Finite Group Databasebeta

← #37545430

Subgroups — #37545430

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#26041941C211yesmaximal, characteristic, complemented
4#26041924C211yesmaximal, characteristic, complemented
5#26041663C211yesmaximal, characteristic, complemented
6#26041509C211yesmaximal, characteristic, complemented
7#26041509C211yesmaximal, characteristic, complemented
8#26041501C211yesmaximal, characteristic, complemented
9#26041494C211yesmaximal, characteristic, complemented
10#26041442C211yesmaximal, characteristic, complemented
11#26041436C211yesmaximal, characteristic, complemented
12#26041427C211yesmaximal, characteristic, complemented
13#26041414C211yesmaximal, characteristic, complemented
14C8×D724C211yesmaximal, characteristic, complemented
15C4×SD16×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic, complemented
17C2×#15668448C2211yescharacteristic
18C2×Q8×D181C2211yescharacteristic
19C2×D4×D181C2211yescharacteristic
20#19153837—24no—
21C2×Dic724C2211yescharacteristic, complemented
22C2×#15668439C2211yescharacteristic, complemented
23C2×#15668438C2211yescharacteristic
24C2×#15668438C2211yescharacteristic, complemented
25C2×#15668437C2211yescharacteristic
26C2×#15668436C2211yescharacteristic
27SD16×D181—24no—
28C2×D1448C2211yescharacteristic
29C2×#15668434C2211yescharacteristic
30C2×C8×D181C2211yescharacteristic
31C2×C8×D181C2211yescharacteristic, complemented
32#19153804—21no—
33#19153804C2211yescharacteristic
34#19153804C2211yescharacteristic, complemented
35#19153800—21no—
36#19153796—21no—
37#19153796C2211yescharacteristic
38#19153796C2211yescharacteristic, complemented
39#19153791—21no—
40#19153791C2211yescharacteristic
41#19153791C2211yescharacteristic
42#19153790—21no—
43#19153786—21no—
44C4×D724—21no—
45C4×D724C2211yescharacteristic
46C2×#15668426—21no—
47C4⋊#13089678C2211yescharacteristic, complemented
48#19153739C2211yescharacteristic, complemented
49#19153739C2211yescharacteristic, complemented
50#19153736C2211yescharacteristic
51#19153736C2211yescharacteristic, complemented
52#19153728C2211yescharacteristic
53#19153728C2211yescharacteristic, complemented
54C2×SD16×C181C2211yescharacteristic
55C4×Q8×C181C2211yescharacteristic, complemented
56C4×D4×C181C2211yescharacteristic
57#106×C181C2211yescharacteristic, complemented
58#103×C181C2211yescharacteristic, complemented
59#102×C181C2211yescharacteristic
60C4×C8×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668450—22no—
63#15668450—41no—
64#15668448—22no—
65#15668448—41no—
66C2×Dic362—21no—
67C2×Dic362C2311yescharacteristic
68C2×Dic362C2311yescharacteristic
69Q8×D181—22no—
70Q8×D181—41no—
71C2×D724—21no—
72C2×D724—21no—
73C2×D724—21no—
74C2×D724D411yescharacteristic
75C2×D724C2311yescharacteristic
76C2×D724C2311yescharacteristic
77C2×C22⋊D181—21no—
78C2×C22⋊D181—21no—
79D4×D181—22no—
80D4×D181—41no—
81C22×Dic181—21no—
82C2×C4×D181—21no—
83C2×C4×D181—21no—
84C2×C4×D181C2311yescharacteristic
85C2×C4×D181C2311yescharacteristic
86Dic724—41no—
87#15668439—22no—
88#15668438—22no—
89#15668438—22no—
90#15668437—22no—
91#15668436—22no—
92D1448—41no—
93#15668434—22no—
94C2×#13089678C2311yescharacteristic
95C2×#13089678C2311yescharacteristic
96C8×D181—22no—
97C8×D181—22no—
98C4⋊Dic181—21no—
99C4⋊Dic181—21no—
100C4⋊Dic181D411yescharacteristic
