Finite Group Databasebeta

← #7157053

Subgroups — #7157053

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C211yesmaximal, characteristic, complemented
2#59821C211yesmaximal, characteristic
3#32576C214yesmaximal, complemented
4#30818C214yesmaximal, complemented
5#29254C211yesmaximal, characteristic
6#28206C211yesmaximal, characteristic, complemented
7#27770C212yesmaximal
8#26691C214yesmaximal, complemented
9#26043C212yesmaximal, complemented
10#25654C214yesmaximal, complemented
11#25277C212yesmaximal
12#23816C211yesmaximal, characteristic, complemented
13#22965C212yesmaximal, complemented
14#22307C211yesmaximal, characteristic, complemented
15#22219C211yesmaximal, characteristic
16C2×#556C2212yes—
17C2×#488C2214yes—
18C2×#486C2211yescharacteristic
19C2×#479C2214yes—
20C2×#477C2212yes—
21#2806—22no—
22#2727—22no—
23#2584C2214yes—
24#2582—22no—
25#2567—21no—
26#2552C2212yes—
27#2552C2212yes—
28#2548C2212yes—
29#2542—21no—
30#2538C2212yes—
31#2512—22no—
32#2507—22no—
33#2448C2214yes—
34#2415—22no—
35#2369C2212yes—
36#2365C2212yes—
37#2354—21no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2100C2212yes—
46#2093C2212yes—
47#2088C2214yes—
48#2083C2212yes—
49#2072C2212yes—
50#2067C2214yes—
51#2059C2214yes—
52#2043C2212yes—
53#2043C2212yes—
54#2042C2214yes—
55#2042C2214yes—
56#2025C2214yes—
57#2024C2212yes—
58#2023C2212yes—
59#1990C2211yescharacteristic
60#1984C2214yes—
61#1983C2214yes—
62#1982C2212yes—
63#1957C2211yescharacteristic
64#1947C2212yes—
65#1942C2214yes—
66#1939C2212yes—
67#1934C2214yes—
68#1928C2214yes—
69#1926C2214yes—
70#1925C2214yes—
71#1921C2214yes—
72#1916C2214yes—
73#1865C2214yes—
74#1864C2214yes—
75#1863C2214yes—
76#1862C2214yes—
77#1844C2211yescharacteristic
78#1831C2211yescharacteristic
79#1803C2212yes—
80C8×C4⋊C4C2212yes—
81C4×#103C2212yes—
82C4×#103C2212yes—
83C4×#103C2214yes—
84#1744C2211yescharacteristic
85#556—48no—
86C2×C4⋊Q8D411yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127C2311yescharacteristic
91C2×#126—24no—
92C2×#125C2314yes—
93C2×#124C2314yes—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2314yes—
101C2×C4×Q8C2312yes—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4C2312yes—
105C2×#117—22no—
106#488—48no—
107#486—24no—
108#479—48no—
109#477—28no—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2314yes—
115C2×#103C2314yes—
116C2×#103C2312yes—
117C2×#103C2314yes—
118C2×#103C2312yes—
119C2×#103C2312yes—
120C2×#103C2312yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2314yes—
128#399—24no—
129#398—24no—
130#398—24no—
131#398—24no—
132#397—24no—
133#396—24no—
134#395—24no—
135#394—24no—
136#391—22no—
137#391—21no—
138#391—21no—
139#391—24no—
140#391D412yes—
141#391C2312yes—
142#390—22no—
143#390—22no—
144#390—21no—
145#390—21no—
146#390C2312yes—
147#390C2314yes—
148#389—22no—
149#389—22no—
150#389—24no—
151#389—22no—
152#389—24no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—24no—
158#389D412yes—
159#389D412yes—
160#389C2312yes—
161#389C2312yes—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#388—24no—
166#388—24no—
167#387—24no—
168#384—22no—
169#384—22no—
170#384—22no—
171#384D411yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#380—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C43C2311yescharacteristic
189#340C2314yes—
190#340C2312yes—
191#340C2312yes—
192#340C2312yes—
193#340C2314yes—
194#340C2312yes—
195#340C2314yes—
196#336C2314yes—
197#336C2312yes—
198#336C2312yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8C2×D412yes—
205C22×Q8C2412yes—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C2412yes—
209C23×C4—22no—
210C22×C8C2412yes—
211C22×C8C2412yes—
212C22×C8C2414yes—
