Finite Group Databasebeta

← #37545491

Subgroups — #37545491

Order 23168.

448 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042095C211yesmaximal, characteristic, complemented
2#26042094C211yesmaximal, characteristic, complemented
3#26042088C211yesmaximal, characteristic, complemented
4#26042087C211yesmaximal, characteristic, complemented
5#26041671C211yesmaximal, characteristic, complemented
6#26041553C211yesmaximal, characteristic, complemented
7#26041552C211yesmaximal, characteristic, complemented
8#26041545C211yesmaximal, characteristic, complemented
9#26041544C211yesmaximal, characteristic, complemented
10#26041461C211yesmaximal, characteristic, complemented
11#26041458C211yesmaximal, characteristic, complemented
12#26041453C211yesmaximal, characteristic, complemented
13#26041398C211yesmaximal, characteristic, complemented
14#26041398C211yesmaximal, characteristic, complemented
15#456×C181C211yesmaximal, characteristic, complemented
16#19153864C2211yescharacteristic
17#19153863C2211yescharacteristic, complemented
18#19153862C2211yescharacteristic
19#19153849C2211yescharacteristic
20#19153848—21no—
21#19153848—21no—
22#19153847—21no—
23#19153847—21no—
24#19153847C2211yescharacteristic
25#19153847C2211yescharacteristic, complemented
26#19153846—21no—
27#19153846C2211yescharacteristic
28#19153845—21no—
29#19153844C2211yescharacteristic
30#19153843—21no—
31#19153841—21no—
32#19153841C2211yescharacteristic
33#19153839—21no—
34#19153837—21no—
35#19153836—21no—
36#19153836—21no—
37#19153836C2211yescharacteristic
38#19153836C2211yescharacteristic
39C2×#15668439—21no—
40C2×#15668439C2211yescharacteristic, complemented
41C2×#15668439C2211yescharacteristic, complemented
42Q16×D181—21no—
43C2×#15668436—21no—
44C2×#15668436C2211yescharacteristic
45C2×#15668436C2211yescharacteristic
46SD16×D181—21no—
47#19153824C2211yescharacteristic
48#19153824C2211yescharacteristic, complemented
49#19153823C2211yescharacteristic
50#19153823C2211yescharacteristic, complemented
51#19153804C2211yescharacteristic
52#19153751C2211yescharacteristic, complemented
53#19153751C2211yescharacteristic, complemented
54#19153744C2211yescharacteristic
55#19153744C2211yescharacteristic, complemented
56#19153735C2211yescharacteristic, complemented
57#19153734C2211yescharacteristic, complemented
58#19153732C2211yescharacteristic
59#19153727C2211yescharacteristic, complemented
602−1+4×C181C2211yescharacteristic
61#137×C181C2211yescharacteristic
62#137×C181C2211yescharacteristic, complemented
63C4⋊Q8×C181C2211yescharacteristic, complemented
64#104×C181C2211yescharacteristic
65#104×C181C2211yescharacteristic, complemented
66#101×C181C2211yescharacteristic, complemented
67#15668450—21no—
68#15668450—21no—
69#15668450—21no—
70#15668449—21no—
71#15668449—21no—
72#15668449—41no—
73#15668449C2311yescharacteristic
74#15668449C2311yescharacteristic
75#15668448—21no—
76#15668448—41no—
77C2×Dic362—21no—
78C2×Dic362C2311yescharacteristic
79Q8×D181—21no—
80Q8×D181—21no—
81Q8×D181—21no—
82C2×D724—21no—
83C2×D724C2311yescharacteristic
84D4×D181—21no—
85D4×D181—41no—
86Dic724—21no—
87#15668439—21no—
88#15668439—21no—
89#15668439—21no—
90#15668439—21no—
91#15668439—21no—
92#15668439—21no—
93#15668439—21no—
94#15668439D411yescharacteristic
95#15668439D411yescharacteristic
96#15668438—21no—
97#15668438—21no—
98#15668437—21no—
99#15668437D411yescharacteristic
100#15668437D411yescharacteristic
101#15668436—21no—
102#15668436—21no—
103#15668436—21no—
104#15668436—21no—
105#15668436—21no—
106#15668436—21no—
107#15668436—21no—
108#15668436D411yescharacteristic
109#15668436D411yescharacteristic
110D1448—21no—
111#15668434—21no—
112#15668434D411yescharacteristic
113#15668434D411yescharacteristic
114#15668433—21no—
115#15668433—21no—
116#15668433C2311yescharacteristic
117#15668433C2311yescharacteristic