101#15668426—21no—
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426—21no—
108#15668426—21no—
109#15668426D411yescharacteristic
110#15668426D411yescharacteristic
111C2×Q8×C181C2311yescharacteristic
112C2×D4×C181C2311yescharacteristic
113C22×C4×C181—21no—
114SD16×C181D414yescomplemented
115C2×C8×C181C2311yescharacteristic
116C2×C8×C181C2311yescharacteristic
117C4⋊C4×C181—21no—
118C4⋊C4×C181C2311yescharacteristic
119C4⋊C4×C181C2311yescharacteristic
120C22⋊C4×C181—21no—
121C42×C181—21no—
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—42no—
126C22×D181—42no—
127C22×D181—81no—
128C22×D181C2×D411yescharacteristic
129C22×D181C2×D411yescharacteristic
130Dic362—21no—
131Dic362—21no—
132Dic362—21no—
133Dic362—21no—
134Dic362—42no—
135D724—22no—
136D724—21no—
137D724—21no—
138D724—21no—
139D724—21no—
140D724—42no—
141D724—42no—
142D724—81no—
143C22⋊D181—42no—
144C22⋊D181—42no—
145C2×Dic181—21no—
146C2×Dic181—21no—
147C2×Dic181—42no—
148C2×Dic181C2×D411yescharacteristic
149C2×Dic181C2×D411yescharacteristic
150C4×D181—21no—
151C4×D181—21no—
152C4×D181—21no—
153C4×D181—21no—
154C4×D181—42no—
155C4×D181—42no—
156#13089678—21no—
157#13089678—21no—
158C23×C181—21no—
159Q8×C181—21no—
160Q8×C181C2×D412yes—
161D4×C181—21no—
162D4×C181C2×D412yes—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—21no—
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—41no—
174C22×C181—41no—
175C22×C181C22×D411yescharacteristic
176C2×D181—21no—
177C2×D181—21no—
178C2×D181—21no—
179C2×D181—21no—
180C2×D181—21no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—41no—
184C2×D181—41no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—84no—
188C2×D181#13512yescomplemented
189C4×C181—22no—
190C4×C181—21no—
191C4×C181—21no—
192C4×C181—41no—
193C4×C181—41no—
194C4×C181C22×D411yescharacteristic
195C4×C181H(5,𝔽2)11yescharacteristic
196Dic181—21no—
197Dic181—21no—
198Dic181—41no—
199Dic181—41no—
200C2×C181—42no—
201C2×C181D4211yescharacteristic
202C2×C181C2×#13511yescharacteristic
203C2×C181#57711yescharacteristic
204D181—22no—
205D181—41no—
206D181—81no—
207D181—81no—
208C181#328211yescharacteristic, complemented, Sylow
209#3282—1811nomaximal, Sylow
210C2×#135—1811no—
211C22×SD16—1811no—
212#548—1811no—
213#546—1811no—
214#495—1811no—
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×SD16D18111yescharacteristic, complemented
224C8×D4—1811no—
225C2×#41—1811no—
226C2×#41—1811no—
227C22×Q8—1811no—
228C22×D4—1811no—
229#135—3624no—
230C2×Q16—1811no—
231C2×Q16—1811no—
232C2×SD16—3624no—
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×SD16—1811no—
237C2×SD16C2×D18111yescharacteristic
238C2×D8—1811no—
239C2×D8—1811no—
240C22×C8—1811no—
241C22×C8—1811no—
242#124—3621no—
243#124—1811no—
244#124—1811no—
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×Q8C2×D18111yescharacteristic, complemented
255C4×D4—3621no—
256C4×D4—1811no—
257C4×D4C2×D18111yescharacteristic
258C2×C22⋊C4—3621no—
259#106C2×D18111yescharacteristic, complemented
260C4⋊C8—1811no—
261#103—1811no—
262#103—1811no—
263#103C2×D18111yescharacteristic, complemented
264#102—1811no—
265#102—1811no—
266#102C2×D18111yescharacteristic
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—1811no—
279C2×Q8—7241no—
280C2×Q8C22×D18111yescharacteristic
281C2×D4—3621no—
282C2×D4—3621no—
283C2×D4—3621no—
