Finite Group Databasebeta

← #37545427

Subgroups — #37545427

Order 23168.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153837C211yesmaximal, characteristic, complemented
2C2×SD16×D181C211yesmaximal, characteristic, complemented
3#26041937C211yesmaximal, characteristic, complemented
4#26041922C211yesmaximal, characteristic, complemented
5#26041505C211yesmaximal, characteristic, complemented
6#26041505C211yesmaximal, characteristic, complemented
7#26041498C211yesmaximal, characteristic, complemented
8#26041491C211yesmaximal, characteristic, complemented
9#26041440C211yesmaximal, characteristic, complemented
10#26041433C211yesmaximal, characteristic, complemented
11#26041421C211yesmaximal, characteristic, complemented
12#26041411C211yesmaximal, characteristic, complemented
13C4×#15668438C211yesmaximal, characteristic, complemented
14C8×D724C211yesmaximal, characteristic, complemented
15#495×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668448C2211yescharacteristic, complemented
18C2×Q8×D181C2211yescharacteristic
19C2×D4×D181C2211yescharacteristic
20#19153837—24no—
21C2×#15668439C2211yescharacteristic, complemented
22C2×#15668438C2211yescharacteristic
23C2×#15668437C2211yescharacteristic
24C2×#15668436C2211yescharacteristic
25SD16×D181—24no—
26C2×#15668434C2211yescharacteristic
27C2×C8×D181C2211yescharacteristic
28C2×C8×D181C2211yescharacteristic, complemented
29#19153802—21no—
30#19153798—21no—
31#19153793—21no—
32#19153792C2211yescharacteristic
33#19153792C2211yescharacteristic, complemented
34#19153789—21no—
35#19153789—21no—
36#19153787C2211yescharacteristic
37#19153787C2211yescharacteristic
38#19153786—21no—
39C4×Dic362C2211yescharacteristic, complemented
40C4×D724C2211yescharacteristic
41C4×D724C2211yescharacteristic
42#19153776—21no—
43C22⋊C4×D181—21no—
44#19153748C2211yescharacteristic, complemented
45C4⋊#13089678C2211yescharacteristic, complemented
46#19153741C2211yescharacteristic, complemented
47#19153740C2211yescharacteristic, complemented
48#19153740C2211yescharacteristic, complemented
49#19153738C2211yescharacteristic
50#19153737C2211yescharacteristic
51#19153737C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic
53#19153728C2211yescharacteristic, complemented
54C2×Q16×C181C2211yescharacteristic, complemented
55C2×SD16×C181C2211yescharacteristic
56C2×SD16×C181C2211yescharacteristic, complemented
57C2×D8×C181C2211yescharacteristic
58#124×C181C2211yescharacteristic
59#124×C181C2211yescharacteristic, complemented
60C4×C8×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668450—22no—
63#15668450—41no—
64#15668448—22no—
65#15668448—41no—
66C2×Dic362—21no—
67C2×Dic362C2311yescharacteristic
68Q8×D181—22no—
69Q8×D181—41no—
70C2×D724—21no—
71C2×D724D411yescharacteristic
72C2×D724C2311yescharacteristic
73C2×C22⋊D181—21no—
74C2×C22⋊D181—21no—
75C2×C22⋊D181—21no—
76C2×C22⋊D181—21no—
77D4×D181—22no—
78D4×D181—41no—
79C22×Dic181—21no—
80C2×C4×D181—21no—
81C2×C4×D181—21no—
82C2×C4×D181—21no—
83C2×C4×D181C2311yescharacteristic
84C2×C4×D181C2311yescharacteristic
85#15668439—22no—
86#15668438D414yescomplemented
87#15668437—22no—
88#15668436—22no—
89#15668434—22no—
90C2×#13089678C2311yescharacteristic
91C2×#13089678C2311yescharacteristic
92C8×D181—22no—
93C8×D181—22no—
94C4⋊Dic181D411yescharacteristic
95C4⋊Dic181C2311yescharacteristic
96C4⋊Dic181C2311yescharacteristic
97#15668428—21no—
98#15668428—21no—
99#15668428—21no—
100C22⋊Dic181—21no—
101C22⋊Dic181—21no—
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426D411yescharacteristic
106#15668426D411yescharacteristic
107C4×Dic181—21no—
