Finite Group Databasebeta

← #6434867

Subgroups — #6434867

Order 512.

1510 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic, complemented
3#30163C214yesmaximal, complemented
4#30161C214yesmaximal, complemented
5#30154C214yesmaximal, complemented
6#30153C214yesmaximal, complemented
7#23669C212yesmaximal, complemented
8#23656C211yesmaximal, characteristic, complemented
9#23628C211yesmaximal, characteristic
10#23614C212yesmaximal
11#23553C211yesmaximal, characteristic, complemented
12C2×#2025C211yesmaximal, characteristic
13#21065C212yesmaximal
14#21035C212yesmaximal, complemented
15#20754C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C2×#428C2212yes—
18C2×#427C2212yes—
19C2×#424C2212yes—
20C2×#418C2212yes—
21C22×#102C2212yes—
22#2860C2214yes—
23#2644C2214yes—
24#2513—22no—
25#2512—22no—
26#2480—24no—
27#2370—22no—
28#2370—22no—
29#2345—24no—
30#2330C2214yescomplemented
31#2317C2211yescharacteristic
32#2316C2212yes—
33#2316C2211yescharacteristic
34#2309C2211yescharacteristic
35#2309C2211yescharacteristic
36C4×#125C2212yescomplemented
37C4×#125C2212yescomplemented
38#2296C2211yescharacteristic
39C2×#390C2211yescharacteristic
40#2285C2211yescharacteristic
41#2274C2212yes—
42C4×#117C2211yescharacteristic
43#2088C2214yescomplemented
44#2088C2214yescomplemented
45#2087C2214yescomplemented
46#2086C2214yescomplemented
47#2085C2212yescomplemented
48#2085C2212yescomplemented
49#2084C2212yescomplemented
50#2083C2212yescomplemented
51#2073C2212yes—
52#2073C2212yes—
53#2071C2214yes—
54#2067C2214yes—
55#2067C2214yes—
56#2065C2214yes—
57#2065C2214yes—
58#2059C2214yescomplemented
59#2059C2214yescomplemented
60#2025C2214yes—
61#2025C2212yes—
62#2025C2212yes—
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—
84C2×#126—24no—
85C2×#125—28no—
86C2×#125—22no—
87C2×#125C2312yes—
88C2×#125C2312yes—
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C2×#117C2312yes—
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C22×C4⋊C4—22no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4—22no—
98C22×C42C2311yescharacteristic
99#428—28no—
100#427—44no—
101C2×#107C2314yes—
102C2×#107C2312yes—
103C2×#107C2312yes—
104C2×#106C2314yes—
105C2×#106C2312yes—
106C2×#106C2312yes—
107#424—44no—
108C2×C4⋊C8C2314yes—
109C2×C4⋊C8C2314yes—
110#418—44no—
111C2×#102—28no—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116#397—24no—
117#397—24no—
118#397—24no—
119#396—24no—
120#390—21no—
121#390—22no—
122#390—21no—
123#390—22no—
124#390C2314yes—
125#390C2314yes—
126#390C2312yes—
127#390C2312yes—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—24no—
132#389—24no—
133#389C2314yes—
134#389C2314yes—
135#389C2312yes—
136#389C2312yes—
137#389C2314yes—
138#389C2314yes—
139#388—24no—
140#388—24no—
141#388—24no—
142#387—24no—
143#387—24no—
144#387—24no—
145#387—24no—
146#385—22no—
147#385—22no—
148#384—21no—
149#384C2312yes—
150#382—21no—
151#382C2311yescharacteristic
152#382C2311yescharacteristic
153#380—22no—
154#380—24no—
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⋊C4—22no—
164C4×C4⋊C4D412yes—
165C4×C4⋊C4D412yes—
166C4×C4⋊C4C2314yes—
167C4×C4⋊C4C2314yes—
168C4×C22⋊C4—24no—
169C4×C22⋊C4D414yes—
170#376C2312yes—
171C43C2312yes—
172#340C2314yes—
173#340C2314yes—
174#340C2312yes—
175#340C2312yes—
176#340C2312yes—
177#340C2312yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#336C2312yes—
183#336C2312yes—
184#336C2311yescharacteristic
185#336C2311yescharacteristic
186#336C2312yes—
187#336C2314yes—
188#336C2314yes—
189C25—21no—
190C22×D4—21no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4C2412yes—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4C2×D412yes—
199C23×C4C2411yescharacteristic
200C2×M16—24no—
201C2×M16—24no—
202C2×M16—24no—
203C22×C8—24no—
204C22×C8—24no—
205C22×C8—24no—
206C22×C8C2412yes—
207C22×C8C2412yes—
208C22×C8C2414yes—
209#126—88no—
210#125—216no—
211#125—28no—
212#125—28no—
213#117—22no—
214#117—24no—
215#117—22no—
216#117—24no—
217#117—24no—
218#117—22no—
219#117—22no—
220#117—24no—
