Finite Group Databasebeta

← #7194271

Subgroups — #7194271

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C211yesmaximal, characteristic, complemented
2#59823C211yesmaximal, characteristic
3#32561C214yesmaximal, complemented
4#31474C214yesmaximal, complemented
5#29277C211yesmaximal, characteristic, complemented
6#28160C211yesmaximal, characteristic
7#27824C212yesmaximal, complemented
8#26918C214yesmaximal, complemented
9#26043C212yesmaximal, complemented
10#25322C212yesmaximal
11#24349C214yesmaximal, complemented
12#23816C211yesmaximal, characteristic, complemented
13#23003C212yesmaximal
14#22311C211yesmaximal, characteristic, complemented
15#22219C211yesmaximal, characteristic
16C2×#554C2212yes—
17C2×#486C2211yescharacteristic
18C2×#479C2214yes—
19C2×#460C2212yes—
20C2×#440C2214yes—
21#2806—22no—
22#2732—22no—
23#2584—22no—
24#2582C2214yes—
25#2567C2212yes—
26#2552—21no—
27#2552C2212yes—
28#2552C2212yes—
29#2542—21no—
30#2538—22no—
31#2538C2212yes—
32#2507—22no—
33#2454C2214yes—
34#2417—22no—
35#2369—21no—
36#2366C2212yes—
37#2365C2212yes—
38Q8×C4⋊C4C2212yes—
39D4×C4⋊C4—21no—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2100C2212yes—
46#2100C2212yes—
47#2099C2214yes—
48#2088C2214yes—
49#2072C2212yes—
50#2071C2212yes—
51#2071C2214yes—
52#2059C2214yes—
53#2057C2212yes—
54#2055C2214yes—
55#2043C2212yes—
56#2026C2214yes—
57#2024C2212yes—
58#2013C2212yes—
59#1992C2214yes—
60#1990C2211yescharacteristic
61#1987C2212yes—
62#1985C2214yes—
63#1957C2211yescharacteristic
64#1947C2212yes—
65#1939C2212yes—
66#1936C2214yes—
67#1935C2214yes—
68#1926C2214yes—
69#1921C2214yes—
70#1917C2214yes—
71#1880C2214yes—
72#1877C2214yes—
73#1877C2214yes—
74#1865C2214yes—
75#1864C2214yes—
76#1862C2214yes—
77#1844C2211yescharacteristic
78#1831C2211yescharacteristic
79#1803C2212yes—
80#1802C2214yes—
81#1770C2212yes—
82C4×#103C2212yes—
83C4×#103C2212yes—
84#1744C2211yescharacteristic
85C22×SD16C2314yes—
86#554—48no—
87C2×C4⋊Q8D411yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#125—22no—
93C2×#125C2314yes—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2314yes—
101C2×C4×Q8C2312yes—
102C2×C4×D4—22no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4C2314yes—
106C2×C4×D4C2312yes—
107C22×C4⋊C4—22no—
108#486—24no—
109#479—48no—
110#460—28no—
111#440—48no—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2314yes—
116C2×#103C2312yes—
117C2×#103C2312yes—
118C2×#103C2312yes—
119C2×#103C2312yes—
120C2×#103C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#97C2314yes—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#399—24no—
128#398—24no—
129#398—24no—
130#398—24no—
131#398—24no—
132#396—24no—
133#396—24no—
134#395—24no—
135#391—24no—
136#391—21no—
137#391—21no—
138#391D412yes—
139#391C2312yes—
140#391C2314yes—
141#390—22no—
142#390—21no—
143#390—21no—
144#390C2312yes—
145#390C2314yes—
146#389—22no—
147#389—22no—
148#389—24no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—24no—
157#389D412yes—
158#389D412yes—
159#389C2312yes—
160#389C2312yes—
161#389C2314yes—
162#389C2314yes—
163#387—24no—
164#386—24no—
165#385—24no—
166#384—22no—
167#384—22no—
168#384D411yescharacteristic
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#384C2314yes—
174#380—22no—
175#380—24no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2312yes—
186C43C2311yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2312yes—
190#340C2314yes—
191#340C2312yes—
192#340C2312yes—
193#340C2314yes—
194#340C2314yes—
195#340C2312yes—
196#336C2312yes—
197#336C2314yes—
198#336C2312yes—
199#336C2312yes—
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—28no—
213C2×SD16—44no—
214C22×C8C2414yes—
215C22×C8C2412yes—
216C22×C8C2412yes—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219C4⋊Q8—24no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222#127—22no—