108C2×Q8×C181C2311yescharacteristic
109C2×Q8×C181C2311yescharacteristic
110C2×D4×C181C2311yescharacteristic
111C2×D4×C181C2311yescharacteristic
112Q16×C181—41no—
113SD16×C181—22no—
114SD16×C181—22no—
115D8×C181—41no—
116C2×C8×C181C2311yescharacteristic
117C2×C8×C181C2311yescharacteristic
118C22⋊C4×C181—21no—
119C22⋊C4×C181—21no—
120C22⋊C4×C181—21no—
121C22⋊C4×C181—21no—
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181—42no—
125C22×D181—42no—
126C22×D181—81no—
127C22×D181C2×D411yescharacteristic
128C22×D181C2×D411yescharacteristic
129Dic362—21no—
130Dic362—42no—
131Dic362C2×D412yes—
132D724—22no—
133D724—21no—
134D724—42no—
135D724C2×D412yes—
136C22⋊D181—42no—
137C22⋊D181—42no—
138C22⋊D181—42no—
139C22⋊D181—81no—
140C2×Dic181—21no—
141C2×Dic181—21no—
142C2×Dic181—21no—
143C2×Dic181—42no—
144C2×Dic181C2×D411yescharacteristic
145C2×Dic181C2×D411yescharacteristic
146C4×D181—21no—
147C4×D181—21no—
148C4×D181—21no—
149C4×D181—21no—
150C4×D181—42no—
151C4×D181—42no—
152C4×D181—81no—
153#13089678—21no—
154#13089678—21no—
155C23×C181—21no—
156C23×C181—21no—
157Q8×C181—21no—
158Q8×C181—21no—
159Q8×C181—21no—
160Q8×C181—21no—
161D4×C181—21no—
162D4×C181—21no—
163D4×C181—21no—
164D4×C181—21no—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2411yescharacteristic
170C8×C181—21no—
171C8×C181C2×D412yes—
172C22×C181—41no—
173C22×C181—41no—
174C22×C181—41no—
175C22×C181—41no—
176C22×C181—41no—
177C22×C181—41no—
178C22×C181C22×D411yescharacteristic
179C2×D181—21no—
180C2×D181—21no—
181C2×D181—21no—
182C2×D181—22no—
183C2×D181—21no—
184C2×D181—21no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—84no—
188C2×D181#13512yescomplemented
189C4×C181—21no—
190C4×C181—21no—
191C4×C181—41no—
192C4×C181—41no—
193C4×C181C22×D411yescharacteristic
194C4×C181H(5,𝔽2)11yescharacteristic
195Dic181—21no—
196Dic181—21no—
197Dic181—22no—
198Dic181—41no—
199Dic181—41no—
200C2×C181—81no—
201C2×C181—81no—
202C2×C181D4211yescharacteristic
203C2×C181C2×#13511yescharacteristic
204C2×C181#57711yescharacteristic
205D181—22no—
206D181—41no—
207D181—42no—
208C181#328211yescharacteristic, complemented, Sylow
209#3282—1811nomaximal, Sylow
210C2×#135—1811no—
211C22×SD16—1811no—
212#548—1811no—
213#546—1811no—
214#495D18111yescharacteristic, complemented
215#465—1811no—
216#465—1811no—
217#464—1811no—
218#463—1811no—
219#452—1811no—
220#451—1811no—
221#449—1811no—
222#447—1811no—
223C4×SD16—1811no—
224C8×D4—1811no—
225C2×#41—1811no—
226C2×#41—1811no—
227C22×Q8—1811no—
228C22×D4—1811no—
229#135—3624no—
230C2×Q16—1811no—
231C2×Q16C2×D18111yescharacteristic, complemented
232C2×SD16—3624no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×SD16C2×D18111yescharacteristic
237C2×SD16C2×D18111yescharacteristic, complemented
238C2×D8—1811no—
239C2×D8C2×D18111yescharacteristic
240C22×C8—1811no—
241C22×C8—1811no—
242#124—3621no—
243#124C2×D18111yescharacteristic
244#124C2×D18111yescharacteristic, complemented
245#123—3621no—
246C22⋊Q8—3621no—
247C22⋊Q8—1811no—
248C22⋊Q8—1811no—
249#121—3621no—
250#121—3621no—
251#121—1811no—
252#121—1811no—
253#120—3621no—
254C4×Q8—1811no—
255C4×D4—3621no—
256C4×D4—1811no—
257C4×D4—1811no—
258C2×C22⋊C4—3621no—
259#106—1811no—
260C4⋊C8—1811no—
261#103—1811no—
262#103—1811no—
263#103—1811no—
264#102—1811no—
265#102—1811no—
266#102—1811no—
267C22⋊C8—1811no—
268C22⋊C8—1811no—
269C4×C8C2×D18111yescharacteristic, complemented
270C24—3621no—
271#41—3622no—
272#41—3622no—
273#41—7241no—
274#41—7241no—
275C2×Q8—3621no—
276C2×Q8—3622no—
277C2×Q8—1811no—
278C2×Q8—7241no—
279C2×Q8C22×D18111yescharacteristic
280C2×Q8C22×D18111yescharacteristic
281C2×D4—3621no—