118#15668432—21no—
119#15668432—21no—
120C2×#13089678—21no—
121C2×#13089678—21no—
122C2×#13089678C2311yescharacteristic
123C2×#13089678C2311yescharacteristic
124C8×D181—21no—
125C8×D181—21no—
126#15668426—41no—
127#41×C181—21no—
128#41×C181—21no—
129#41×C181—41no—
130#41×C181C2311yescharacteristic
131#41×C181C2311yescharacteristic
132C2×Q8×C181—21no—
133C2×Q8×C181—21no—
134C2×Q8×C181C2311yescharacteristic
135C2×Q8×C181C2311yescharacteristic
136Q16×C181—21no—
137Q16×C181—21no—
138SD16×C181—21no—
139SD16×C181—21no—
140M16×C181C2311yescharacteristic
141M16×C181C2311yescharacteristic
142C4⋊C4×C181—41no—
143C42×C181C2311yescharacteristic
144C22×D181—21no—
145C22×D181—41no—
146Dic362—21no—
147Dic362—21no—
148Dic362—41no—
149Dic362—41no—
150Dic362C2×D411yescharacteristic
151Dic362C2×D411yescharacteristic
152D724—21no—
153D724—21no—
154D724—41no—
155D724—41no—
156D724C2×D411yescharacteristic
157D724C2×D411yescharacteristic
158C22⋊D181—21no—
159C22⋊D181—21no—
160C22⋊D181—41no—
161C22⋊D181—41no—
162C2×Dic181—21no—
163C2×Dic181—41no—
164C4×D181—21no—
165C4×D181—21no—
166C4×D181—41no—
167C4×D181—41no—
168C4×D181—41no—
169C4×D181—41no—
170#13089678—21no—
171#13089678—21no—
172#13089678C2×D411yescharacteristic
173#13089678C2×D411yescharacteristic
174#13089678C2×D411yescharacteristic
175#13089678C2×D411yescharacteristic
176Q8×C181—21no—
177Q8×C181—21no—
178Q8×C181—21no—
179Q8×C181—21no—
180Q8×C181—41no—
181Q8×C181C2×D411yescharacteristic
182Q8×C181C2×D411yescharacteristic
183D4×C181—21no—
184D4×C181—21no—
185D4×C181—41no—
186D4×C181C2×D411yescharacteristic
187D4×C181C2×D411yescharacteristic
188C2×C4×C181—21no—
189C2×C4×C181—21no—
190C2×C4×C181—21no—
191C2×C4×C181—21no—
192C2×C4×C181—21no—
193C2×C4×C181—41no—
194C2×C4×C181—41no—
195C2×C4×C181C2411yescharacteristic
196C8×C181—21no—
197C8×C181—21no—
198C22×C181—21no—
199C22×C181—21no—
200C22×C181H(5,𝔽2)11yescharacteristic
201C2×D181—21no—
202C2×D181—21no—
203C2×D181—41no—
204C2×D181—81no—
205C2×D181—161no—
206C4×C181—21no—
207C4×C181—21no—
208C4×C181—41no—
209C4×C181—41no—
210C4×C181—41no—
211C4×C181C22×D411yescharacteristic
212C4×C181C22×D411yescharacteristic
213Dic181—21no—
214Dic181—21no—
215Dic181—41no—
216C2×C181—21no—
217C2×C181—41no—
218C2×C181—41no—
219C2×C181D4211yescharacteristic
220D181—41no—
221D181—41no—
222D181—81no—
223C181#328911yescharacteristic, complemented, Sylow
224#3289—1811nomaximal, Sylow
225#578—1811no—
226#578—1811no—
227#577—1811no—
228#577—1811no—
229#497—1811no—
230#473—1811no—
231#473—1811no—
232#471—1811no—
233#471—1811no—
234#456—1811no—
235#456D18111yescharacteristic, complemented
236#455—1811no—
237#454—1811no—
238#444—1811no—
239#444—1811no—
2402−1+4—1811no—
2412−1+4—1811no—
2422−1+4C2×D18111yescharacteristic
243H(5,𝔽2)—1811no—
244#137—3621no—
245#137—3621no—
246#137—3621no—
247#137—3621no—
248#137—3621no—
249#137—3621no—
250#137—1811no—
251#137—1811no—
252#137—1811no—
253#137—1811no—
254#137C2×D18111yescharacteristic
255#137C2×D18111yescharacteristic, complemented
256AGL(1,ℤ/8)—3621no—
257AGL(1,ℤ/8)—3621no—
258AGL(1,ℤ/8)—1811no—
259AGL(1,ℤ/8)—1811no—
260#135—3621no—
261#135—3621no—
262#135—3621no—
263#135—3621no—
264#135—1811no—
265#135—1811no—
266C2×Q16—3621no—
267C2×Q16—3621no—
268C2×Q16—1811no—
269C2×Q16—1811no—
270C2×SD16—3621no—
271C2×SD16—3621no—
272C2×SD16—1811no—
273C2×SD16—1811no—
274#131—1811no—
275#131—1811no—
276#131—1811no—
277#131—1811no—
278C4⋊Q8C2×D18111yescharacteristic, complemented
279#124—1811no—
280#108—1811no—
281#108—1811no—
282#104—1811no—
283#104—1811no—
284#104C2×D18111yescharacteristic
285#104C2×D18111yescharacteristic, complemented
286#101—1811no—
287#101—1811no—
288#101C2×D18111yescharacteristic, complemented
289#100—1811no—
290#97—1811no—
291#41—21no—
292#41—21no—
293#41—41no—
294#41—3621no—
295#41—3621no—
296#41—3621no—
297#41—3621no—
