Finite Group Databasebeta

← #37545265

Subgroups — #37545265

Order 23168.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153840C211yesmaximal, characteristic, complemented
2C2×#19153836C211yesmaximal, characteristic, complemented
3#26041947C211yesmaximal, characteristic, complemented
4#26041917C211yesmaximal, characteristic, complemented
5#26041665C211yesmaximal, characteristic, complemented
6#26041528C211yesmaximal, characteristic, complemented
7#26041513C211yesmaximal, characteristic, complemented
8#26041504C211yesmaximal, characteristic, complemented
9#26041472C211yesmaximal, characteristic, complemented
10#26041438C211yesmaximal, characteristic, complemented
11#26041426C211yesmaximal, characteristic, complemented
12#26041414C211yesmaximal, characteristic, complemented
13#26041376C211yesmaximal, characteristic, complemented
14#26041347C211yesmaximal, characteristic, complemented
15#449×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic, complemented
17C2×#15668448C2211yescharacteristic
18C2×D4×D181C2211yescharacteristic
19#19153840—24no—
20#19153836—24no—
21C2×#15668439C2211yescharacteristic, complemented
22C2×#15668438C2211yescharacteristic
23C2×#15668437C2211yescharacteristic
24C2×#15668436C2211yescharacteristic, complemented
25C2×D1448C2211yescharacteristic
26C2×#15668434C2211yescharacteristic
27C2×#15668434C2211yescharacteristic
28C2×#15668432C2211yescharacteristic
29C22×#13089678C2211yescharacteristic, complemented
30#19153812—21no—
31#19153812C2211yescharacteristic, complemented
32#19153805C2211yescharacteristic
33#19153799—21no—
34#19153792C2211yescharacteristic
35#19153791C2211yescharacteristic
36#19153790C2211yescharacteristic
37#19153789—21no—
38#19153788—21no—
39#19153788—21no—
40#19153784—21no—
41#19153781C2211yescharacteristic, complemented
42D4×Dic181C2211yescharacteristic
43C4×C22⋊D181C2211yescharacteristic
44#19153776—21no—
45C2×#15668426—21no—
46#19153749C2211yescharacteristic, complemented
47#19153745C2211yescharacteristic, complemented
48#19153741C2211yescharacteristic, complemented
49#19153738C2211yescharacteristic, complemented
50#19153737C2211yescharacteristic, complemented
51#19153736C2211yescharacteristic
52#19153728C2211yescharacteristic
53#19153726C2211yescharacteristic, complemented
54C2×#41×C181C2211yescharacteristic
55C2×SD16×C181C2211yescharacteristic, complemented
56C2×D8×C181C2211yescharacteristic
57#121×C181C2211yescharacteristic, complemented
58#103×C181C2211yescharacteristic, complemented
59#102×C181C2211yescharacteristic
60C22⋊C8×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668449—22no—
63#15668449—41no—
64#15668448—22no—
65#15668448—41no—
66C2×Dic362C2311yescharacteristic
67C2×D724—21no—
68C2×D724C2311yescharacteristic
69C2×D724C2311yescharacteristic
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C22⋊D181—21no—
73C2×C22⋊D181—21no—
74C2×C22⋊D181—21no—
75C2×C22⋊D181—21no—
76C2×C22⋊D181D411yescharacteristic
77D4×D181—22no—
78D4×D181—41no—
79C22×Dic181—21no—
80C22×Dic181—21no—
81C2×C4×D181—21no—
82C2×C4×D181C2311yescharacteristic
83#15668439—22no—
84#15668438—22no—
85#15668437D414yescomplemented
86#15668436—22no—
87D1448—41no—
88#15668434—22no—
89#15668434—22no—
90#15668432—22no—
91C2×#13089678—22no—
92C2×#13089678C2311yescharacteristic
93C2×#13089678C2311yescharacteristic
94C4⋊Dic181C2311yescharacteristic
95#15668428—21no—
96#15668428—21no—
97#15668428D411yescharacteristic
98C22⋊Dic181—21no—
99C22⋊Dic181D411yescharacteristic
100#15668426—21no—
101#15668426—21no—
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426D411yescharacteristic
