Finite Group Databasebeta

← #37547378

Subgroups — #37547378

Order 23168.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153864C211yesmaximal, characteristic, complemented
2C2×#41×D181C211yesmaximal, characteristic, complemented
3#26041951C211yesmaximal, characteristic, complemented
4#26041950C213yesmaximal, complemented
5#26041948C211yesmaximal, characteristic, complemented
6#26041947C213yesmaximal, complemented
7#26041940C211yesmaximal, characteristic, complemented
8#26041939C213yesmaximal, complemented
9#26041937C213yesmaximal, complemented
10#26041847C213yesmaximal, complemented
11#26041845C213yesmaximal, complemented
12#26041792C213yesmaximal, complemented
13#26041788C213yesmaximal, complemented
14#26041755C211yesmaximal, characteristic, complemented
15#548×C181C211yesmaximal, characteristic, complemented
16#19153864—28no—
17C2×#15668450C2213yes—
18C2×#15668450C2211yescharacteristic
19C2×#15668450C2211yescharacteristic, complemented
20C2×#15668449C2213yes—
21C2×#15668449C2213yes—
22C2×#15668449C2213yescomplemented
23C2×#15668448C2213yes—
24#41×D181—28no—
25C2×Q8×D181C2211yescharacteristic
26C2×Q8×D181C2213yescomplemented
27C2×D4×D181C2213yes—
28C22×C4×D181C2213yes—
29#19153813C2213yescomplemented
30#19153812C2213yescomplemented
31#19153812C2216yescomplemented
32#19153805C2213yescomplemented
33#19153802C2216yescomplemented
34#19153801C2213yes—
35#19153798C2211yescharacteristic
36#19153794C2213yescomplemented
37#19153793C2213yescomplemented
38#19153792C2213yescomplemented
39#19153791C2213yescomplemented
40#19153791C2213yescomplemented
41#19153790C2213yescomplemented
42#19153790C2213yescomplemented
43#19153789C2216yescomplemented
44#19153789C2216yescomplemented
45#19153789C2213yescomplemented
46#19153788C2213yescomplemented
47#19153787C2213yescomplemented
48#19153786C2213yes—
49#19153786C2216yes—
50#19153784C2213yes—
51Q8×Dic181C2211yescharacteristic, complemented
52C4×Dic362C2213yescomplemented
53#19153781C2213yescomplemented
54C4×D724C2213yescomplemented
55#19153778C2216yescomplemented
56C4×C22⋊D181C2213yescomplemented
57C4×C22⋊D181C2213yescomplemented
58#19153776C2216yescomplemented
59#19153774C2213yes—
60#19153773C2213yes—
61C2×#41×C181C2211yescharacteristic
62C22×Q8×C181C2211yescharacteristic, complemented
63#124×C181C2213yescomplemented
64C22⋊Q8×C181C2213yescomplemented
65#121×C181C2213yescomplemented
66C4×Q8×C181C2211yescharacteristic, complemented
67C4×D4×C181C2213yescomplemented
68C23×D181C2313yes—
69#15668450—212no—
70#15668450—24no—
71#15668450—24no—
72#15668449—212no—
73#15668449—212no—
74#15668449—212no—
75#15668448—212no—
76C2×Dic362C2313yes—
77C2×Dic362C2313yes—
78Q8×D181—212no—
79Q8×D181D418yescomplemented
80C2×D724C2313yes—
81C2×D724C2313yes—
82C2×D724C2313yes—
83C2×D724C2316yes—
84C2×D724C2313yes—
85C2×C22⋊D181C2311yescharacteristic
86C2×C22⋊D181C2313yes—