298#41—3621no—
299#41—3621no—
300#41—1811no—
301#41—1811no—
302#41—7241no—
303#41—7241no—
304#41C22×D18111yescharacteristic
305#41C22×D18111yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—3621no—
309C2×Q8—3621no—
310C2×Q8—3621no—
311C2×Q8—3621no—
312C2×Q8—1811no—
313C2×Q8C22×D18111yescharacteristic
314C2×Q8C22×D18111yescharacteristic
315C2×D4—3621no—
316C2×D4—3621no—
317C2×D4—1811no—
318C2×D4—7241no—
319Q16—21no—
320Q16—21no—
321Q16—3621no—
322Q16—3621no—
323Q16—3621no—
324Q16—3621no—
325Q16—3621no—
326Q16—3621no—
327Q16—3621no—
328Q16—3621no—
329Q16—1811no—
330Q16—1811no—
331SD16—21no—
332SD16—21no—
333SD16—3621no—
334SD16—3621no—
335SD16—3621no—
336SD16—3621no—
337SD16—3621no—
338SD16—3621no—
339SD16—3621no—
340SD16—3621no—
341SD16—3621no—
342SD16—3621no—
343SD16—1811no—
344SD16—1811no—
345SD16—1811no—
346SD16—1811no—
347D8—3621no—
348D8—3621no—
349D8—1811no—
350D8—1811no—
351M16—3621no—
352M16—3621no—
353M16—3621no—
354M16—3621no—
355M16—1811no—
356M16—1811no—
357M16C22×D18111yescharacteristic
358M16C22×D18111yescharacteristic
359C2×C8—3621no—
360C2×C8—3621no—
361C2×C8—3621no—
362C2×C8—3621no—
363C2×C8—1811no—
364C2×C8—1811no—
365C4⋊C4—41no—
366C22⋊C4—7241no—
367C42C22×D18111yescharacteristic
368C23—3621no—
369C23—7241no—
370Q8—21no—
371Q8—21no—
372Q8—21no—
373Q8—21no—
374Q8—41no—
375Q8—3621no—
376Q8—3621no—
377Q8—1811no—
378Q8—1811no—
379Q8—7241no—
380Q8—7241no—
381Q8D4×D18111yescharacteristic
382Q8D4×D18111yescharacteristic
383D4—21no—
384D4—21no—
385D4—41no—
386D4—3621no—
387D4—3621no—
388D4—3621no—
389D4—3621no—
390D4—1811no—
391D4—1811no—
392D4—7241no—
393D4—7241no—
394D4—7241no—
395D4—7241no—
396D4D4×D18111yescharacteristic
397D4D4×D18111yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—3621no—
406C2×C4—3621no—
407C2×C4—3621no—
408C2×C4—7241no—
409C2×C4—7241no—
410C2×C4—7241no—
411C2×C4—7241no—
412C2×C4—7241no—
413C2×C4C23×D18111yescharacteristic
414C8—21no—
415C8—21no—
416C8—3621no—
417C8—3621no—
418C8—1811no—
419C8—1811no—
420C8—1811no—
421C8—1811no—
422C22—21no—
423C22—21no—
424C22—28961no—
425C22—3621no—
426C22—3621no—
427C22—14481no—
428C22—7241no—
429C22#1915386111yescharacteristic
430C4—21no—
431C4—21no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—3621no—
436C4—3621no—
437C4—7241no—
438C4C2×D4×D18111yescharacteristic
439C4C2×D4×D18111yescharacteristic
440C2—21no—
441C2—41no—
442C2—41no—
443C2—14481no—
444C2—7241no—
445C2—7241no—
446C2#2604191111yescharacteristic
subgroupG/H
centerC2#26041911
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#456×C181C2
socleC2×C181D42
hypercenter#456D181
hypocenterC181#3289
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #26042095C2semidirect
2. #26042094C2semidirect
3. #26042088C2semidirect
4. #26042087C2semidirect
5. #26041671C2semidirect
6. #26041553C2semidirect
7. #26041552C2semidirect
8. #26041545C2semidirect
9. #26041544C2semidirect
10. #26041461C2semidirect
11. #26041458C2semidirect
12. #26041453C2semidirect
13. #26041398C2semidirect
14. #26041398C2semidirect
15. #456×C181C2semidirect
17. #19153863C22semidirect
25. #19153847C22semidirect
40. C2×#15668439C22semidirect
41. C2×#15668439C22semidirect
48. #19153824C22semidirect
50. #19153823C22semidirect
52. #19153751C22semidirect
53. #19153751C22semidirect
55. #19153744C22semidirect
56. #19153735C22semidirect
57. #19153734C22semidirect
59. #19153727C22semidirect
62. #137×C181C22semidirect
63. C4⋊Q8×C181C22semidirect
65. #104×C181C22semidirect
66. #101×C181C22semidirect
223. C181#3289semidirect
235. #456D181semidirect
255. #137C2×D181semidirect
278. C4⋊Q8C2×D181semidirect
285. #104C2×D181semidirect
288. #101C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420952
#260420942
#260420882
#260420872
#260416712
#260415532
#260415522
#260415452
#260415442
#260414612
#260414582
#260414532
#260413982
#456×C1812
#3289181