Finite Group Databasebeta

← #6461735

Subgroups — #6461735

Order 512.

1676 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34255C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic, complemented
3#29406C214yesmaximal, complemented
4#29404C214yesmaximal, complemented
5#29397C214yesmaximal, complemented
6#29396C214yesmaximal, complemented
7#23656C211yesmaximal, characteristic, complemented
8#23628C211yesmaximal, characteristic
9#23553C211yesmaximal, characteristic, complemented
10C2×#2025C211yesmaximal, characteristic
11#21063C212yesmaximal
12#21039C212yesmaximal, complemented
13#21036C212yesmaximal, complemented
14#20879C212yesmaximal
15#20774C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C2×#518C2211yescharacteristic
18C2×#428C2212yes—
19C2×#427C2212yes—
20C2×#418C2212yes—
21C22×#102C2212yes—
22#2370—22no—
23#2370—22no—
24#2357—22no—
25#2354—22no—
26#2345—24no—
27#2342—24no—
28#2330C2214yescomplemented
29#2319C2214yes—
30#2317C2211yescharacteristic
31#2316C2212yes—
32#2309C2211yescharacteristic
33C4×#125C2212yescomplemented
34C4×#125C2212yescomplemented
35#2296C2211yescharacteristic
36#2293C2214yes—
37C2×#390C2211yescharacteristic
38#2285C2211yescharacteristic
39#2274C2211yescharacteristic
40#2274C2212yes—
41C4×#117C2211yescharacteristic
42#2085C2212yescomplemented
43#2085C2212yescomplemented
44#2084C2212yescomplemented
45#2083C2212yescomplemented
46#2073C2212yes—
47#2073C2212yes—
48#2071C2214yes—
49#2059C2214yescomplemented
50#2059C2214yescomplemented
51#2025C2214yes—
52#2025C2212yes—
53#2025C2212yes—
54#1926C2214yes—
55#1926C2214yes—
56#1925C2214yes—
57#1925C2214yes—
58#1921C2214yescomplemented
59#1921C2214yescomplemented
60#1920C2214yescomplemented
61#1920C2214yescomplemented
62#1919C2214yescomplemented
63#1919C2214yescomplemented
64#1918C2214yescomplemented
65#1918C2214yescomplemented
66#1864C2214yes—
67#1864C2214yes—
68#1862C2214yes—
69#1862C2214yes—
70#1840C2212yescomplemented
71#1840C2212yescomplemented
72#1839C2214yescomplemented
73#1736C2212yes—
74#1736C2212yes—
75#1735C2212yes—
76C2×#340C2212yes—
77#1729C2212yes—
78#1729C2212yes—
79#1728C2211yescharacteristic
80C2×#336C2211yescharacteristic
81C23×D4C2311yescharacteristic
82C22×M16C2312yes—
83C23×C8C2312yes—
84C2×#125—28no—
85C2×#125—22no—
86C2×#125C2312yes—
87C2×#125C2312yes—
88#518—24no—
89#518—24no—
90C2×C4×D4—22no—
91C2×C4×D4C2314yes—
92C2×#117—24no—
93C2×#117C2311yescharacteristic
94C2×#117C2312yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C22×C4⋊C4—22no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4—22no—
102C22×C22⋊C4—22no—
103C22×C42C2311yescharacteristic
104#428—28no—
105#427—28no—
106C2×#107C2314yes—
107C2×#107C2312yes—
108C2×#107C2312yes—
109C2×#106C2314yes—
110C2×#106C2312yes—
111C2×#106C2312yes—
112#418—28no—
113C2×#102—28no—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118#390—21no—
119#390—21no—
120#390—22no—
121#390C2314yes—
122#390C2312yes—
123#390C2312yes—
124#389—22no—
125#389—22no—
126#389—24no—
127#389—24no—
128#389C2314yes—
129#389C2314yes—
130#389C2312yes—
131#389C2312yes—
132#389C2314yes—
133#388—24no—
134#388—24no—
135#388—24no—
136#388—24no—
137#387—24no—
138#387—24no—
139#387—24no—
140#387—24no—
141#385—22no—
142#385—22no—
143#384—21no—
144#384C2312yes—
145#382—21no—
146#382C2311yescharacteristic
147#382C2311yescharacteristic
148#381—24no—
149#380—22no—
150#380—24no—
151#380—24no—
152#379—22no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4D412yes—
161C4×C4⋊C4D412yes—
162C4×C4⋊C4C2314yes—
163C4×C4⋊C4C2314yes—
164C4×C4⋊C4C2314yes—
165C4×C22⋊C4—22no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4—24no—
168C4×C22⋊C4D414yes—
169#376C2312yes—
170C43C2312yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2312yes—
176#340C2312yes—
177#340C2312yes—
178#340C2312yes—
179#340C2314yes—
180#340C2314yes—
181#336C2312yes—
182#336C2312yes—
183#336C2311yescharacteristic
184#336C2311yescharacteristic
185#336C2312yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189#336C2314yes—
190C25—21no—
191C22×D4—21no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4#4114yes—
196C22×D4C2412yes—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—21no—
200C23×C4C2×D412yes—
201C23×C4C2411yescharacteristic
202C2×M16—24no—
203C2×M16—24no—
204C2×M16—24no—
205C22×C8—24no—
206C22×C8—24no—
207C22×C8—24no—
208C22×C8C2412yes—
209C22×C8C2412yes—
210C22×C8C2414yes—
211#125—216no—
212#125—28no—