221C2×C4⋊C4—216no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4#4118yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2414yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—84no—
256C2×C22⋊C4C2×D412yes—
257C2×C22⋊C4C2×D412yes—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42C2×D412yes—
265C2×C42C2×D414yes—
266C2×C42C2×D414yes—
267C2×C42C2412yes—
268C2×C42C2412yes—
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271#107—28no—
272#107—42no—
273#107—42no—
274#106—28no—
275#106—42no—
276#106—42no—
277C4⋊C8—44no—
278C4⋊C8—44no—
279#102—28no—
280#102—28no—
281#102—44no—
282#102—44no—
283#95—24no—
284#95—24no—
285#95—22no—
286#95—22no—
287#95—24no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—24no—
292#95—24no—
293#95C2×D414yes—
294#95C2×D414yes—
295C24—22no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—82no—
300C24—82no—
301C24C22×D411yescharacteristic
302C2×D4—24no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—24no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—48no—
310C2×D4—48no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—28no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4C22×D412yes—
326C22×C4C22×D414yes—
327C22×C4C22×D412yes—
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4114yes—
332C22×C4H(5,𝔽2)12yes—
333C22×C42−1+414yes—
334C22×C42−1+412yes—
335C22×C4C2511yescharacteristic
336M16—48no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—24no—
346C2×C8—48no—
347C4⋊C4—216no—
348C4⋊C4—216no—
349C4⋊C4—28no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—84no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—84no—
369C4⋊C4—82no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4C2×#4118yes—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—48no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—84no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C22⋊C4—84no—
384C42—24no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—24no—
389C42—22no—
390C42—44no—
391C42—44no—
392C42—42no—
393C42—44no—
394C42—42no—
395C42—84no—
396C42—84no—
397C23—22no—
398C23—22no—
399C23—44no—
400C23—44no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—88no—
405C23—84no—
406C23—164no—
407C23—162no—
408C23—162no—
409C23#52512yes—
410C23#53612yes—
411C23#53712yes—
412C23C23×D411yescharacteristic
413D4—44no—
414D4—44no—
415D4—84no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—28no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—84no—
447C2×C4—88no—
448C2×C4—162no—
449C2×C4#52512yes—
450C2×C4#53612yes—
451C2×C4#53712yes—
452C2×C4#54118yes—
453C2×C4#54218yes—
454C2×C4C22×#4112yes—
455C2×C4C2×H(5,𝔽2)11yescharacteristic
456C2×C4C2×2−1+411yescharacteristic
457C2×C4C2×2−1+412yes—
458C8—42no—
459C8—42no—
460C8—44no—
461C22—24no—
462C22—24no—
463C22—22no—
464C22—22no—
465C22—84no—
466C22—82no—
467C22—84no—
468C22—82no—
469C22—82no—
470C22—168no—
471C22#263714yes—
472C22C2×#52511yescharacteristic
473C22C2×#53611yescharacteristic
474C22C2×#53711yescharacteristic
475C22#358412yes—
476C22#358512yes—
477C4—44no—
478C4—42no—
479C4—42no—
480C4—48no—
481C4—42no—
482C4—42no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4—82no—
487C4—82no—
488C4#263714yes—
489C4C2×#54112yes—
490C4C2×#54212yes—
491C2—24no—
492C2—162no—
493C2C2×#263711yescharacteristic
494C2#6058012yes—
495C2#6087911yescharacteristic
496C2#6088311yescharacteristic
497C2#6093312yes—
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. #34217C2semidirect
3. #30163C2semidirect
4. #30161C2semidirect
5. #30154C2semidirect
6. #30153C2semidirect
7. #23669C2semidirect
8. #23656C2semidirect
11. #23553C2semidirect
14. #21035C2semidirect
15. #20754C2semidirect
30. #2330C22semidirect
36. C4×#125C22semidirect
37. C4×#125C22semidirect
43. #2088C22semidirect
44. #2088C22semidirect
45. #2087C22semidirect
46. #2086C22semidirect
47. #2085C22semidirect
48. #2085C22semidirect
49. #2084C22semidirect
50. #2083C22semidirect
58. #2059C22semidirect
59. #2059C22semidirect
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
#342172
#301632
#301612
#301542
#301532
#236692
#236562
#236282
#236142
#235532
C2×#20252
#210652
#210352
#207542