87C2×C22⋊D181C2316yes—
88C2×C22⋊D181C2316yes—
89C2×C22⋊D181C2313yes—
90C2×C22⋊D181C2313yes—
91D4×D181—212no—
92C22×Dic181C2313yes—
93C2×C4×D181—212no—
94C2×C4×D181C2316yes—
95C2×C4×D181C2313yes—
96C2×C4×D181C2313yes—
97C2×C4×D181C2313yes—
98C2×C4×D181C2311yescharacteristic
99C2×C4×D181C2313yes—
100C4⋊Dic181C2313yes—
101C4⋊Dic181C2311yescharacteristic
102#15668428C2313yes—
103#15668428C2313yes—
104#15668428C2313yes—
105#15668428C2311yescharacteristic
106C22⋊Dic181C2311yescharacteristic
107C22⋊Dic181C2313yes—
108#15668426C2311yescharacteristic
109#15668426C2313yes—
110#15668426C2313yes—
111#15668426C2313yes—
112#15668426C2313yes—
113#15668426C2316yes—
114#15668426C2313yes—
115C4×Dic181C2313yes—
116C4×Dic181C2316yes—
117#41×C181—24no—
118C2×Q8×C181—24no—
119C2×Q8×C181C2313yes—
120C2×Q8×C181C2311yescharacteristic
121C2×D4×C181C2313yes—
122C2×D4×C181C2313yes—
123C22×C4×C181C2313yes—
124C22×C4×C181C2313yes—
125C4⋊C4×C181C2313yes—
126C4⋊C4×C181C2313yes—
127C22⋊C4×C181C2311yescharacteristic
128C22⋊C4×C181C2313yes—
129C22⋊C4×C181C2316yes—
130C42×C181C2313yes—
131C22×D181—26no—
132C22×D181—43no—
133C22×D181—43no—
134C22×D181C2413yes—
135C22×D181C2413yes—
136C22×D181C2411yescharacteristic
137Dic362—26no—
138Dic362C2×D4112yes—
139D724—26no—
140D724—26no—
141D724—26no—
142D724—46no—
143D724—43no—
144C22⋊D181—22no—
145C22⋊D181—212no—
146C22⋊D181—26no—
147C22⋊D181—46no—
148C22⋊D181—43no—
149C22⋊D181—43no—
150C2×Dic181—26no—
151C2×Dic181C2411yescharacteristic
152C2×Dic181C2413yes—
153C2×Dic181C2411yescharacteristic
154C2×Dic181C2411yescharacteristic
155C2×Dic181C2413yes—
156C4×D181—22no—
157C4×D181—26no—
158C4×D181—26no—
159C4×D181—26no—
160C4×D181—212no—
161C4×D181C2×D4112yes—
162C23×C181C2413yes—
163C23×C181C2411yescharacteristic
164Q8×C181—26no—
165Q8×C181C2×D414yes—
166D4×C181—26no—
167D4×C181—43no—
168C2×C4×C181—26no—
169C2×C4×C181—26no—
170C2×C4×C181C2411yescharacteristic
171C2×C4×C181C2413yes—
172C2×C4×C181C2413yes—
173C2×C4×C181C2411yescharacteristic
174C2×C4×C181C2413yes—
175C22×C181—21no—
176C22×C181—23no—
177C22×C181—21no—
178C22×C181—23no—
179C22×C181—21no—
180C22×C181—23no—
181C22×C181C2511yescharacteristic
182C2×D181—21no—
183C2×D181—21no—
184C2×D181—23no—
185C2×D181—23no—
186C2×D181—23no—
187C2×D181—23no—
188C2×D181—23no—
189C2×D181—23no—
190C2×D181—412no—
191C2×D181C22×D412yes—
192C4×C181—21no—
193C4×C181—23no—
194C4×C181—23no—
195C4×C181—21no—
196C4×C181C22×D416yes—
197Dic181—21no—
198Dic181—21no—
199Dic181—21no—
200Dic181—23no—
201Dic181C22×D416yes—
202C2×C181—43no—
203C2×C181—41no—
204C2×C181C23×D411yescharacteristic
205C2×C181#58511yescharacteristic
206C2×C181#58511yescharacteristic
207D181—22no—
208D181—43no—
209D181—43no—
210C181#348711yescharacteristic, complemented, Sylow
211#3487—1811nomaximal, Sylow
212C2×2−1+4—1811no—
213C22×#41—1811no—
214#550—1811no—
215#550—1813no—
