Finite Group Databasebeta

← #37545328

Subgroups — #37545328

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×#19153841C211yesmaximal, characteristic, complemented
3#26041925C211yesmaximal, characteristic, complemented
4#26041661C211yesmaximal, characteristic, complemented
5#26041527C211yesmaximal, characteristic, complemented
6#26041523C211yesmaximal, characteristic, complemented
7#26041494C211yesmaximal, characteristic, complemented
8#26041491C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041439C211yesmaximal, characteristic, complemented
11#26041413C211yesmaximal, characteristic, complemented
12#26041412C211yesmaximal, characteristic, complemented
13C4×#15668434C211yesmaximal, characteristic, complemented
14#26041357C211yesmaximal, characteristic, complemented
15#546×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×D724C2211yescharacteristic, complemented
18#19153842—24no—
19#19153841—24no—
20C2×#15668439C2211yescharacteristic, complemented
21C2×#15668437C2211yescharacteristic, complemented
22C2×#15668437C2211yescharacteristic, complemented
23C2×#15668436C2211yescharacteristic, complemented
24C2×#15668436C2211yescharacteristic, complemented
25C2×#15668434C2211yescharacteristic
26C2×#15668434C2211yescharacteristic
27C2×#15668434C2211yescharacteristic
28C2×#15668434C2211yescharacteristic
29C2×#15668433C2211yescharacteristic, complemented
30C2×#15668433C2211yescharacteristic, complemented
31#19153804C2211yescharacteristic, complemented
32#19153801—21no—
33#19153795C2211yescharacteristic, complemented
34#19153790C2211yescharacteristic
35#19153789—21no—
36#19153784—21no—
37C4×D724C2211yescharacteristic, complemented
38C22⋊C4×D181—21no—
39#19153749C2211yescharacteristic, complemented
40C4⋊#13089678C2211yescharacteristic, complemented
41#19153740C2211yescharacteristic, complemented
42#19153739C2211yescharacteristic, complemented
43#19153737C2211yescharacteristic, complemented
44#19153737C2211yescharacteristic, complemented
45#19153736C2211yescharacteristic, complemented
46#19153736C2211yescharacteristic, complemented
47C22⋊#13089678C2211yescharacteristic, complemented
48C22⋊#13089678C2211yescharacteristic, complemented
49C4×#13089678C2211yescharacteristic, complemented
50C2×#41×C181C2211yescharacteristic
51C22×D4×C181C2211yescharacteristic, complemented
52#124×C181C2211yescharacteristic, complemented
53#123×C181—21no—
54C22⋊Q8×C181C2211yescharacteristic, complemented
55#121×C181—21no—
56#121×C181C2211yescharacteristic
57#120×C181—21no—
58C4×D4×C181C2211yescharacteristic
59C4×D4×C181C2211yescharacteristic, complemented
60C2×C22⋊C4×C181—21no—
61C23×D181—21no—
62#15668449—22no—
63#15668449—41no—
64C2×Dic362C2311yescharacteristic
65C2×D724—22no—
66C2×D724—21no—
67C2×D724—41no—
68C2×D724C2311yescharacteristic
69C2×D724C2311yescharacteristic
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C4×D181—21no—
73C2×C4×D181—21no—
74#15668439—41no—
75#15668437—22no—
76#15668437—22no—
77#15668436—22no—
78#15668436—22no—
79#15668434—22no—
80#15668434—22no—
81#15668434—41no—
82#15668434D414yescomplemented
83#15668433—22no—
84#15668433—22no—
85C2×#13089678C2311yescharacteristic
86C2×#13089678C2311yescharacteristic
87C2×#13089678C2311yescharacteristic
88C2×#13089678C2311yescharacteristic
89C4⋊Dic181C2311yescharacteristic
90#15668428—21no—
91C22⋊Dic181—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96C24×C181—21no—
97#41×C181—22no—
98#41×C181—41no—
99C2×Q8×C181C2311yescharacteristic
100C2×D4×C181—22no—
