Finite Group Databasebeta

← #37545247

Subgroups — #37545247

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×#19153841C211yesmaximal, characteristic, complemented
2C2×#19153836C211yesmaximal, characteristic, complemented
3#26041925C211yesmaximal, characteristic, complemented
4#26041671C211yesmaximal, characteristic, complemented
5#26041535C211yesmaximal, characteristic, complemented
6#26041510C211yesmaximal, characteristic, complemented
7#26041494C211yesmaximal, characteristic, complemented
8#26041481C211yesmaximal, characteristic, complemented
9#26041441C211yesmaximal, characteristic, complemented
10#26041439C211yesmaximal, characteristic, complemented
11#26041429C211yesmaximal, characteristic, complemented
12#26041424C211yesmaximal, characteristic, complemented
13#26041389C211yesmaximal, characteristic, complemented
14#26041351C211yesmaximal, characteristic, complemented
15#547×C181C211yesmaximal, characteristic, complemented
16C2×#15668449C2211yescharacteristic
17C22×D724C2211yescharacteristic
18#19153841—24no—
19#19153836—24no—
20C2×#15668439C2211yescharacteristic, complemented
21C2×#15668439C2211yescharacteristic, complemented
22C2×#15668437C2211yescharacteristic, complemented
23C2×#15668436C2211yescharacteristic
24C2×#15668436C2211yescharacteristic
25C2×#15668436C2211yescharacteristic
26C2×#15668436C2211yescharacteristic
27C2×#15668434C2211yescharacteristic
28C2×#15668434C2211yescharacteristic
29C2×#15668433C2211yescharacteristic
30C22×#13089678C2211yescharacteristic, complemented
31#19153804C2211yescharacteristic
32#19153801—21no—
33#19153795C2211yescharacteristic
34#19153790C2211yescharacteristic
35#19153789—21no—
36#19153784—21no—
37C4×D724C2211yescharacteristic
38C22⋊C4×D181—21no—
39#19153747C2211yescharacteristic, complemented
40C4⋊#13089678C2211yescharacteristic, complemented
41#19153740C2211yescharacteristic, complemented
42#19153740C2211yescharacteristic, complemented
43#19153739C2211yescharacteristic, complemented
44#19153737C2211yescharacteristic, complemented
45#19153736C2211yescharacteristic
46#19153736C2211yescharacteristic
47C22⋊#13089678C2211yescharacteristic
48C22⋊#13089678C2211yescharacteristic, complemented
49#19153727C2211yescharacteristic, complemented
50C2×#41×C181C2211yescharacteristic
51C2×#41×C181C2211yescharacteristic, complemented
52C4⋊Q8×C181C2211yescharacteristic, complemented
53#123×C181—21no—
54#123×C181—21no—
55C22⋊Q8×C181C2211yescharacteristic
56C22⋊Q8×C181C2211yescharacteristic, complemented
57#121×C181—21no—
58C4×D4×C181C2211yescharacteristic
59C4×D4×C181C2211yescharacteristic, complemented
60C2×C4⋊C4×C181—21no—
61C23×D181—21no—
62#15668449—22no—
63#15668449—41no—
64C2×Dic362C2311yescharacteristic
65C2×D724—22no—
66C2×D724—21no—
67C2×D724—41no—
68C2×D724C2311yescharacteristic
69C2×D724C2311yescharacteristic
70C2×C22⋊D181—21no—
71C2×C22⋊D181—21no—
72C2×C4×D181—21no—
73C2×C4×D181—21no—
74#15668439—22no—
75#15668439—41no—
76#15668437—22no—
77#15668436—22no—
78#15668436—22no—
79#15668436—22no—
80#15668436—41no—
81#15668434—22no—
82#15668434D414yescomplemented
83#15668433—22no—
84C2×#13089678—22no—
85C2×#13089678C2311yescharacteristic
86C2×#13089678C2311yescharacteristic
87C2×#13089678C2311yescharacteristic
88C2×#13089678C2311yescharacteristic
89C4⋊Dic181C2311yescharacteristic
90#15668428—21no—
91C22⋊Dic181—21no—
92#15668426—21no—
93#15668426—21no—
94#15668426—21no—
95#15668426—21no—
96#41×C181—22no—
97#41×C181—22no—
98#41×C181—41no—
99#41×C181—41no—
