Finite Group Databasebeta

← #7186906

Subgroups — #7186906

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59870C211yesmaximal, characteristic, complemented
2#59823C211yesmaximal, characteristic
3#32251C214yesmaximal, complemented
4#31534C214yesmaximal, complemented
5#29277C211yesmaximal, characteristic, complemented
6#28157C211yesmaximal, characteristic
7#27442C214yesmaximal, complemented
8#27081C214yesmaximal
9#25934C212yesmaximal, complemented
10#25301C212yesmaximal, complemented
11#25056C212yesmaximal, complemented
12#23815C211yesmaximal, characteristic
13#22522C212yesmaximal, complemented
14#22242C211yesmaximal, characteristic, complemented
15#22221C211yesmaximal, characteristic, complemented
16C2×#554C2212yes—
17C2×#487C2211yescharacteristic
18C2×#461C2212yes—
19C2×#446C2212yes—
20#2805—22no—
21#2732—22no—
22#2581—22no—
23#2571—21no—
24#2568C2212yes—
25#2567C2212yes—
26#2552C2212yes—
27#2542—21no—
28#2538—22no—
29#2538C2212yes—
30#2511C2214yes—
31#2509—22no—
32#2454C2214yes—
33#2417—22no—
34#2366C2212yes—
35#2365—21no—
36#2355C2212yes—
37Q8×C4⋊C4C2212yes—
38D4×C4⋊C4—21no—
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44#2100C2212yes—
45#2099C2212yes—
46#2093C2214yes—
47#2083C2214yes—
48#2072C2212yes—
49#2071C2212yes—
50#2068C2214yes—
51#2057C2212yes—
52#2055C2214yes—
53#2041C2212yes—
54#2026C2214yes—
55#2026C2214yes—
56#2023C2212yes—
57#2014C2212yes—
58#1992C2214yes—
59#1990C2211yescharacteristic
60#1989C2214yes—
61#1987C2212yes—
62#1984C2212yes—
63#1973C2214yes—
64#1956C2211yescharacteristic
65#1947C2212yes—
66#1935C2214yes—
67#1928C2214yes—
68#1925C2214yes—
69#1921C2214yes—
70#1918C2214yes—
71#1917C2214yes—
72#1917C2214yes—
73#1880C2214yes—
74#1877C2214yes—
75#1877C2214yes—
76#1863C2214yes—
77#1845C2211yescharacteristic
78#1831C2211yescharacteristic
79#1802C2214yes—
80#1802C2212yes—
81#1762C2214yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84#1744C2211yescharacteristic
85C22×SD16C2314yes—
86#554—28no—
87#554—44no—
88C2×C4⋊Q8D411yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#125C2314yes—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97C2×C4×Q8—22no—
98C2×C4×Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×D4—22no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4C2312yes—
106C2×C4×D4C2314yes—
107C22×C4⋊C4—22no—
108#487—24no—
109#461—44no—
110#446—44no—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#102C2312yes—
119C2×#102C2312yes—
120C2×#102C2314yes—
121C2×#102C2312yes—
122C2×#102C2312yes—
123C2×#97C2314yes—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#400—24no—
127#400—24no—
128#400—24no—
129#399—24no—
130#398—24no—
131#396—24no—
132#396—24no—
133#395—24no—
134#391—22no—
135#391—21no—
136#391—21no—
137#391D412yes—
138#391C2312yes—
139#391C2314yes—
140#391C2314yes—
141#390—22no—
142#390—21no—
143#390—21no—
144#390C2312yes—
145#390C2314yes—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—24no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389D412yes—
156#389D412yes—
157#389C2312yes—
158#389C2312yes—
159#389C2314yes—
160#387—24no—
161#387—24no—
162#387—24no—
163#386—24no—
164#385—24no—
165#384—22no—
166#384D411yescharacteristic
167#384C2311yescharacteristic
168#384C2311yescharacteristic
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2314yes—
172#382—22no—
173#382—22no—
174#380—22no—
175#380—24no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2314yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C43C2311yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2312yes—
192#340C2312yes—
193#340C2312yes—
194#340C2312yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200#336C2314yes—
201#336C2312yes—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8C2×D412yes—
205C22×Q8C2412yes—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C2412yes—
209C23×C4—22no—
210C23×C4—24no—
211C2×SD16—28no—
212C2×SD16—44no—
