Finite Group Databasebeta

← #6434877

Subgroups — #6434877

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic, complemented
3#30158C214yesmaximal, complemented
4#30157C214yesmaximal, complemented
5#30154C214yesmaximal, complemented
6#30153C214yesmaximal, complemented
7#23671C212yesmaximal, complemented
8#23659C211yesmaximal, characteristic, complemented
9C2×#2073C211yesmaximal, characteristic
10#23612C212yesmaximal
11#23551C211yesmaximal, characteristic, complemented
12#23454C211yesmaximal, characteristic
13#21063C212yesmaximal
14#21039C212yesmaximal, complemented
15#20754C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#428C2212yes—
18C2×#427C2212yes—
19C2×#424C2212yes—
20C2×#418C2212yes—
21C22×#102C2212yes—
22#2860C2214yes—
23#2644C2214yes—
24#2513—22no—
25#2512—22no—
26#2480—24no—
27#2368—24no—
28#2355—22no—
29#2355—22no—
30#2331C2214yescomplemented
31#2317C2211yescharacteristic
32#2317C2212yes—
33#2316C2211yescharacteristic
34#2309C2211yescharacteristic
35#2309C2211yescharacteristic
36C4×#125C2212yescomplemented
37C4×#125C2212yescomplemented
38#2296C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#382C2211yescharacteristic
41#2274C2212yes—
42C4×#117C2211yescharacteristic
43#2088C2214yescomplemented
44#2088C2214yescomplemented
45#2087C2214yescomplemented
46#2086C2214yescomplemented
47#2085C2214yescomplemented
48#2085C2214yescomplemented
49#2073C2214yes—
50#2073C2212yes—
51#2073C2212yes—
52#2067C2214yes—
53#2067C2214yes—
54#2065C2214yes—
55#2065C2214yes—
56#2059C2212yescomplemented
57#2059C2212yescomplemented
58#2058C2212yescomplemented
59#2057C2212yescomplemented
60#2025C2212yes—
61#2025C2212yes—
62#2023C2214yes—
63#1926C2214yes—
64#1926C2214yes—
65#1925C2214yes—
66#1925C2214yes—
67#1921C2214yescomplemented
68#1920C2214yescomplemented
69#1919C2214yescomplemented
70#1918C2214yescomplemented
71#1832C2212yescomplemented
72#1832C2212yescomplemented
73#1831C2214yescomplemented
74#1736C2212yes—
75#1736C2212yes—
76#1735C2212yes—
77C2×#340C2212yes—
78#1729C2212yes—
79#1728C2211yescharacteristic
80C2×#336C2211yescharacteristic
81C23×D4C2311yescharacteristic
82C22×M16C2312yes—
83C23×C8C2312yes—
84#533—28no—
85C2×C4⋊Q8C2312yes—
86C2×#126—24no—
87C2×#125C2311yescharacteristic
88C2×#125C2311yescharacteristic
89C2×C22⋊Q8—22no—
90C2×C4×Q8—22no—
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C2×#117C2312yes—
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4—22no—
99C22×C42C2311yescharacteristic
100#428—44no—
101#427—28no—
102C2×#107C2312yes—
103C2×#107C2312yes—
104C2×#107C2314yes—
105C2×#106C2312yes—
106C2×#106C2312yes—
107C2×#106C2314yes—
108#424—44no—
109C2×C4⋊C8C2314yes—
110C2×C4⋊C8C2314yes—
111#418—28no—
112C2×#102—44no—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117#397—24no—
118#397—24no—
119#397—24no—
120#396—24no—
121#391—24no—
122#391—24no—
123#390—21no—
124#390—22no—
125#390—21no—
126#390—22no—
127#390C2314yes—
128#390C2314yes—
129#390C2312yes—
130#390C2312yes—
131#389—22no—
132#389C2314yes—
133#389C2314yes—
134#389C2312yes—
135#389C2312yes—
136#389C2314yes—
137#389C2314yes—
138#388—24no—
139#388—24no—
140#388—24no—
141#387—24no—
142#387—24no—
143#387—24no—
144#387—24no—
145#385—22no—
146#385—24no—
147#385—22no—
148#384—22no—
149#384—21no—
150#382—22no—
151#382—21no—
152#382—22no—
153#382C2312yes—
154#382C2312yes—
155#380—22no—
156#379—22no—
157#379—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4D412yes—
164C4×C4⋊C4D412yes—
165C4×C4⋊C4C2314yes—
166C4×C4⋊C4C2314yes—
167C4×C22⋊C4—24no—
168C4×C22⋊C4D414yes—
169#376C2312yes—
170C2×#95—22no—
171C43C2312yes—
172#340C2312yes—
173#340C2312yes—
174#340C2312yes—
175#340C2312yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#336C2311yescharacteristic
183#336C2311yescharacteristic
184#336C2312yes—
185#336C2312yes—
186#336C2312yes—
187#336C2314yes—
188#336C2314yes—
189C25—21no—
190C22×Q8—22no—
191C22×D4—21no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4C2412yes—
197C23×C4—21no—
198C23×C4—22no—
199C23×C4C2×D412yes—
200C23×C4C2411yescharacteristic
201C2×M16—24no—
202C2×M16—24no—
203C2×M16—24no—
204C22×C8—24no—
205C22×C8—24no—
206C22×C8—24no—
207C22×C8C2412yes—
208C22×C8C2412yes—