213C4⋊Q8—24no—
214C4⋊Q8—24no—
215C4⋊Q8—24no—
216C4⋊Q8—22no—
217C4⋊Q8#4114yes—
218#127—22no—
219#127—22no—
220#126—416no—
221#125—44no—
222#125—44no—
223#124—28no—
224#124—44no—
225C22⋊Q8—48no—
226C22⋊Q8—84no—
227C4×Q8—28no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—44no—
231C4×Q8—44no—
232C4×Q8—44no—
233C4×Q8—82no—
234C4×Q8—84no—
235C4×D4—44no—
236C4×D4—44no—
237C4×D4—82no—
238#117—48no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2414yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292#107—28no—
293#107—44no—
294#106—44no—
295C4⋊C8—24no—
296C4⋊C8—44no—
297#103—24no—
298#103—28no—
299#103—24no—
300#103—28no—
301#103—24no—
302#103—24no—
303#102—28no—
304#102—28no—
305#102—44no—
306#102—44no—
307C4×C8—22no—
308C4×C8—22no—
309C4×C8—44no—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—22no—
319#95—24no—
320#95C2×D414yes—
321#95C2×D414yes—
322C24—22no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—81no—
336C2×Q8—81no—
337C2×Q8H(5,𝔽2)14yes—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—22no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—81no—
347C2×D4—81no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D414yes—
360C22×C4C22×D414yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2511yescharacteristic
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—22no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—48no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—81no—
402C4⋊C4—81no—
403C4⋊C4—84no—
404C4⋊C4—81no—
405C4⋊C4—81no—
406C4⋊C4—82no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4C2×#4114yes—
412C4⋊C42−1+414yes—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—84no—
418C22⋊C4—82no—
419C22⋊C4—84no—
420C42—22no—
421C42—24no—
422C42—24no—
423C42—22no—
424C42—22no—
425C42—41no—
426C42—41no—
427C42—48no—
428C42—44no—
429C42—44no—
430C42—44no—
431C42—84no—
432C42#13514yes—
433C42C2×#4112yes—
434C422−1+412yes—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23C23×D411yescharacteristic
443Q8—24no—
444Q8—24no—
445Q8—48no—
446Q8—42no—
447Q8—42no—
448Q8—84no—
449Q8—84no—
450D4—44no—
451D4—44no—
452D4—84no—
453D4—84no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—88no—
494C2×C4#54812yes—
495C2×C4#54812yes—
496C2×C4C2×#13512yes—
497C2×C4C2×#13512yes—
498C2×C4C2×#13512yes—
499C2×C4#57812yes—
500C2×C4C22×#4111yescharacteristic
501C2×C4C2×H(5,𝔽2)12yes—
502C2×C4C2×2−1+412yes—
503C2×C4C2×2−1+411yescharacteristic
504C8—42no—
505C8—44no—
506C8—42no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22C2×#54811yescharacteristic
515C22#348411yescharacteristic
516C22#348611yescharacteristic
517C22C22×#13511yescharacteristic
518C22C2×#57811yescharacteristic
519C22#358511yescharacteristic
520C22#358511yescharacteristic
521C4—22no—
522C4—22no—
523C4—22no—
524C4—44no—
525C4—44no—
526C4—44no—
527C4—44no—
528C4—41no—
529C4—41no—
530C4—44no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4#336414yes—
536C4#352212yes—
537C4#354412yes—
538C2—162no—
539C2#4493711yescharacteristic
540C2C2×#336411yescharacteristic
541C2#6133211yescharacteristic
542C2#6156611yescharacteristic
543C2#6168711yescharacteristic
544C2#6181911yescharacteristic
545C2#6185411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59871C2semidirect
3. #32576C2semidirect
4. #30818C2semidirect
6. #28206C2semidirect
8. #26691C2semidirect
9. #26043C2semidirect
10. #25654C2semidirect
12. #23816C2semidirect
13. #22965C2semidirect
14. #22307C2semidirect

Maximal subgroups

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

subgroupindex
#598712
#598212
#325762
#308182
#292542
#282062
#277702
#266912
#260432
#256542
#252772
#238162
#229652
#223072
#222192