Finite Group Databasebeta

← #37545248

Subgroups — #37545248

Order 23168.

500 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×#19153840C211yesmaximal, characteristic, complemented
2C2×#19153835C211yesmaximal, characteristic, complemented
3#26041933C211yesmaximal, characteristic, complemented
4#26041924C211yesmaximal, characteristic, complemented
5#26041673C211yesmaximal, characteristic, complemented
6#26041538C211yesmaximal, characteristic, complemented
7#26041509C211yesmaximal, characteristic, complemented
8#26041494C211yesmaximal, characteristic, complemented
9#26041484C211yesmaximal, characteristic, complemented
10#26041442C211yesmaximal, characteristic, complemented
11#26041442C211yesmaximal, characteristic, complemented
12#26041431C211yesmaximal, characteristic, complemented
13#26041425C211yesmaximal, characteristic, complemented
14#26041352C211yesmaximal, characteristic, complemented
15#442×C181C211yesmaximal, characteristic, complemented
16C2×#15668448C2211yescharacteristic
17C2×#15668448C2211yescharacteristic
18C2×#15668448C2211yescharacteristic, complemented
19C2×D4×D181C2211yescharacteristic
20#19153840—24no—
21#19153835—24no—
22C2×Dic724C2211yescharacteristic, complemented
23C2×Dic724C2211yescharacteristic, complemented
24C2×#15668438C2211yescharacteristic
25C2×#15668438C2211yescharacteristic
26C2×#15668437C2211yescharacteristic
27C2×#15668437C2211yescharacteristic
28C2×#15668437C2211yescharacteristic, complemented
29C2×#15668434C2211yescharacteristic
30C2×#15668432C2211yescharacteristic
31C2×C8×D181C2211yescharacteristic, complemented
32#19153816C2211yescharacteristic, complemented
33#19153804C2211yescharacteristic
34#19153800—21no—
35#19153800—21no—
36#19153800—21no—
37#19153796C2211yescharacteristic
38#19153796C2211yescharacteristic
39#19153796C2211yescharacteristic, complemented
40#19153791C2211yescharacteristic
41#19153790—21no—
42#19153790—21no—
43#19153786—21no—
44C4×D724C2211yescharacteristic
45C2×C4⋊Dic181—21no—
46C2×#15668426—21no—
47C4⋊#13089678C2211yescharacteristic, complemented
48#19153739C2211yescharacteristic, complemented
49#19153739C2211yescharacteristic, complemented
50#19153739C2211yescharacteristic, complemented
51#19153736C2211yescharacteristic
52#19153728C2211yescharacteristic
53#19153728C2211yescharacteristic, complemented
54C2×D8×C181C2211yescharacteristic
55C4×D4×C181C2211yescharacteristic
56C4×D4×C181C2211yescharacteristic, complemented
57#106×C181C2211yescharacteristic, complemented
58#102×C181C2211yescharacteristic
59#102×C181C2211yescharacteristic, complemented
60#97×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668448—22no—
63#15668448—22no—
64#15668448—22no—
65#15668448—41no—
66#15668448—41no—
67#15668448—41no—
68C2×Dic362C2311yescharacteristic
69C2×Dic362C2311yescharacteristic
70C2×Dic362C2311yescharacteristic
71C2×D724—21no—
72C2×D724D411yescharacteristic
73C2×D724C2311yescharacteristic
74C2×C22⋊D181—21no—
75C2×C22⋊D181—21no—
76C2×C22⋊D181—21no—
77C2×C22⋊D181—21no—
78D4×D181—22no—
79D4×D181—41no—
80C22×Dic181—21no—
81C22×Dic181—21no—
82C22×Dic181—21no—
83C2×C4×D181C2311yescharacteristic
84C2×C4×D181C2311yescharacteristic
85Dic724—22no—
86Dic724—41no—
87#15668438—22no—
88#15668438—41no—
89#15668437—22no—
90#15668437—22no—
91#15668437—22no—
92#15668434—22no—
93#15668432—22no—
94C2×#13089678C2311yescharacteristic
95C2×#13089678C2311yescharacteristic
96C8×D181—22no—
97C4⋊Dic181—21no—