213#125—28no—
214C4×D4—216no—
215C4×D4—28no—
216C4×D4—28no—
217#117—22no—
218#117—22no—
219#117—22no—
220#117—24no—
221#117—24no—
222#117—24no—
223#117—24no—
224#117—88no—
225#117#4114yes—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4#4118yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2412yes—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—84no—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D412yes—
262C2×C42—21no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42C2×D412yes—
270C2×C42C2×D414yes—
271C2×C42C2×D414yes—
272C2×C42C2412yes—
273C2×C42C2412yes—
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276#107—24no—
277#107—24no—
278#107—28no—
279#106—24no—
280#106—24no—
281#106—28no—
282#102—28no—
283#102—28no—
284#102—28no—
285#102—28no—
286#95—22no—
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#95—22no—
296#95C2×D414yes—
297#95C2×D414yes—
298C24—22no—
299C24—24no—
300C24—42no—
301C24—42no—
302C24—82no—
303C24—82no—
304C24C22×D411yescharacteristic
305C2×D4—24no—
306C2×D4—216no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—48no—
313C2×D4C2×#4114yes—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—216no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4C22×D412yes—
332C22×C4C22×D414yes—
333C22×C4C22×D412yes—
334C22×C4C22×D412yes—
335C22×C4C22×D412yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4112yes—
338C22×C4C2×#4114yes—
339C22×C4C2×#4112yes—
340C22×C42−1+414yes—
341C22×C4C2511yescharacteristic
342M16—48no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—24no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—48no—
353C4⋊C4—216no—
354C4⋊C4—216no—
355C4⋊C4—28no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—84no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—84no—
374C4⋊C4—82no—
375C4⋊C4—84no—
376C4⋊C4C2×#4118yes—
377C4⋊C4C2×#4114yes—
378C22⋊C4—216no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—48no—
383C22⋊C4—44no—
384C22⋊C4—42no—
385C22⋊C4—84no—
386C22⋊C4—84no—
387C22⋊C4—84no—
388C42—24no—
389C42—24no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—24no—
395C42—44no—
396C42—44no—
397C42—44no—
398C42—42no—
399C42—84no—
400C42—84no—
401C42—84no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—44no—
406C23—44no—
407C23—42no—
408C23—42no—
409C23—48no—
410C23—88no—
411C23—164no—
412C23—162no—
413C23—162no—
414C23#52512yes—
415C23#52512yes—
416C23#53612yes—
417C23C23×D411yescharacteristic
418D4—28no—
419D4—48no—
420D4—44no—
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—28no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—84no—
452C2×C4—88no—
453C2×C4—816no—
454C2×C4—162no—
455C2×C4#52512yes—
456C2×C4#52512yes—
457C2×C4#53612yes—
458C2×C4#55218yes—
459C2×C4#55618yes—
460C2×C4C22×#4111yescharacteristic
461C2×C4C22×#4112yes—
462C2×C4C22×#4111yescharacteristic
463C2×C4C2×2−1+412yes—
464C8—44no—
465C8—42no—
466C8—42no—
467C22—24no—
468C22—24no—
469C22—22no—
470C22—22no—
471C22—44no—
472C22—48no—
473C22—84no—
474C22—82no—
475C22—82no—
476C22—168no—
477C22#259914yes—
478C22C2×#52511yescharacteristic
479C22C2×#52511yescharacteristic
480C22C2×#53611yescharacteristic
481C22#358412yes—
482C22#358512yes—
483C4—42no—
484C4—42no—
485C4—48no—
486C4—44no—
487C4—44no—
488C4—42no—
489C4—42no—
490C4—82no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4#259914yes—
495C4C2×#55212yes—
496C4C2×#55612yes—
497C2—24no—
498C2—84no—
499C2C2×#259911yescharacteristic
500C2#6058012yes—
501C2#6060112yes—
502C2#6087911yescharacteristic
503C2#6088311yescharacteristic
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. #29406C2semidirect
4. #29404C2semidirect
5. #29397C2semidirect
6. #29396C2semidirect
7. #23656C2semidirect
9. #23553C2semidirect
12. #21039C2semidirect
13. #21036C2semidirect
15. #20774C2semidirect
28. #2330C22semidirect
33. C4×#125C22semidirect
34. C4×#125C22semidirect
42. #2085C22semidirect
43. #2085C22semidirect
44. #2084C22semidirect
45. #2083C22semidirect
49. #2059C22semidirect
50. #2059C22semidirect
58. #1921C22semidirect
59. #1921C22semidirect
60. #1920C22semidirect
61. #1920C22semidirect
62. #1919C22semidirect
63. #1919C22semidirect
64. #1918C22semidirect
65. #1918C22semidirect
70. #1840C22semidirect
71. #1840C22semidirect
72. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#342552
#342172
#294062
#294042
#293972
#293962
#236562
#236282
#235532
C2×#20252
#210632
#210392
#210362
#208792
#207742