223#127#4114yes—
224#125—28no—
225#125—44no—
226#125—84no—
227C22⋊Q8—48no—
228C22⋊Q8—416no—
229C22⋊Q8—84no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—44no—
233C4×Q8—44no—
234C4×Q8—44no—
235C4×Q8—82no—
236C4×Q8—82no—
237C4×D4—28no—
238C4×D4—44no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4—84no—
242C4×D4—82no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2414yes—
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296#107—28no—
297#106—44no—
298C4⋊C8—24no—
299C4⋊C8—44no—
300#103—24no—
301#103—24no—
302#103—24no—
303#103—24no—
304#103—44no—
305#102—28no—
306#102—28no—
307#102—44no—
308#97—44no—
309C4×C8—22no—
310C4×C8—22no—
311#95—22no—
312#95—24no—
313#95—22no—
314#95—24no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95C2×D414yes—
322#95C2×D414yes—
323C24—22no—
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—81no—
335C2×Q8—81no—
336C2×Q8—81no—
337C2×Q8—81no—
338C2×Q82−1+414yes—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—81no—
347C2×D4—81no—
348C2×D4C2×#4114yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—48no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D414yes—
364C22×C4C22×D414yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2511yescharacteristic
371SD16—416no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—24no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—81no—
408C4⋊C4—84no—
409C4⋊C4—81no—
410C4⋊C4—84no—
411C4⋊C4—81no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C4⋊C4—81no—
415C4⋊C4—84no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4H(5,𝔽2)14yes—
419C22⋊C4—48no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—82no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C42—22no—
426C42—24no—
427C42—24no—
428C42—22no—
429C42—22no—
430C42—41no—
431C42—41no—
432C42—44no—
433C42—44no—
434C42—44no—
435C42—84no—
436C42#13714yes—
437C42C2×#4112yes—
438C422−1+412yes—
439C23—24no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23C23×D411yescharacteristic
447Q8—24no—
448Q8—24no—
449Q8—42no—
450Q8—42no—
451Q8—84no—
452Q8—84no—
453Q8—84no—
454D4—28no—
455D4—48no—
456D4—44no—
457D4—84no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
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—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—88no—
498C2×C4—168no—
499C2×C4#54812yes—
500C2×C4#54812yes—
501C2×C4C2×#13712yes—
502C2×C4C2×#13712yes—
503C2×C4C2×#13712yes—
504C2×C4#57712yes—
505C2×C4C22×#4111yescharacteristic
506C2×C4C2×H(5,𝔽2)12yes—
507C2×C4C2×2−1+412yes—
508C2×C4C2×2−1+411yescharacteristic
509C8—42no—
510C8—44no—
511C8—42no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22C2×#54811yescharacteristic
520C22#348411yescharacteristic
521C22#348611yescharacteristic
522C22C22×#13711yescharacteristic
523C22C2×#57711yescharacteristic
524C22#358511yescharacteristic
525C22#358511yescharacteristic
526C4—22no—
527C4—22no—
528C4—22no—
529C4—44no—
530C4—44no—
531C4—41no—
532C4—44no—
533C4—41no—
534C4—44no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4#337214yes—
541C4#352012yes—
542C4#355212yes—
543C2—84no—
544C2#4533711yescharacteristic
545C2C2×#337211yescharacteristic
546C2#6133211yescharacteristic
547C2#6158811yescharacteristic
548C2#6168511yescharacteristic
549C2#6183011yescharacteristic
550C2#6185111yescharacteristic
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. #32561C2semidirect
4. #31474C2semidirect
5. #29277C2semidirect
7. #27824C2semidirect
8. #26918C2semidirect
9. #26043C2semidirect
11. #24349C2semidirect
12. #23816C2semidirect
14. #22311C2semidirect

Maximal subgroups

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

subgroupindex
#598712
#598232
#325612
#314742
#292772
#281602
#278242
#269182
#260432
#253222
#243492
#238162
#230032
#223112
#222192