98C4⋊Dic181—21no—
99C4⋊Dic181—21no—
100C4⋊Dic181—21no—
101C4⋊Dic181—21no—
102C4⋊Dic181D411yescharacteristic
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426—21no—
108#15668426—21no—
109#15668426D411yescharacteristic
110#15668426D411yescharacteristic
111C2×D4×C181C2311yescharacteristic
112C2×D4×C181C2311yescharacteristic
113C22×C4×C181—21no—
114C22×C4×C181—21no—
115D8×C181D414yescomplemented
116C2×C8×C181C2311yescharacteristic
117C2×C8×C181C2311yescharacteristic
118C4⋊C4×C181C2311yescharacteristic
119C4⋊C4×C181C2311yescharacteristic
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—21no—
131Dic362—21no—
132Dic362—21no—
133Dic362—21no—
134Dic362—21no—
135D724—21no—
136D724—21no—
137D724—41no—
138D724—81no—
139C22⋊D181—42no—
140C22⋊D181—42no—
141C22⋊D181—42no—
142C22⋊D181—42no—
143C2×Dic181—21no—
144C2×Dic181—21no—
145C2×Dic181—21no—
146C2×Dic181—42no—
147C2×Dic181—42no—
148C2×Dic181—42no—
149C2×Dic181C2×D411yescharacteristic
150C2×Dic181C2×D411yescharacteristic
151C4×D181—21no—
152C4×D181—21no—
153C4×D181—21no—
154C4×D181—21no—
155#13089678—21no—
156#13089678—21no—
157C23×C181—21no—
158C23×C181—21no—
159D4×C181—21no—
160D4×C181—21no—
161D4×C181C2×D412yes—
162D4×C181C2×D412yes—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—42no—
166C2×C4×C181—81no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2411yescharacteristic
170C8×C181—21no—
171C8×C181C2×D412yes—
172C22×C181—22no—
173C22×C181—22no—
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—21no—
183C2×D181—21no—
184C2×D181—21no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—41no—
188C2×D181—84no—
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—41no—
198Dic181—41no—
199Dic181—41no—
200C2×C181—42no—
201C2×C181—42no—
202C2×C181D4211yescharacteristic
203C2×C181#57511yescharacteristic
204C2×C181#57711yescharacteristic
205D181—41no—
206D181—41no—
207D181—81no—
208C181#326711yescharacteristic, complemented, Sylow
209#3267—1811nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1811no—
211C2×#135—1811no—
212#547—1811no—
213#546—1811no—
214#497—1811no—
215#470—1811no—
216#465—1811no—
217#463—1811no—
218#461—1811no—
219#452—1811no—
220#452—1811no—
221#450—1811no—
222#449—1811no—
223#442D18111yescharacteristic, complemented
224#435—1811no—
225C2×#41—1811no—
226C2×#41—1811no—
227C2×#41—1811no—
228C22×D4—1811no—
229AGL(1,ℤ/8)—3624no—
230#135—3624no—
231C2×Q16—1811no—
232C2×Q16—1811no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×SD16—1811no—
237C2×SD16—1811no—
238C2×D8—1811no—
239C2×D8C2×D18111yescharacteristic
240C2×M16—1811no—
241C22×C8—1811no—
242C4⋊Q8—1811no—
243#124—1811no—
244#123—3621no—
245#123—3621no—
246#123—3621no—
247C22⋊Q8—1811no—
248C22⋊Q8—1811no—
249C22⋊Q8—1811no—
250#121—3621no—
251#121—3621no—
252#121—1811no—
253#120—3621no—
254C4×D4—1811no—
255C4×D4C2×D18111yescharacteristic
256C4×D4C2×D18111yescharacteristic, complemented
257C2×C4⋊C4—3621no—
258C2×C22⋊C4—3621no—
259#106C2×D18111yescharacteristic, complemented
260C4⋊C8—1811no—
261#103—1811no—
262#103—1811no—
263#103—1811no—
264#102—1811no—
265#102C2×D18111yescharacteristic
266#102C2×D18111yescharacteristic, complemented
267C22⋊C8—1811no—
268C22⋊C8—1811no—
269#97C2×D18111yescharacteristic, complemented
270C24—3621no—
271#41—3622no—
272#41—3622no—
273#41—3622no—
274#41—7241no—
275#41—7241no—
276#41—7241no—
277C2×Q8—1811no—
278C2×Q8—1811no—
279C2×Q8—1811no—
280C2×D4—3621no—