282C2×D4—3622no—
283C2×D4—3621no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—1811no—
288C2×D4—1811no—
289C2×D4—7241no—
290C2×D4C22×D18111yescharacteristic
291C2×D4C22×D18111yescharacteristic
292C22×C4—3621no—
293C22×C4—3621no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4—1811no—
297C22×C4—1811no—
298Q16—41no—
299Q16—3622no—
300SD16—22no—
301SD16—22no—
302SD16—3622no—
303SD16—3622no—
304SD16—1814no—
305D8—41no—
306D8—3622no—
307C2×C8—3622no—
308C2×C8—3622no—
309C2×C8—1811no—
310C2×C8—1811no—
311C2×C8C22×D18111yescharacteristic
312C2×C8C22×D18111yescharacteristic
313C4⋊C4—3621no—
314C4⋊C4—3621no—
315C4⋊C4—3621no—
316C4⋊C4—1811no—
317C4⋊C4—1811no—
318C4⋊C4—1811no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4—1811no—
329C22⋊C4—1811no—
330C42—3621no—
331C42C22×D18111yescharacteristic
332C23—21no—
333C23—21no—
334C23—3621no—
335C23—14481no—
336C23—1811no—
337C23—1811no—
338C23—7242no—
339C23—7242no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—3621no—
345Q8—1812no—
346Q8—7242no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—3622no—
352D4—3621no—
353D4—14481no—
354D4—1812no—
355D4—7242no—
356D4—7242no—
357D4—7242no—
358D4—7242no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—3621no—
362C2×C4—3621no—
363C2×C4—3621no—
364C2×C4—3621no—
365C2×C4—3621no—
366C2×C4—3621no—
367C2×C4—3621no—
368C2×C4—14481no—
369C2×C4—1811no—
370C2×C4—1811no—
371C2×C4—7242no—
372C2×C4—7242no—
373C2×C4—7242no—
374C2×C4D4×D18111yescharacteristic
375C2×C4D4×D18111yescharacteristic
376C2×C4C23×D18111yescharacteristic
377C8—21no—
378C8—3621no—
379C8—3621no—
380C8D4×D18112yes—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3621no—
388C22—3621no—
389C22—3621no—
390C22—3622no—
391C22—3621no—
392C22—3621no—
393C22—14484no—
394C22—1812no—
395C22—7241no—
396C22—7241no—
397C22C2×D4×D18111yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3621no—
403C4—3621no—
404C4—3622no—
405C4—7241no—
406C4—7241no—
407C4C2×D4×D18111yescharacteristic
408C4#1915386111yescharacteristic
409C2—81no—
410C2—81no—
411C2—3622no—
412C2—7241no—
413C2—7242no—
414C2#2604191411yescharacteristic
415C2#135×D18111yescharacteristic
416C2#2604208611yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#495×C181C2
socleC22×C181C22×D4
hypercenter#495D181
hypocenterC181#3282
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#19153837C2semidirect
2. C2×SD16×D181C2semidirect
3. #26041937C2semidirect
4. #26041922C2semidirect
5. #26041505C2semidirect
6. #26041505C2semidirect
7. #26041498C2semidirect
8. #26041491C2semidirect
9. #26041440C2semidirect
10. #26041433C2semidirect
11. #26041421C2semidirect
12. #26041411C2semidirect
13. C4×#15668438C2semidirect
14. C8×D724C2semidirect
15. #495×C181C2semidirect
17. C2×#15668448C22semidirect
21. C2×#15668439C22semidirect
28. C2×C8×D181C22semidirect
33. #19153792C22semidirect
39. C4×Dic362C22semidirect
44. #19153748C22semidirect
45. C4⋊#13089678C22semidirect
46. #19153741C22semidirect
47. #19153740C22semidirect
48. #19153740C22semidirect
51. #19153737C22semidirect
53. #19153728C22semidirect
54. C2×Q16×C181C22semidirect
56. C2×SD16×C181C22semidirect
59. #124×C181C22semidirect
60. C4×C8×C181C22semidirect
86. #15668438D4semidirect
188. C2×D181#135semidirect
208. C181#3282semidirect
214. #495D181semidirect
231. C2×Q16C2×D181semidirect
237. C2×SD16C2×D181semidirect
244. #124C2×D181semidirect
269. C4×C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538372
C2×SD16×D1812
#260419372
#260419222
#260415052
#260414982
#260414912
#260414402
#260414332
#260414212
#260414112
C4×#156684382
C8×D7242
#495×C1812
#3282181