284C2×D4—3622no—
285C2×D4—3621no—
286C2×D4—3621no—
287C2×D4—1811no—
288C2×D4—1811no—
289C2×D4—1811no—
290C2×D4—7241no—
291C2×D4C22×D18111yescharacteristic
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—
304SD16D72414yescomplemented
305D8—3622no—
306D8—7241no—
307C2×C8—3622no—
308C2×C8—3622no—
309C2×C8—1811no—
310C2×C8—1811no—
311C2×C8C22×D18111yescharacteristic
312C2×C8C22×D18111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3621no—
315C4⋊C4—3621no—
316C4⋊C4—1811no—
317C4⋊C4C22×D18111yescharacteristic
318C4⋊C4C22×D18111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—3621no—
321C22⋊C4—3621no—
322C22⋊C4—3621no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4—1811no—
329C22⋊C4—1811no—
330C42—21no—
331C42C22×D18111yescharacteristic
332C23—21no—
333C23—3621no—
334C23—3621no—
335C23—14481no—
336C23—1811no—
337C23—1811no—
338C23—7242no—
339C23—7242no—
340Q8—21no—
341Q8—3621no—
342Q8—3621no—
343Q8—3621no—
344Q8—3621no—
345Q8—7242no—
346Q8C2×D72412yes—
347D4—21no—
348D4—3622no—
349D4—3621no—
350D4—3621no—
351D4—3621no—
352D4—3621no—
353D4—14481no—
354D4—7242no—
355D4—7242no—
356D4—7242no—
357D4—7242no—
358D4C2×D72412yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—81no—
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—41no—
383C22—41no—
384C22—3621no—
385C22—3621no—
386C22—3621no—
387C22—3621no—
388C22—3621no—
389C22—14484no—
390C22—1812no—
391C22—7241no—
392C22—7241no—
393C22—7241no—
394C22—7241no—
395C22—7241no—
396C22—7241no—
397C22C2×D4×D18111yescharacteristic
398C4—22no—
399C4—21no—
400C4—21no—
401C4—41no—
402C4—41no—
403C4—3621no—
404C4—3621no—
405C4—7241no—
406C4—7241no—
407C4C22×D72411yescharacteristic
408C4#1915386211yescharacteristic
409C2—42no—
410C2—3622no—
411C2—14481no—
412C2—14481no—
413C2—7241no—
414C2D4×D72411yescharacteristic
415C2#135×D18111yescharacteristic
416C2#2604208511yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC4×SD16×C181C2
socleC22×C181C22×D4
hypercenterC4×SD16D181
hypocenterC181#3282
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19153837C2semidirect
2. C2×SD16×D181C2semidirect
3. #26041941C2semidirect
4. #26041924C2semidirect
5. #26041663C2semidirect
6. #26041509C2semidirect
7. #26041509C2semidirect
8. #26041501C2semidirect
9. #26041494C2semidirect
10. #26041442C2semidirect
11. #26041436C2semidirect
12. #26041427C2semidirect
13. #26041414C2semidirect
14. C8×D724C2semidirect
15. C4×SD16×C181C2semidirect
16. C2×#15668450C22semidirect
21. C2×Dic724C22semidirect
22. C2×#15668439C22semidirect
24. C2×#15668438C22semidirect
31. C2×C8×D181C22semidirect
34. #19153804C22semidirect
38. #19153796C22semidirect
47. C4⋊#13089678C22semidirect
48. #19153739C22semidirect
49. #19153739C22semidirect
51. #19153736C22semidirect
53. #19153728C22semidirect
55. C4×Q8×C181C22semidirect
57. #106×C181C22semidirect
58. #103×C181C22semidirect
60. C4×C8×C181C22semidirect
114. SD16×C181D4semidirect
188. C2×D181#135semidirect
208. C181#3282semidirect
223. C4×SD16D181semidirect
254. C4×Q8C2×D181semidirect
259. #106C2×D181semidirect
263. #103C2×D181semidirect
269. C4×C8C2×D181semidirect
304. SD16D724semidirect

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
#260419412
#260419242
#260416632
#260415092
#260415012
#260414942
#260414422
#260414362
#260414272
#260414142
C8×D7242
C4×SD16×C1812
#3282181