209C22×C8C2414yes—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212#126—88no—
213#125—24no—
214#125—24no—
215C22⋊Q8—216no—
216C4×Q8—216no—
217#117—24no—
218#117—22no—
219#117—24no—
220#117—22no—
221#117—24no—
222#117—22no—
223#117—22no—
224#117#4118yes—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—84no—
258C2×C22⋊C4C2×D412yes—
259C2×C22⋊C4C2×D412yes—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42C2×D412yes—
267C2×C42C2×D414yes—
268C2×C42C2×D414yes—
269C2×C42C2412yes—
270C2×C42C2412yes—
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273#107—24no—
274#107—24no—
275#107—44no—
276#106—24no—
277#106—24no—
278#106—44no—
279C4⋊C8—44no—
280C4⋊C8—44no—
281#102—28no—
282#102—28no—
283#102—44no—
284#102—44no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—24no—
295#95C2×D414yes—
296#95C2×D414yes—
297C24—22no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—82no—
302C24—82no—
303C24C22×D411yescharacteristic
304C2×Q8—28no—
305C2×Q8—82no—
306C2×Q8—82no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—48no—
315C2×D4—48no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4C22×D412yes—
331C22×C4C22×D412yes—
332C22×C4C22×D414yes—
333C22×C4C22×D412yes—
334C22×C4C22×D412yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4114yes—
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+414yes—
339C22×C42−1+412yes—
340C22×C4C2511yescharacteristic
341M16—48no—
342C2×C8—24no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—48no—
352C4⋊C4—28no—
353C4⋊C4—28no—
354C4⋊C4—28no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4—84no—
371C4⋊C4—82no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4C2×#4114yes—
376C4⋊C4C2×#4114yes—
377C22⋊C4—28no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—48no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C22⋊C4—84no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C42—24no—
389C42—28no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—24no—
394C42—22no—
395C42—44no—
396C42—44no—
397C42—42no—
398C42—42no—
399C42—84no—
400C42—84no—
401C23—22no—
402C23—22no—
403C23—44no—
404C23—44no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—88no—
409C23—84no—
410C23—164no—
411C23—162no—
412C23—162no—
413C23#52512yes—
414C23#53612yes—
415C23#53712yes—
416C23C23×D411yescharacteristic
417Q8—88no—
418D4—44no—
419D4—44no—
420D4—84no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—84no—
452C2×C4—162no—
453C2×C4—164no—
454C2×C4#52512yes—
455C2×C4#53612yes—
456C2×C4#53712yes—
457C2×C4#54118yes—
458C2×C4#54218yes—
459C2×C4C22×#4112yes—
460C2×C4C2×H(5,𝔽2)11yescharacteristic
461C2×C4C2×2−1+411yescharacteristic
462C2×C4C2×2−1+412yes—
463C8—42no—
464C8—42no—
465C8—44no—
466C22—24no—
467C22—22no—
468C22—22no—
469C22—24no—
470C22—82no—
471C22—84no—
472C22—82no—
473C22—82no—
474C22—84no—
475C22—168no—
476C22#263714yes—
477C22C2×#52511yescharacteristic
478C22C2×#53611yescharacteristic
479C22C2×#53711yescharacteristic
480C22#358412yes—
481C22#358512yes—
482C4—42no—
483C4—42no—
484C4—44no—
485C4—44no—
486C4—42no—
487C4—42no—
488C4—44no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—84no—
493C4#263714yes—
494C4C2×#54112yes—
495C4C2×#54212yes—
496C2—24no—
497C2—162no—
498C2C2×#263711yescharacteristic
499C2#6058712yes—
500C2#6087911yescharacteristic
501C2#6088311yescharacteristic
502C2#6092612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34215C2semidirect
3. #30158C2semidirect
4. #30157C2semidirect
5. #30154C2semidirect
6. #30153C2semidirect
7. #23671C2semidirect
8. #23659C2semidirect
11. #23551C2semidirect
14. #21039C2semidirect
15. #20754C2semidirect
30. #2331C22semidirect
36. C4×#125C22semidirect
37. C4×#125C22semidirect
43. #2088C22semidirect
44. #2088C22semidirect
45. #2087C22semidirect
46. #2086C22semidirect
47. #2085C22semidirect
48. #2085C22semidirect
56. #2059C22semidirect
57. #2059C22semidirect
58. #2058C22semidirect
59. #2057C22semidirect
67. #1921C22semidirect
68. #1920C22semidirect
69. #1919C22semidirect
70. #1918C22semidirect
71. #1832C22semidirect
72. #1832C22semidirect
73. #1831C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#342152
#301582
#301572
#301542
#301532
#236712
#236592
C2×#20732
#236122
#235512
#234542
#210632
#210392
#207542