Finite Group Databasebeta

← #6499309

Subgroups — #6499309

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic, complemented
2#34280C211yesmaximal, characteristic
3#33074C214yesmaximal, complemented
4#29982C212yesmaximal, complemented
5#29980C212yesmaximal, complemented
6#29283C212yesmaximal, complemented
7#29280C212yesmaximal, complemented
8#29263C212yesmaximal, complemented
9#29254C212yesmaximal, complemented
10#29175C214yesmaximal, complemented
11#29163C212yesmaximal, complemented
12#29157C212yesmaximal, complemented
13#21968C212yesmaximal, complemented
14#21897C212yesmaximal
15#20754C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C2×#424C2212yes—
18#2868C2211yescharacteristic
19#2862C2211yescharacteristic
20#2857C2212yes—
21#2647C2212yes—
22#2628C2212yes—
23#2511—22no—
24#2492—22no—
25#2451—22no—
26#2423—22no—
27#2374—22no—
28#2364—22no—
29#2363—22no—
30#2359—22no—
31#2331C2212yes—
32#2330C2211yescharacteristic
33#2328C2211yescharacteristic
34#2324C2212yes—
35#2314C2212yes—
36#2314C2211yescharacteristic
37C4×C4⋊Q8C2212yes—
38C4×#127C2211yescharacteristic
39C4×#125C2211yescharacteristic
40C4×#121C2212yes—
41#2293C2212yes—
42#2275C2212yes—
43C4×#117C2211yescharacteristic
44#2101C2214yes—
45#2099C2212yes—
46#2099C2212yes—
47#2093C2212yes—
48#2093C2212yes—
49#2092C2214yes—
50#2058C2212yes—
51#2057C2214yes—
52#2057C2212yes—
53#2023C2214yes—
54#2023C2212yes—
55#2023C2212yes—
56#1989C2214yes—
57#1982C2214yes—
58#1949C2214yes—
59#1949C2214yes—
60#1947C2214yes—
61#1947C2214yes—
62#1921C2214yes—
63#1921C2214yes—
64#1920C2214yes—
65#1918C2214yes—
66#1895C2214yes—
67#1893C2214yes—
68#1864C2214yes—
69#1862C2214yes—
70#1862C2214yes—
71#1862C2214yes—
72#1832C2212yes—
73#1832C2212yes—
74#1831C2212yes—
75#1831C2212yes—
76#1809C2214yes—
77#1808C2214yes—
78#1804C2214yes—
79#1802C2214yes—
80#1729C2212yes—
81#1728C2211yescharacteristic
82C2×#336C2211yescharacteristic
83C22×M16C2312yes—
84C23×C8C2312yes—
85#540—24no—
86#540Q818yescomplemented
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91C2×#125C2311yescharacteristic
92C2×#123—22no—
93C2×C22⋊Q8—22no—
94C2×#121—22no—
95C2×#121C2312yes—
96C2×C4×Q8—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×#117—22no—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C2×#107C2314yes—
107C2×#107C2314yes—
108C2×#107C2314yes—
109C2×#106C2314yes—
110#424—28no—
111C2×C4⋊C8C2314yes—
112C2×C4⋊C8C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117#400—22no—
118#400—24no—
119#399—22no—
120#398—24no—
121#395—22no—
122#393—22no—
123#391—22no—
124#391—22no—
125#391—22no—
126#391—22no—
127#391—24no—
128#390—22no—
129#390—22no—
130#390C2312yes—
131#390C2312yes—
132#389—22no—
133#389—22no—
134#389—22no—
135#389C2312yes—
136#389C2314yes—
137#389C2312yes—
138#389C2314yes—
139#388—22no—
140#388—22no—
141#388—24no—
142#388—22no—
143#388—22no—
144#388—22no—
145#387—22no—
146#387—24no—
147#387—22no—
148#387—22no—
149#387—24no—
150#386—22no—
151#386—22no—
152#386—22no—
153#386—22no—
154#385—22no—
155#384—22no—
156#384C2311yescharacteristic
157#384C2312yes—
158#384C2312yes—
159#382C2311yescharacteristic
160#382C2312yes—
161#382C2312yes—
162#380—22no—
163#380C2312yes—
164#380C2314yes—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4D412yes—
168C4×C4⋊C4D412yes—
169C4×C4⋊C4C2312yes—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C22⋊C4—22no—
174C4×C22⋊C4—22no—
175C4×C22⋊C4—22no—
176C4×C22⋊C4—22no—
177C4×C22⋊C4—22no—
178C4×C22⋊C4D412yes—
179C4×C22⋊C4D412yes—
180#376C2311yescharacteristic
181#376C2311yescharacteristic
182C2×#95—22no—
183C43C2311yescharacteristic
184C43C2311yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2312yes—
194#336C2314yes—
195#336C2312yes—
196#336C2314yes—
197#336C2312yes—
198#336C2312yes—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4C2412yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4C2×D412yes—
