Finite Group Databasebeta

← #6461725

Subgroups — #6461725

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34255C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic, complemented
3#29401C214yesmaximal, complemented
4#29400C214yesmaximal, complemented
5#29397C214yesmaximal, complemented
6#29396C214yesmaximal, complemented
7#23659C211yesmaximal, characteristic, complemented
8C2×#2073C211yesmaximal, characteristic
9#23551C211yesmaximal, characteristic, complemented
10#23454C211yesmaximal, characteristic
11#21065C212yesmaximal
12#21039C212yesmaximal, complemented
13#21035C212yesmaximal, complemented
14#20876C212yesmaximal
15#20774C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#518C2211yescharacteristic
18C2×#428C2212yes—
19C2×#427C2212yes—
20C2×#418C2212yes—
21C22×#102C2212yes—
22#2368—24no—
23#2357—22no—
24#2355—22no—
25#2355—22no—
26#2354—22no—
27#2342—24no—
28#2331C2214yescomplemented
29#2319C2214yes—
30#2317C2211yescharacteristic
31#2317C2212yes—
32#2309C2211yescharacteristic
33C4×#125C2212yescomplemented
34C4×#125C2212yescomplemented
35#2296C2211yescharacteristic
36#2293C2214yes—
37C2×#390C2211yescharacteristic
38C2×#382C2211yescharacteristic
39#2274C2212yes—
40#2274C2211yescharacteristic
41C4×#117C2211yescharacteristic
42#2085C2214yescomplemented
43#2085C2214yescomplemented
44#2073C2214yes—
45#2073C2212yes—
46#2073C2212yes—
47#2059C2212yescomplemented
48#2059C2212yescomplemented
49#2058C2212yescomplemented
50#2057C2212yescomplemented
51#2025C2212yes—
52#2025C2212yes—
53#2023C2214yes—
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—
84#533—28no—
85C2×C4⋊Q8C2312yes—
86C2×#125C2311yescharacteristic
87C2×#125C2311yescharacteristic
88C2×C22⋊Q8—22no—
89#518—24no—
90#518—24no—
91C2×C4×Q8—22no—
92C2×C4×D4—22no—
93C2×C4×D4C2314yes—
94C2×#117—24no—
95C2×#117C2311yescharacteristic
96C2×#117C2312yes—
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—22no—
103C22×C22⋊C4—22no—
104C22×C42C2311yescharacteristic
105#428—28no—
106#427—28no—
107C2×#107C2312yes—
108C2×#107C2312yes—
109C2×#107C2314yes—
110C2×#106C2312yes—
111C2×#106C2312yes—
112C2×#106C2314yes—
113#418—28no—
114C2×#102—28no—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119#391—24no—
120#391—24no—
121#390—21no—
122#390—21no—
123#390—22no—
124#390C2314yes—
125#390C2312yes—
126#390C2312yes—
127#389C2314yes—
128#389C2314yes—
129#389C2312yes—
130#389C2312yes—
131#389C2314yes—
132#388—24no—
133#388—24no—
134#388—24no—
135#388—24no—
136#387—24no—
137#387—24no—
138#387—24no—
139#387—24no—
140#385—22no—
141#385—22no—
142#385—24no—
143#384—21no—
144#384—22no—
145#382—22no—
146#382—21no—
147#382—22no—
148#382C2312yes—
149#382C2312yes—
150#381—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⋊C4D412yes—
160C4×C4⋊C4D412yes—
161C4×C4⋊C4C2314yes—
162C4×C4⋊C4C2314yes—
163C4×C4⋊C4C2314yes—
164C4×C22⋊C4—22no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4D414yes—
168#376C2312yes—
169C2×#95—22no—
170C43C2312yes—
171#340C2312yes—
172#340C2312yes—
173#340C2312yes—
174#340C2312yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#336C2311yescharacteristic
182#336C2311yescharacteristic
183#336C2312yes—
184#336C2312yes—
185#336C2312yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189#336C2314yes—
190C25—21no—
191C22×Q8—22no—
192C22×D4—21no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4#4114yes—
197C22×D4C2412yes—
198C23×C4—24no—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4C2×D412yes—
202C23×C4C2411yescharacteristic
203C2×M16—24no—
204C2×M16—24no—
205C2×M16—24no—
206C22×C8—24no—
207C22×C8—24no—
208C22×C8—24no—
209C22×C8C2412yes—
210C22×C8C2412yes—
211C22×C8C2414yes—
212C4⋊Q8—24no—
213C4⋊Q8—24no—
214#125—24no—
215#125—24no—
216C22⋊Q8—216no—
217C4×Q8—216no—
218C4×D4—216no—
219C4×D4—28no—
220C4×D4—28no—
221#117—22no—
222#117—22no—
223#117—22no—
224#117—24no—
225#117—24no—
226#117—24no—