213C2×SD16—44no—
214C22×C8C2414yes—
215C22×C8C2412yes—
216C22×C8C2412yes—
217C4⋊Q8—24no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—42no—
222C4⋊Q8#4114yes—
223#127—24no—
224#125—28no—
225#125—44no—
226C22⋊Q8—48no—
227C22⋊Q8—416no—
228C22⋊Q8—84no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—44no—
232C4×Q8—82no—
233C4×Q8—82no—
234C4×Q8—84no—
235C4×D4—28no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—44no—
239C4×D4—44no—
240C4×D4—82no—
241C4×D4—84no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2414yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293#107—44no—
294#107—44no—
295C4⋊C8—24no—
296C4⋊C8—44no—
297#103—28no—
298#103—44no—
299#103—44no—
300#102—24no—
301#102—24no—
302#102—42no—
303#102—42no—
304#102—44no—
305#97—44no—
306C4×C8—22no—
307C4×C8—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95C2×D414yes—
321#95C2×D414yes—
322C24—22no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—42no—
332C2×Q8—81no—
333C2×Q8—81no—
334C2×Q8—81no—
335C2×Q8—81no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—81no—
346C2×D4—81no—
347C2×D42−1+414yes—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—48no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D414yes—
363C22×C4C22×D414yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2511yescharacteristic
370SD16—88no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—24no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—28no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—81no—
407C4⋊C4—84no—
408C4⋊C4—81no—
409C4⋊C4—81no—
410C4⋊C4—82no—
411C4⋊C4—81no—
412C4⋊C4—82no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4C2×#4114yes—
416C22⋊C4—48no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—84no—
420C22⋊C4—82no—
421C22⋊C4—84no—
422C42—22no—
423C42—24no—
424C42—24no—
425C42—22no—
426C42—22no—
427C42—41no—
428C42—41no—
429C42—44no—
430C42—44no—
431C42—44no—
432C42—84no—
433C42#13714yes—
434C42C2×#4112yes—
435C42H(5,𝔽2)12yes—
436C23—24no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23C23×D411yescharacteristic
444Q8—48no—
445Q8—44no—
446Q8—44no—
447Q8—84no—
448Q8—84no—
449D4—24no—
450D4—24no—
451D4—48no—
452D4—42no—
453D4—42no—
454D4—84no—
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—21no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—42no—
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—42no—
494C2×C4—88no—
495C2×C4—168no—
496C2×C4#54612yes—
497C2×C4#54612yes—
498C2×C4C2×#13712yes—
499C2×C4C2×#13712yes—
500C2×C4C2×#13712yes—
501C2×C4#57612yes—
502C2×C4C22×#4111yescharacteristic
503C2×C4C2×H(5,𝔽2)11yescharacteristic
504C2×C4C2×2−1+412yes—
505C2×C4#58512yes—
506C8—42no—
507C8—42no—
508C8—44no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22C2×#54611yescharacteristic
517C22#348611yescharacteristic
518C22#348711yescharacteristic
519C22C22×#13711yescharacteristic
520C22C2×#57611yescharacteristic
521C22#358511yescharacteristic
522C22#358611yescharacteristic
523C4—22no—
524C4—22no—
525C4—22no—
526C4—44no—
527C4—44no—
528C4—41no—
529C4—41no—
530C4—44no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4#333314yes—
538C4#351612yes—
539C4#356112yes—
540C2—84no—
541C2#4526811yescharacteristic
542C2C2×#333311yescharacteristic
543C2#6121911yescharacteristic
544C2#6182611yescharacteristic
545C2#6184911yescharacteristic
546C2#6189311yescharacteristic
547C2#6191011yescharacteristic
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. #59870C2semidirect
3. #32251C2semidirect
4. #31534C2semidirect
5. #29277C2semidirect
7. #27442C2semidirect
9. #25934C2semidirect
10. #25301C2semidirect
11. #25056C2semidirect
13. #22522C2semidirect
14. #22242C2semidirect
15. #22221C2semidirect

Maximal subgroups

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

subgroupindex
#598702
#598232
#322512
#315342
#292772
#281572
#274422
#270812
#259342
#253012
#250562
#238152
#225222
#222422
#222212