirect
55. C2×SD16×C181C22semidirect
56. C2×SD16×C181C22semidirect
58. #102×C181C22semidirect
59. #102×C181C22semidirect
88. #15668434D4semidirect
184. C22×C181#135semidirect
215. C181#3280semidirect
227. #450D181semidirect
240. C2×SD16C2×D181semidirect
241. C2×SD16C2×D181semidirect
272. #102C2×D181semidirect
273. #102C2×D181semidirect
410. C22#19153837semidirect

Maximal subgroups

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

subgroupindex
C2×#191538372
C22×#156684342
#260419232
#260419202
#260416602
#260415182
#260415142
#260414962
#260414932
#260414322
#260414252
#260414232
#260413592
#260413422
#450×C1812
#3280181