106C4×Dic181—21no—
107C4×Dic181C2311yescharacteristic
108#41×C181—22no—
109#41×C181—41no—
110C2×Q8×C181C2311yescharacteristic
111C2×D4×C181—21no—
112C2×D4×C181—21no—
113C2×D4×C181C2311yescharacteristic
114C2×D4×C181C2311yescharacteristic
115C22×C4×C181—21no—
116C22×C4×C181C2311yescharacteristic
117SD16×C181—41no—
118D8×C181—22no—
119C2×C8×C181C2311yescharacteristic
120C2×C8×C181C2311yescharacteristic
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—
130Dic362C2×D412yes—
131D724—21no—
132D724—21no—
133D724—21no—
134D724—21no—
135D724—81no—
136C22⋊D181—41no—
137C22⋊D181—42no—
138C22⋊D181—42no—
139C22⋊D181—42no—
140C22⋊D181—42no—
141C22⋊D181—81no—
142C22⋊D181—81no—
143C2×Dic181—21no—
144C2×Dic181—21no—
145C2×Dic181—42no—
146C2×Dic181—81no—
147C2×Dic181C2×D411yescharacteristic
148C2×Dic181C2×D411yescharacteristic
149C2×Dic181C2×D411yescharacteristic
150C4×D181—21no—
151C4×D181—21no—
152C4×D181—42no—
153#13089678—21no—
154#13089678C2×D412yes—
155C23×C181—21no—
156C23×C181—21no—
157C23×C181C2×D411yescharacteristic
158Q8×C181—21no—
159Q8×C181—21no—
160D4×C181—21no—
161D4×C181—21no—
162D4×C181—21no—
163D4×C181—42no—
164D4×C181—81no—
165D4×C181C2×D412yes—
166C2×C4×C181—21no—
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×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×C181C22×D411yescharacteristic
185C2×D181—21no—
186C2×D181—21no—
187C2×D181—21no—
188C2×D181—41no—
189C2×D181—41no—
190C2×D181—41no—
191C2×D181—41no—
192C2×D181—41no—
193C2×D181—41no—
194C2×D181—84no—
195C4×C181—21no—
196C4×C181—41no—
197C4×C181—41no—
198C4×C181C22×D411yescharacteristic
199C4×C181H(5,𝔽2)11yescharacteristic
200Dic181—21no—
201Dic181—22no—
202Dic181—21no—
203Dic181—21no—
204Dic181—41no—
205C2×C181—42no—
206C2×C181—41no—
207C2×C181—81no—
208C2×C181D4211yescharacteristic
209C2×C181#57511yescharacteristic
210C2×C181#57611yescharacteristic
211D181—41no—
212D181—81no—
213D181—81no—
214C181#326811yescharacteristic, complemented, Sylow
215#3268—1811nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1811no—
217C2×#135—1811no—
218#550—1811no—
219#546—1811no—
220#496—1811no—
221#468—1811no—
222#466—1811no—
223#464—1811no—
224#459—1811no—
225#452—1811no—
226#449—1811no—
227#449D18111yescharacteristic, complemented
228#447—1811no—
229#440—1811no—
230#435—1811no—
231C2×#41—1811no—
232C2×#41—1811no—
233C2×#41C2×D18111yescharacteristic
234C22×D4—1811no—
235AGL(1,ℤ/8)—3624no—
236#135—3624no—
237C2×Q16—1811no—
238C2×SD16—1811no—
239C2×SD16—1811no—
240C2×SD16—1811no—
241C2×SD16C2×D18111yescharacteristic, complemented
242C2×D8—1811no—
243C2×D8—1811no—
244C2×D8—1811no—
245C2×D8C2×D18111yescharacteristic
246C2×M16—1811no—
247C22×C8—1811no—
248#127—3621no—
249#127—1811no—
250#124—1811no—
251#123—3621no—
252C22⋊Q8—1811no—
253#121—3621no—
254#121—3621no—
255#121—3621no—
256#121—1811no—
257#121—1811no—
258#121C2×D18111yescharacteristic, complemented
259#120—3621no—
260C4×Q8—1811no—
261C4×D4—3621no—
262C4×D4—1811no—
263C4×D4—1811no—
264C2×C22⋊C4—3621no—
265#107—1811no—
266C4⋊C8—1811no—
267#103—1811no—
268#103C2×D18111yescharacteristic, complemented
269#102—1811no—
270#102—1811no—
271#102—1811no—
272#102C2×D18111yescharacteristic
273C22⋊C8—1811no—
274C22⋊C8C2×D18111yescharacteristic, complemented
275#97—1811no—
276C24—3621no—
277#41—22no—
278#41—41no—
279#41—3622no—
280#41—3622no—
281#41—7241no—
282#41—7241no—
283C2×Q8—1811no—
284C2×Q8C22×D18111yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—3621no—
288C2×D4—3622no—
289C2×D4—3621no—
290C2×D4—3621no—
291C2×D4—3621no—
292C2×D4—3621no—
293C2×D4—3621no—
294C2×D4—3621no—