100C2×Q8×C181C2311yescharacteristic
101C2×Q8×C181C2311yescharacteristic
102C2×D4×C181—21no—
103C2×D4×C181—21no—
104C2×D4×C181D411yescharacteristic
105C2×D4×C181C2311yescharacteristic
106C22×C4×C181—21no—
107C22×C4×C181—21no—
108C22×C4×C181—21no—
109C22×C4×C181C2311yescharacteristic
110C22×C4×C181C2311yescharacteristic
111C4⋊C4×C181—21no—
112C4⋊C4×C181—21no—
113C4⋊C4×C181—21no—
114C4⋊C4×C181—21no—
115C4⋊C4×C181D411yescharacteristic
116C4⋊C4×C181C2311yescharacteristic
117C22⋊C4×C181—21no—
118C22⋊C4×C181—21no—
119C22⋊C4×C181—21no—
120C22⋊C4×C181D411yescharacteristic
121C22⋊C4×C181D411yescharacteristic
122C42×C181C2311yescharacteristic
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—42no—
126C22×D181—42no—
127C22×D181—81no—
128Dic362—21no—
129Dic362—21no—
130D724—21no—
131D724—21no—
132D724—21no—
133D724—42no—
134D724C2×D412yes—
135C22⋊D181—42no—
136C22⋊D181—81no—
137C2×Dic181—21no—
138C2×Dic181—21no—
139C4×D181—42no—
140C4×D181—81no—
141#13089678—21no—
142#13089678—21no—
143#13089678—21no—
144#13089678C2×D412yes—
145C23×C181—21no—
146C23×C181C2×D411yescharacteristic
147C23×C181C2×D411yescharacteristic
148Q8×C181—21no—
149Q8×C181—21no—
150Q8×C181—21no—
151Q8×C181—21no—
152D4×C181—21no—
153D4×C181—41no—
154D4×C181—42no—
155D4×C181—42no—
156D4×C181C2×D412yes—
157C2×C4×C181—21no—
158C2×C4×C181—21no—
159C2×C4×C181—21no—
160C2×C4×C181—21no—
161C2×C4×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—42no—
165C2×C4×C181—42no—
166C2×C4×C181—42no—
167C2×C4×C181C2×D411yescharacteristic
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2×D411yescharacteristic
170C2×C4×C181C2×D411yescharacteristic
171C2×C4×C181C2411yescharacteristic
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—22no—
175C22×C181—21no—
176C22×C181—21no—
177C22×C181—21no—
178C22×C181—21no—
179C22×C181—41no—
180C22×C181—41no—
181C22×C181C22×D411yescharacteristic
182C2×D181—22no—
183C2×D181—41no—
184C2×D181—41no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—41no—
188C2×D181—84no—
189C4×C181—21no—
190C4×C181—21no—
191C4×C181—21no—
192C4×C181—21no—
193C4×C181—41no—
194C4×C181—41no—
195C4×C181—41no—
196C4×C181C22×D411yescharacteristic
197C4×C181H(5,𝔽2)11yescharacteristic
198Dic181—41no—
199Dic181—41no—
200C2×C181—41no—
201C2×C181—41no—
202C2×C181—42no—
203C2×C181D4211yescharacteristic
204C2×C181#57511yescharacteristic
205C2×C181#57711yescharacteristic
206D181—42no—
207D181—81no—
208C181#326711yescharacteristic, complemented, Sylow
209#3267—1811nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1811no—
211C2×#135—1811no—
212#547D18111yescharacteristic, complemented
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#442—1811no—
224#435—1811no—
225C2×#41—1811no—
226C2×#41C2×D18111yescharacteristic
227C2×#41C2×D18111yescharacteristic, complemented
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×D8—1811no—
240C2×M16—1811no—
241C22×C8—1811no—
242C4⋊Q8C2×D18111yescharacteristic, complemented
243#124—1811no—
244#123—21no—
245#123—21no—
246#123—3621no—
247C22⋊Q8—1811no—
248C22⋊Q8C2×D18111yescharacteristic
249C22⋊Q8C2×D18111yescharacteristic, complemented
250#121—21no—
251#121—3621no—
252#121—1811no—
253#120—3621no—
254C4×D4—1811no—
255C4×D4C2×D18111yescharacteristic
256C4×D4C2×D18111yescharacteristic, complemented
257C2×C4⋊C4—21no—
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—
269#97—1811no—
270C24—3621no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—3622no—
276#41—7241no—
277C2×Q8—1811no—
278C2×Q8C22×D18111yescharacteristic