101C2×D4×C181—21no—
102C2×D4×C181—21no—
103C2×D4×C181—21no—
104C2×D4×C181—41no—
105C2×D4×C181D411yescharacteristic
106C2×D4×C181C2311yescharacteristic
107C2×D4×C181C2311yescharacteristic
108C22×C4×C181—21no—
109C22×C4×C181—21no—
110C22×C4×C181C2311yescharacteristic
111C22×C4×C181C2311yescharacteristic
112C4⋊C4×C181—21no—
113C4⋊C4×C181D411yescharacteristic
114C4⋊C4×C181C2311yescharacteristic
115C22⋊C4×C181—21no—
116C22⋊C4×C181—21no—
117C22⋊C4×C181—21no—
118C22⋊C4×C181—21no—
119C22⋊C4×C181—21no—
120C22⋊C4×C181D411yescharacteristic
121C22⋊C4×C181D411yescharacteristic
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—42no—
126C22×D181—42no—
127C22×D181—81no—
128Dic362—21no—
129Dic362—21no—
130D724—21no—
131D724—21no—
132D724—21no—
133D724—42no—
134D724C2×D412yes—
135C22⋊D181—42no—
136C22⋊D181—81no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C4×D181—42no—
140C4×D181—81no—
141#13089678—21no—
142#13089678—21no—
143#13089678—21no—
144#13089678C2×D412yes—
145C23×C181—21no—
146C23×C181—21no—
147C23×C181—42no—
148C23×C181—42no—
149C23×C181—81no—
150C23×C181C2×D411yescharacteristic
151C23×C181C2×D411yescharacteristic
152Q8×C181—21no—
153Q8×C181—21no—
154D4×C181—21no—
155D4×C181—21no—
156D4×C181—21no—
157D4×C181—41no—
158D4×C181—42no—
159D4×C181—42no—
160D4×C181—81no—
161D4×C181C2×D412yes—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181—21no—
168C2×C4×C181—42no—
169C2×C4×C181—81no—
170C2×C4×C181C2×D411yescharacteristic
171C2×C4×C181C2×D411yescharacteristic
172C2×C4×C181C2×D411yescharacteristic
173C2×C4×C181C2×D411yescharacteristic
174C2×C4×C181C2411yescharacteristic
175C22×C181—21no—
176C22×C181—21no—
177C22×C181—22no—
178C22×C181—21no—
179C22×C181—21no—
180C22×C181—21no—
181C22×C181—21no—
182C22×C181—41no—
183C22×C181—41no—
184C22×C181—41no—
185C22×C181—41no—
186C22×C181—41no—
187C22×C181—84no—
188C22×C181C22×D411yescharacteristic
189C2×D181—22no—
190C2×D181—41no—
191C2×D181—41no—
192C2×D181—41no—
193C2×D181—41no—
194C2×D181—41no—
195C2×D181—84no—
196C4×C181—21no—
197C4×C181—21no—
198C4×C181—21no—
199C4×C181—21no—
200C4×C181—41no—
201C4×C181—41no—
202C4×C181C22×D411yescharacteristic
203C4×C181H(5,𝔽2)11yescharacteristic
204Dic181—41no—
205Dic181—41no—
206C2×C181—41no—
207C2×C181—41no—
208C2×C181—42no—
209C2×C181—81no—
210C2×C181D4211yescharacteristic
211C2×C181#57511yescharacteristic
212C2×C181#57611yescharacteristic
213D181—42no—
214D181—81no—
215C181#327311yescharacteristic, complemented, Sylow
216#3273—1811nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1811no—
218C2×AGL(1,ℤ/8)—1811no—
219#546—1811no—
220#546D18111yescharacteristic, complemented
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)—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×D8—1811no—
246C2×D8—1811no—
247C2×M16—1811no—
248C2×M16—1811no—
249#124—1811no—
250#124C2×D18111yescharacteristic, complemented
251#123—21no—
252#123—3621no—
253C22⋊Q8—1811no—
254C22⋊Q8C2×D18111yescharacteristic, complemented
255#121—21no—
256#121—3621no—
257#121—1811no—
258#121C2×D18111yescharacteristic
259#120—21no—
260#120—3621no—
261C4×D4—1811no—
262C4×D4C2×D18111yescharacteristic
263C4×D4C2×D18111yescharacteristic, complemented
264C2×C22⋊C4—21no—
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—21no—
289C2×D4—41no—
290C2×D4—3622no—
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—21no—
302C22×C4—3621no—
303C22×C4—3621no—
304C22×C4C22×D18111yescharacteristic
305C22×C4C22×D18111yescharacteristic
306Q16—7241no—
307SD16—3622no—