281C2×D4—3622no—
282C2×D4—3621no—
283C2×D4—3621no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—1811no—
287C2×D4—1811no—
288C2×D4—7241no—
289C2×D4C22×D18111yescharacteristic
290C2×D4C22×D18111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—3621no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4—1811no—
297C22×C4—1811no—
298Q16—3622no—
299Q16—7241no—
300SD16—3622no—
301SD16—3622no—
302SD16—3622no—
303SD16—3622no—
304SD16—7241no—
305D8—3622no—
306D8D72414yescomplemented
307M16—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—3621no—
317C4⋊C4—3621no—
318C4⋊C4—1811no—
319C4⋊C4C22×D18111yescharacteristic
320C4⋊C4C22×D18111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4—3621no—
329C22⋊C4—1811no—
330C22⋊C4—1811no—
331C42C22×D18111yescharacteristic
332C23—21no—
333C23—21no—
334C23—3621no—
335C23—14481no—
336C23—1811no—
337C23—1811no—
338C23—7242no—
339C23—7242no—
340Q8—3621no—
341Q8—3621no—
342Q8—3621no—
343Q8—3621no—
344Q8—3621no—
345Q8—3621no—
346D4—21no—
347D4—21no—
348D4—3621no—
349D4—3621no—
350D4—14481no—
351D4—7241no—
352D4—7242no—
353D4—7242no—
354D4—7242no—
355D4—7242no—
356D4C2×D72412yes—
357D4C2×D72412yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—3621no—
363C2×C4—3621no—
364C2×C4—3621no—
365C2×C4—3621no—
366C2×C4—3621no—
367C2×C4—3621no—
368C2×C4—3621no—
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—
380C8C2×D72412yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3621no—
388C22—3621no—
389C22—3621no—
390C22—3621no—
391C22—3621no—
392C22—3621no—
393C22—14484no—
394C22—7241no—
395C22—7241no—
396C22—7241no—
397C22C2×D4×D18111yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3621no—
403C4—3621no—
404C4—7241no—
405C4—7241no—
406C4—7241no—
407C4C22×D72411yescharacteristic
408C4#1915386211yescharacteristic
409C2—42no—
410C2—42no—
411C2—14481no—
412C2—7241no—
413C2—7241no—
414C2D4×D72411yescharacteristic
415C2#2604207511yescharacteristic
416C2#2604208911yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#442×C181C2
socleC22×C181C22×D4
hypercenter#442D181
hypocenterC181#3267
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#19153840C2semidirect
2. C2×#19153835C2semidirect
3. #26041933C2semidirect
4. #26041924C2semidirect
5. #26041673C2semidirect
6. #26041538C2semidirect
7. #26041509C2semidirect
8. #26041494C2semidirect
9. #26041484C2semidirect
10. #26041442C2semidirect
11. #26041442C2semidirect
12. #26041431C2semidirect
13. #26041425C2semidirect
14. #26041352C2semidirect
15. #442×C181C2semidirect
18. C2×#15668448C22semidirect
22. C2×Dic724C22semidirect
23. C2×Dic724C22semidirect
28. C2×#15668437C22semidirect
31. C2×C8×D181C22semidirect
32. #19153816C22semidirect
39. #19153796C22semidirect
47. C4⋊#13089678C22semidirect
48. #19153739C22semidirect
49. #19153739C22semidirect
50. #19153739C22semidirect
53. #19153728C22semidirect
56. C4×D4×C181C22semidirect
57. #106×C181C22semidirect
59. #102×C181C22semidirect
60. #97×C181C22semidirect
115. D8×C181D4semidirect
208. C181#3267semidirect
223. #442D181semidirect
256. C4×D4C2×D181semidirect
259. #106C2×D181semidirect
266. #102C2×D181semidirect
269. #97C2×D181semidirect
306. D8D724semidirect

Maximal subgroups

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

subgroupindex
C2×#191538402
C2×#191538352
#260419332
#260419242
#260416732
#260415382
#260415092
#260414942
#260414842
#260414422
#260414312
#260414252
#260413522
#442×C1812
#3267181