279C2×Q8C22×D18111yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—3622no—
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—21no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4C22×D18111yescharacteristic
297C22×C4C22×D18111yescharacteristic
298Q16—3622no—
299Q16—7241no—
300SD16—3622no—
301SD16—3622no—
302SD16—3622no—
303SD16—3622no—
304SD16—7241no—
305D8—3622no—
306D8—1814no—
307M16—3622no—
308C2×C8—3622no—
309C2×C8—1811no—
310C2×C8—1811no—
311C2×C8—1811no—
312C2×C8—1811no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—3621no—
318C4⋊C4—1811no—
319C4⋊C4C22⋊D18111yescharacteristic
320C4⋊C4C22×D18111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4—3621no—
329C22⋊C4C22⋊D18111yescharacteristic
330C22⋊C4C22⋊D18111yescharacteristic
331C42C22×D18111yescharacteristic
332C23—21no—
333C23—3621no—
334C23—3621no—
335C23—14481no—
336C23—7242no—
337C23—7242no—
338C23C2×C22⋊D18111yescharacteristic
339C23C2×C22⋊D18111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—3621no—
345Q8—3621no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—3621no—
351D4—3621no—
352D4—3621no—
353D4—14481no—
354D4—1812no—
355D4—7242no—
356D4—7242no—
357D4D4×D18112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—3621no—
369C2×C4—3621no—
370C2×C4—14481no—
371C2×C4—7242no—
372C2×C4C2×C22⋊D18111yescharacteristic
373C2×C4C2×C22⋊D18111yescharacteristic
374C2×C4C2×C22⋊D18111yescharacteristic
375C2×C4C2×C22⋊D18111yescharacteristic
376C2×C4C23×D18111yescharacteristic
377C8—3621no—
378C8—3621no—
379C8—3621no—
380C8—1812no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—3622no—
391C22—14484no—
392C22—7241no—
393C22—7241no—
394C22—7241no—
395C22—7241no—
396C22—7241no—
397C22C22×C22⋊D18111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—7241no—
406C4—7241no—
407C4C2×D4×D18111yescharacteristic
408C4#1915386111yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—14481no—
413C2—7242no—
414C2D4×C22⋊D18111yescharacteristic
415C2#2604207711yescharacteristic
416C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#547×C181C2
socleC22×C181C22×D4
hypercenter#547D181
hypocenterC181#3267
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#19153841C2semidirect
2. C2×#19153836C2semidirect
3. #26041925C2semidirect
4. #26041671C2semidirect
5. #26041535C2semidirect
6. #26041510C2semidirect
7. #26041494C2semidirect
8. #26041481C2semidirect
9. #26041441C2semidirect
10. #26041439C2semidirect
11. #26041429C2semidirect
12. #26041424C2semidirect
13. #26041389C2semidirect
14. #26041351C2semidirect
15. #547×C181C2semidirect
20. C2×#15668439C22semidirect
21. C2×#15668439C22semidirect
22. C2×#15668437C22semidirect
30. C22×#13089678C22semidirect
39. #19153747C22semidirect
40. C4⋊#13089678C22semidirect
41. #19153740C22semidirect
42. #19153740C22semidirect
43. #19153739C22semidirect
44. #19153737C22semidirect
48. C22⋊#13089678C22semidirect
49. #19153727C22semidirect
51. C2×#41×C181C22semidirect
52. C4⋊Q8×C181C22semidirect
56. C22⋊Q8×C181C22semidirect
59. C4×D4×C181C22semidirect
82. #15668434D4semidirect
208. C181#3267semidirect
212. #547D181semidirect
227. C2×#41C2×D181semidirect
242. C4⋊Q8C2×D181semidirect
249. C22⋊Q8C2×D181semidirect
256. C4×D4C2×D181semidirect

Maximal subgroups

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

subgroupindex
C2×#191538412
C2×#191538362
#260419252
#260416712
#260415352
#260415102
#260414942
#260414812
#260414412
#260414392
#260414292
#260414242
#260413892
#260413512
#547×C1812
#3267181