308SD16—3622no—
309SD16—3622no—
310SD16—3622no—
311D8—3622no—
312D8—3622no—
313D8—1814no—
314D8—7241no—
315M16—3622no—
316M16—3622no—
317C2×C8—1811no—
318C2×C8—1811no—
319C2×C8—1811no—
320C2×C8—1811no—
321C4⋊C4—21no—
322C4⋊C4—3621no—
323C4⋊C4—1811no—
324C4⋊C4C22⋊D18111yescharacteristic
325C4⋊C4C22×D18111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—3621no—
332C22⋊C4—3621no—
333C22⋊C4—3621no—
334C22⋊C4—3621no—
335C22⋊C4—3621no—
336C22⋊C4C22⋊D18111yescharacteristic
337C22⋊C4C22⋊D18111yescharacteristic
338C42C22×D18111yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—3621no—
345C23—3621no—
346C23—14481no—
347C23—7242no—
348C23—7242no—
349C23C2×C22⋊D18111yescharacteristic
350C23C2×C22⋊D18111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3621no—
354Q8—3621no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—3621no—
363D4—3621no—
364D4—3621no—
365D4—14481no—
366D4—1812no—
367D4—7242no—
368D4—7242no—
369D4D4×D18112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—3621no—
379C2×C4—3621no—
380C2×C4—14481no—
381C2×C4—7242no—
382C2×C4C2×C22⋊D18111yescharacteristic
383C2×C4C2×C22⋊D18111yescharacteristic
384C2×C4C2×C22⋊D18111yescharacteristic
385C2×C4C2×C22⋊D18111yescharacteristic
386C2×C4C23×D18111yescharacteristic
387C8—3621no—
388C8—3621no—
389C8—3621no—
390C8—1812no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—3622no—
405C22—14484no—
406C22—7241no—
407C22—7241no—
408C22—7241no—
409C22—7241no—
410C22—7241no—
411C22C22×C22⋊D18111yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—7241no—
419C4—7241no—
420C4C2×D4×D18111yescharacteristic
421C4#1915386111yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—14481no—
427C2—7242no—
428C2D4×C22⋊D18111yescharacteristic
429C2#2604207711yescharacteristic
430C2#2604208211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#546×C181C2
socleC22×C181C22×D4
hypercenter#546D181
hypocenterC181#3273
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#19153842C2semidirect
2. C2×#19153841C2semidirect
3. #26041925C2semidirect
4. #26041661C2semidirect
5. #26041527C2semidirect
6. #26041523C2semidirect
7. #26041494C2semidirect
8. #26041491C2semidirect
9. #26041441C2semidirect
10. #26041439C2semidirect
11. #26041413C2semidirect
12. #26041412C2semidirect
13. C4×#15668434C2semidirect
14. #26041357C2semidirect
15. #546×C181C2semidirect
17. C22×D724C22semidirect
20. C2×#15668439C22semidirect
21. C2×#15668437C22semidirect
22. C2×#15668437C22semidirect
23. C2×#15668436C22semidirect
24. C2×#15668436C22semidirect
29. C2×#15668433C22semidirect
30. C2×#15668433C22semidirect
31. #19153804C22semidirect
33. #19153795C22semidirect
37. C4×D724C22semidirect
39. #19153749C22semidirect
40. C4⋊#13089678C22semidirect
41. #19153740C22semidirect
42. #19153739C22semidirect
43. #19153737C22semidirect
44. #19153737C22semidirect
45. #19153736C22semidirect
46. #19153736C22semidirect
47. C22⋊#13089678C22semidirect
48. C22⋊#13089678C22semidirect
49. C4×#13089678C22semidirect
51. C22×D4×C181C22semidirect
52. #124×C181C22semidirect
54. C22⋊Q8×C181C22semidirect
59. C4×D4×C181C22semidirect
82. #15668434D4semidirect
215. C181#3273semidirect
220. #546D181semidirect
235. C22×D4C2×D181semidirect
250. #124C2×D181semidirect
254. C22⋊Q8C2×D181semidirect
263. C4×D4C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538422
C2×#191538412
#260419252
#260416612
#260415272
#260415232
#260414942
#260414912
#260414412
#260414392
#260414132
#260414122
C4×#156684342
#260413572
#546×C1812
#3273181