208C23×C4C2411yescharacteristic
209C2×M16—24no—
210C2×M16—42no—
211C2×M16—42no—
212C22×C8—24no—
213C22×C8—42no—
214C22×C8—42no—
215C22×C8C2414yes—
216C22×C8C2414yes—
217C4⋊Q8—44no—
218C4⋊Q8—44no—
219C4⋊Q8—84no—
220#127—22no—
221#127C2×Q814yes—
222#125—22no—
223#125C2×Q814yes—
224#123—216no—
225C22⋊Q8—84no—
226#121—24no—
227#121—216no—
228#121C2×Q818yes—
229C4×Q8—84no—
230C4×D4—24no—
231C4×D4—44no—
232C4×D4—44no—
233C4×D4—84no—
234C4×D4—84no—
235C4×D4C2×Q818yes—
236#117—42no—
237#117—84no—
238#117C2×Q814yes—
239#117#4114yes—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2412yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4C2×D411yescharacteristic
270C2×C22⋊C4C2×D411yescharacteristic
271C2×C22⋊C4C2×D412yes—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—42no—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288#107—44no—
289#107—44no—
290#107—44no—
291#106—44no—
292C4⋊C8—28no—
293C4⋊C8—28no—
294#102—44no—
295#102—44no—
296#102—44no—
297#102—44no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—24no—
311#95—22no—
312#95C2×D412yes—
313#95C2×D412yes—
314#95C2×D412yes—
315#95C2×D412yes—
316C24—22no—
317C24C22×D411yescharacteristic
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—84no—
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—82no—
329C2×D4—82no—
330C2×D4—82no—
331C2×D4—82no—
332C2×D4C22×Q814yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—28no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D412yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C22×D412yes—
361C22×C4C22×Q812yes—
362C22×C4C2×#4112yes—
363C22×C4C2511yescharacteristic
364M16—84no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—84no—
372C4⋊C4—28no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C4⋊C4C22⋊Q818yes—
393C4⋊C4C22×Q814yes—
394C22⋊C4—28no—
395C22⋊C4—28no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C22⋊C4—84no—
406C22⋊C4C22⋊Q818yes—
407C42—24no—
408C42—24no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—82no—
417C42—82no—
418C42—82no—
419C42—84no—
420C42C22×Q812yes—
421C42C22×Q812yes—
422C42C2×#4112yes—
423C42C2×#4112yes—
424C23—22no—
425C23—21no—
426C23—21no—
427C23—22no—
428C23—24no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23C2×C22⋊Q812yes—
436C23C23×D411yescharacteristic
437Q8—88no—
438Q8—164no—
439D4—28no—
440D4—44no—
441D4—88no—
442D4—88no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—84no—
480C2×C4—88no—
481C2×C4—164no—
482C2×C4—164no—
483C2×C4C2×C22⋊Q814yes—
484C2×C4C2×C22⋊Q814yes—
485C2×C4C2×C22⋊Q812yes—
486C2×C4#57614yes—
487C2×C4C23×Q811yescharacteristic
488C2×C4C22×#4111yescharacteristic
489C2×C4#58512yes—
490C2×C4#58512yes—
491C8—44no—
492C8—44no—
493C22—22no—
494C22—24no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22C22×C22⋊Q811yescharacteristic
506C22#344911yescharacteristic
507C22#344911yescharacteristic
508C22C2×#57612yes—
509C22#358612yes—
510C4—22no—
511C4—24no—
512C4—24no—
513C4—24no—
514C4—42no—
515C4—44no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—82no—
521C4—84no—
522C4—82no—
523C4—84no—
524C4#350412yes—
525C4#351012yes—
526C2—42no—
527C2—84no—
528C2#3533411yescharacteristic
529C2#6049412yes—
530C2#6097612yes—
531C2#6101111yescharacteristic
532C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
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
1. #35283C2semidirect
3. #33074C2semidirect
4. #29982C2semidirect
5. #29980C2semidirect
6. #29283C2semidirect
7. #29280C2semidirect
8. #29263C2semidirect
9. #29254C2semidirect
10. #29175C2semidirect
11. #29163C2semidirect
12. #29157C2semidirect
13. #21968C2semidirect
15. #20754C2semidirect
86. #540Q8semidirect

Maximal subgroups

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

subgroupindex
#352832
#342802
#330742
#299822
#299802
#292832
#292802
#292632
#292542
#291752
#291632
#291572
#219682
#218972
#207542