216#550—1811no—
217#550—1813no—
218#548—1811no—
219#548—1813no—
220#548—1813no—
221#548D18111yescharacteristic, complemented
222#535—1813no—
223#535—1813no—
224#525—1813no—
225#525—1813no—
226#519—1811no—
2272−1+4—3628no—
228C2×#41—3628no—
229C2×#41—1813no—
230C2×#41—1811no—
231C2×#41—1811no—
232C2×#41—1813no—
233C2×#41—1813no—
234C2×#41—1813no—
235C2×#41—1813no—
236C2×#41C2×D18111yescharacteristic
237C22×Q8—1813no—
238C22×Q8—1811no—
239C22×Q8C2×D18111yescharacteristic, complemented
240C22×D4—1813no—
241C23×C4—1813no—
242#127—1816no—
243#127—1813no—
244#127—1813no—
245#124—1813no—
246#124—1816no—
247#124C2×D18113yescomplemented
248#123—1813no—
249#123—1811no—
250C22⋊Q8—1813no—
251C22⋊Q8—1813no—
252C22⋊Q8—1813no—
253C22⋊Q8C2×D18113yescomplemented
254#121—1813no—
255#121—1813no—
256#121—1813no—
257#121—1813no—
258#121—1813no—
259#121—1813no—
260#121—1813no—
261#121—1816no—
262#121—1816no—
263#121C2×D18113yescomplemented
264#120—1813no—
265#120—1816no—
266#120—1813no—
267C4×Q8—1811no—
268C4×Q8—1813no—
269C4×Q8—1813no—
270C4×Q8C2×D18111yescharacteristic, complemented
271C4×D4—1813no—
272C4×D4—1813no—
273C4×D4—1816no—
274C4×D4—1816no—
275C4×D4—1813no—
276C4×D4C2×D18113yescomplemented
277#117—1813no—
278#117—1813no—
279C24—1813no—
280#41—24no—
281#41—36212no—
282#41—36212no—
283#41—36212no—
284#41—36212no—
285#41—3624no—
286#41—3624no—
287#41—36212no—
288C2×Q8—24no—
289C2×Q8—36212no—
290C2×Q8—1818no—
291C2×Q8—1813no—
292C2×Q8—1813no—
293C2×Q8C22×D18113yes—
294C2×Q8C22×D18111yescharacteristic
295C2×D4—36212no—
296C2×D4—1811no—
297C2×D4—1816no—
298C2×D4—1813no—
299C2×D4—1813no—
300C2×D4—1813no—
301C2×D4—1816no—
302C2×D4—1813no—
303C2×D4—1813no—
304C2×D4—1813no—
305C2×D4—1816no—
306C2×D4—1813no—
307C2×D4C22×D18113yes—
308C2×D4C22×D18113yes—
309C22×C4—36212no—
310C22×C4—1813no—
311C22×C4—1816no—
312C22×C4—1813no—
313C22×C4—1813no—
314C22×C4—1813no—
315C22×C4—1811no—
316C22×C4—1813no—
317C22×C4C22×D18113yes—
318C22×C4C22×D18113yes—
319C4⋊C4—1813no—
320C4⋊C4—1813no—
321C4⋊C4—1813no—
322C4⋊C4—1811no—
323C4⋊C4—1813no—
324C4⋊C4—1811no—
325C4⋊C4C22×D18113yes—
326C4⋊C4C22×D18113yes—
327C22⋊C4—1811no—
328C22⋊C4—1813no—
329C22⋊C4—1813no—
330C22⋊C4—1811no—
331C22⋊C4—1813no—
332C22⋊C4—1813no—
333C22⋊C4—1813no—
334C22⋊C4—1816no—
335C22⋊C4—1813no—
336C22⋊C4C22×D18116yes—
337C22⋊C4C22×D18111yescharacteristic
338C22⋊C4C22×D18113yes—
339C42—1813no—
340C42—1816no—
341C42C22×D18113yes—
342C23—3626no—
343C23—1813no—
344C23—1813no—
345C23—1811no—
346C23—7243no—
347C23—7243no—
348C23C23×D18113yes—
349C23C23×D18111yescharacteristic
350Q8—26no—
351Q8—3626no—
352Q8—18112no—
353Q8D4×D18114yes—
354D4—26no—
355D4—43no—
356D4—3626no—
357D4—3622no—
358D4—36212no—
359D4—3626no—
360D4—3626no—
361D4—3626no—
362D4—7246no—
363D4—7243no—
364D4—7243no—
365D4—7243no—
366D4—7246no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—3622no—