227#117—88no—
228#117#4114yes—
229#117#4118yes—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2412yes—
252C2×C22⋊C4—216no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—84no—
262C2×C22⋊C4C2×D412yes—
263C2×C22⋊C4C2×D412yes—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—21no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42C2×D412yes—
272C2×C42C2×D414yes—
273C2×C42C2×D414yes—
274C2×C42C2412yes—
275C2×C42C2412yes—
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278#107—24no—
279#107—24no—
280#107—28no—
281#106—24no—
282#106—24no—
283#106—28no—
284#102—28no—
285#102—28no—
286#102—28no—
287#102—28no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95C2×D414yes—
299#95C2×D414yes—
300C24—22no—
301C24—24no—
302C24—42no—
303C24—42no—
304C24—82no—
305C24—82no—
306C24C22×D411yescharacteristic
307C2×Q8—28no—
308C2×Q8—82no—
309C2×Q8—82no—
310C2×D4—24no—
311C2×D4—216no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—24no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—48no—
318C2×D4C2×#4114yes—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—216no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D414yes—
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4112yes—
343C22×C4C2×#4114yes—
344C22×C4C2×#4112yes—
345C22×C42−1+414yes—
346C22×C4C2511yescharacteristic
347M16—48no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—48no—
358C4⋊C4—28no—
359C4⋊C4—28no—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—84no—
376C4⋊C4—82no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4C2×#4114yes—
380C4⋊C4C2×#4114yes—
381C4⋊C4C2×#4114yes—
382C22⋊C4—216no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—28no—
386C22⋊C4—24no—
387C22⋊C4—48no—
388C22⋊C4—42no—
389C22⋊C4—84no—
390C22⋊C4—84no—
391C22⋊C4—84no—
392C42—24no—
393C42—24no—
394C42—28no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—24no—
400C42—44no—
401C42—44no—
402C42—42no—
403C42—84no—
404C42—84no—
405C42—84no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—44no—
410C23—44no—
411C23—42no—
412C23—42no—
413C23—48no—
414C23—88no—
415C23—164no—
416C23—162no—
417C23—162no—
418C23#52512yes—
419C23#52512yes—
420C23#53612yes—
421C23C23×D411yescharacteristic
422Q8—88no—
423D4—28no—
424D4—48no—
425D4—44no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—84no—
457C2×C4—816no—
458C2×C4—164no—
459C2×C4—162no—
460C2×C4#52512yes—
461C2×C4#52512yes—
462C2×C4#53612yes—
463C2×C4#55218yes—
464C2×C4#55618yes—
465C2×C4C22×#4111yescharacteristic
466C2×C4C22×#4112yes—
467C2×C4C22×#4111yescharacteristic
468C2×C4C2×2−1+412yes—
469C8—42no—
470C8—42no—
471C8—44no—
472C22—24no—
473C22—24no—
474C22—22no—
475C22—22no—
476C22—44no—
477C22—48no—
478C22—84no—
479C22—82no—
480C22—82no—
481C22—168no—
482C22#259914yes—
483C22C2×#52511yescharacteristic
484C22C2×#52511yescharacteristic
485C22C2×#53611yescharacteristic
486C22#358412yes—
487C22#358512yes—
488C4—42no—
489C4—42no—
490C4—44no—
491C4—44no—
492C4—44no—
493C4—44no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—84no—
498C4—82no—
499C4#259914yes—
500C4C2×#55212yes—
501C4C2×#55612yes—
502C2—24no—
503C2—84no—
504C2C2×#259911yescharacteristic
505C2#6058712yes—
506C2#6059212yes—
507C2#6087911yescharacteristic
508C2#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. #34215C2semidirect
3. #29401C2semidirect
4. #29400C2semidirect
5. #29397C2semidirect
6. #29396C2semidirect
7. #23659C2semidirect
9. #23551C2semidirect
12. #21039C2semidirect
13. #21035C2semidirect
15. #20774C2semidirect
28. #2331C22semidirect
33. C4×#125C22semidirect
34. C4×#125C22semidirect
42. #2085C22semidirect
43. #2085C22semidirect
47. #2059C22semidirect
48. #2059C22semidirect
49. #2058C22semidirect
50. #2057C22semidirect
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
#342152
#294012
#294002
#293972
#293962
#236592
C2×#20732
#235512
#234542
#210652
#210392
#210352
#208762
#207742