295C2×D4—1811no—
296C2×D4—1811no—
297C2×D4—1811no—
298C2×D4—7241no—
299C2×D4C22×D18111yescharacteristic
300C2×D4C22×D18111yescharacteristic
301C22×C4—21no—
302C22×C4—3621no—
303C22×C4—3621no—
304C22×C4—3621no—
305C22×C4—1811no—
306C22×C4C22×D18111yescharacteristic
307Q16—3622no—
308SD16—41no—
309SD16—3622no—
310SD16—3622no—
311SD16—1814no—
312D8—22no—
313D8—3622no—
314D8—3622no—
315D8—7241no—
316M16—3622no—
317C2×C8—3622no—
318C2×C8—1811no—
319C2×C8—1811no—
320C2×C8C22×D18111yescharacteristic
321C2×C8C22×D18111yescharacteristic
322C4⋊C4—3621no—
323C4⋊C4—3621no—
324C4⋊C4—1811no—
325C4⋊C4—1811no—
326C4⋊C4C22×D18111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—3621no—
329C22⋊C4—3621no—
330C22⋊C4—3621no—
331C22⋊C4—3621no—
332C22⋊C4—3621no—
333C22⋊C4—3621no—
334C22⋊C4—1811no—
335C22⋊C4—1811no—
336C42—3621no—
337C42—1811no—
338C23—21no—
339C23—21no—
340C23—3621no—
341C23—3621no—
342C23—14481no—
343C23—1811no—
344C23—7242no—
345C23—7242no—
346C23D4×D18111yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—3621no—
350Q8—1812no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—81no—
356D4—3621no—
357D4—3621no—
358D4—3621no—
359D4—3621no—
360D4—14481no—
361D4—14481no—
362D4—14481no—
363D4—7241no—
364D4—7242no—
365D4—7242no—
366D4—7242no—
367D4—7242no—
368D4D4×D18112yes—
369C2×C4—21no—
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—14481no—
379C2×C4—1811no—
380C2×C4—1811no—
381C2×C4—1811no—
382C2×C4—7242no—
383C2×C4—7242no—
384C2×C4D4×D18111yescharacteristic
385C2×C4C23×D18111yescharacteristic
386C8—21no—
387C8—21no—
388C8—3621no—
389C8—1812no—
390C22—21no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—3621no—
400C22—3621no—
401C22—3621no—
402C22—14484no—
403C22—7241no—
404C22—7241no—
405C22—7241no—
406C22—7241no—
407C22—7241no—
408C22—7241no—
409C22C2×D4×D18111yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—3621no—
414C4—3622no—
415C4—3621no—
416C4—3621no—
417C4—7241no—
418C4C2×D4×D18111yescharacteristic
419C4#1915386111yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—14481no—
424C2—14481no—
425C2—7241no—
426C2#2604191111yescharacteristic
427C2#2604207611yescharacteristic
428C2#2604208311yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#449×C181C2
socleC22×C181C22×D4
hypercenter#449D181
hypocenterC181#3268
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#19153840C2semidirect
2. C2×#19153836C2semidirect
3. #26041947C2semidirect
4. #26041917C2semidirect
5. #26041665C2semidirect
6. #26041528C2semidirect
7. #26041513C2semidirect
8. #26041504C2semidirect
9. #26041472C2semidirect
10. #26041438C2semidirect
11. #26041426C2semidirect
12. #26041414C2semidirect
13. #26041376C2semidirect
14. #26041347C2semidirect
15. #449×C181C2semidirect
16. C2×#15668449C22semidirect
21. C2×#15668439C22semidirect
24. C2×#15668436C22semidirect
29. C22×#13089678C22semidirect
31. #19153812C22semidirect
41. #19153781C22semidirect
46. #19153749C22semidirect
47. #19153745C22semidirect
48. #19153741C22semidirect
49. #19153738C22semidirect
50. #19153737C22semidirect
53. #19153726C22semidirect
55. C2×SD16×C181C22semidirect
57. #121×C181C22semidirect
58. #103×C181C22semidirect
60. C22⋊C8×C181C22semidirect
85. #15668437D4semidirect
214. C181#3268semidirect
227. #449D181semidirect
241. C2×SD16C2×D181semidirect
258. #121C2×D181semidirect
268. #103C2×D181semidirect
274. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538402
C2×#191538362
#260419472
#260419172
#260416652
#260415282
#260415132
#260415042
#260414722
#260414382
#260414262
#260414142
#260413762
#260413472
#449×C1812
#3268181