370C2×C4—3626no—
371C2×C4—36212no—
372C2×C4—3626no—
373C2×C4—3626no—
374C2×C4—3626no—
375C2×C4—1811no—
376C2×C4—1813no—
377C2×C4—1811no—
378C2×C4—1811no—
379C2×C4—1813no—
380C2×C4—18112no—
381C2×C4C23×D18113yes—
382C2×C4C23×D18111yescharacteristic
383C2×C4C23×D18113yes—
384C2×C4C23×D18111yescharacteristic
385C2×C4C23×D18113yes—
386C22—21no—
387C22—23no—
388C22—21no—
389C22—23no—
390C22—21no—
391C22—23no—
392C22—3621no—
393C22—3621no—
394C22—3623no—
395C22—3623no—
396C22—3623no—
397C22—3623no—
398C22—3623no—
399C22—3623no—
400C22—1812no—
401C22—72412no—
402C22C24×D18111yescharacteristic
403C4—23no—
404C4—21no—
405C4—21no—
406C4—23no—
407C4—3621no—
408C4—3623no—
409C4—3621no—
410C4—3621no—
411C4—1816no—
412C4C2×D4×D18116yes—
413C2—43no—
414C2—41no—
415C2—3622no—
416C2—7243no—
417C2—7243no—
418C2C22×D4×D18111yescharacteristic
419C2#2604211511yescharacteristic
420C2#2604211611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#548×C181C2
hypercenter#548D181
hypocenterC181#3487
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#19153864C2semidirect
2. C2×#41×D181C2semidirect
3. #26041951C2semidirect
4. #26041950C2semidirect
5. #26041948C2semidirect
6. #26041947C2semidirect
7. #26041940C2semidirect
8. #26041939C2semidirect
9. #26041937C2semidirect
10. #26041847C2semidirect
11. #26041845C2semidirect
12. #26041792C2semidirect
13. #26041788C2semidirect
14. #26041755C2semidirect
15. #548×C181C2semidirect
19. C2×#15668450C22semidirect
22. C2×#15668449C22semidirect
26. C2×Q8×D181C22semidirect
29. #19153813C22semidirect
30. #19153812C22semidirect
31. #19153812C22semidirect
32. #19153805C22semidirect
33. #19153802C22semidirect
36. #19153794C22semidirect
37. #19153793C22semidirect
38. #19153792C22semidirect
39. #19153791C22semidirect
40. #19153791C22semidirect
41. #19153790C22semidirect
42. #19153790C22semidirect
43. #19153789C22semidirect
44. #19153789C22semidirect
45. #19153789C22semidirect
46. #19153788C22semidirect
47. #19153787C22semidirect
51. Q8×Dic181C22semidirect
52. C4×Dic362C22semidirect
53. #19153781C22semidirect
54. C4×D724C22semidirect
55. #19153778C22semidirect
56. C4×C22⋊D181C22semidirect
57. C4×C22⋊D181C22semidirect
58. #19153776C22semidirect
62. C22×Q8×C181C22semidirect
63. #124×C181C22semidirect
64. C22⋊Q8×C181C22semidirect
65. #121×C181C22semidirect
66. C4×Q8×C181C22semidirect
67. C4×D4×C181C22semidirect
79. Q8×D181D4semidirect
210. C181#3487semidirect
221. #548D181semidirect
239. C22×Q8C2×D181semidirect
247. #124C2×D181semidirect
253. C22⋊Q8C2×D181semidirect
263. #121C2×D181semidirect
270. C4×Q8C2×D181semidirect
276. C4×D4C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538642
C2×#41×D1812
#260419512
#260419502
#260419482
#260419472
#260419402
#260419392
#260419372
#260418472
#260418452
#260417922
#260417882
#260417552
#548×C1812
#3487181