Finite Group Databasebeta

← #7194229

Subgroups — #7194229

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59832C211yesmaximal, characteristic, complemented
2#59823C211yesmaximal, characteristic
3#32546C214yesmaximal, complemented
4#31456C214yesmaximal, complemented
5#29276C211yesmaximal, characteristic, complemented
6#28157C211yesmaximal, characteristic
7#27813C212yesmaximal, complemented
8#26897C214yesmaximal
9#26044C212yesmaximal, complemented
10#25320C212yesmaximal, complemented
11#24329C214yesmaximal, complemented
12#23815C211yesmaximal, characteristic
13#22996C212yesmaximal, complemented
14#22307C211yesmaximal, characteristic, complemented
15#22218C211yesmaximal, characteristic, complemented
16C2×Q82C2212yes—
17C2×#554—21no—
18C2×#487C2211yescharacteristic
19C2×#476C2214yes—
20C2×#461C2212yes—
21C2×C4×SD16C2214yes—
22#2734—22no—
23#2732—22no—
24#2582C2212yes—
25#2552C2212yes—
26#2551—21no—
27#2551C2212yes—
28#2546—22no—
29#2542C2212yes—
30#2538—21no—
31#2538C2214yes—
32#2454—22no—
33#2451—22no—
34#2451C2214yes—
35#2417—22no—
36#2366C2212yes—
37#2365—21no—
38#2365C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2100C2212yes—
46#2099C2212yes—
47#2099C2214yes—
48#2088C2214yes—
49#2072C2214yes—
50#2072C2212yes—
51#2071C2212yes—
52#2058C2212yes—
53#2058C2214yes—
54#2055C2214yes—
55#2041C2212yes—
56#2023C2212yes—
57#2023C2214yes—
58#2014C2212yes—
59#1990C2214yes—
60#1990C2211yescharacteristic
61#1988C2212yes—
62#1984C2214yes—
63#1956C2211yescharacteristic
64#1948C2212yes—
65#1939C2212yes—
66#1934C2214yes—
67#1933C2214yes—
68#1927C2214yes—
69#1920C2214yes—
70#1916C2214yes—
71#1880C2214yes—
72#1880C2214yes—
73#1877C2214yes—
74#1865C2214yes—
75#1865C2214yes—
76#1864C2214yes—
77#1844C2211yescharacteristic
78#1831C2211yescharacteristic
79#1803C2214yes—
80#1802C2212yes—
81C8×C4⋊C4C2212yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84#1744C2211yescharacteristic
85C22×SD16C2314yes—
86Q82—48no—
87#554—48no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8D411yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2314yes—
96C2×#127C2311yescharacteristic
97C2×#125—22no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2314yes—
112C22×C4⋊C4—22no—
113#487—24no—
114#476—48no—
115#461—28no—
116C4×SD16—48no—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#102C2312yes—
125C2×#102C2312yes—
126C2×#102C2312yes—
127C2×#102C2312yes—
128C2×#102C2314yes—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2314yes—
132#399—24no—
133#398—24no—
134#398—24no—
135#398—24no—
136#396—24no—
137#396—24no—
138#395—24no—
139#395—24no—
140#391—22no—
141#391—21no—
142#391—21no—
143#391—24no—
144#391D412yes—
145#391C2312yes—
146#391C2314yes—
147#390—22no—
148#390—21no—
149#390—21no—
150#390C2312yes—
151#390C2314yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389D412yes—
161#389D412yes—
162#389C2312yes—
163#389C2312yes—
164#389C2314yes—
165#389C2314yes—
166#387—24no—
167#387—24no—
168#386—24no—
169#385—24no—
170#384D411yescharacteristic
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2314yes—
176#380—22no—
177#380—24no—
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⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2312yes—
191#340C2314yes—
192#340C2312yes—
193#340C2312yes—
194#340C2314yes—
195#340C2312yes—
196#340C2314yes—
197#336C2314yes—
198#336C2312yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202#336C2312yes—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8—24no—
206C22×Q8C2×D412yes—
207C22×Q8C2412yes—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C2412yes—
211C23×C4—22no—
212C23×C4—24no—
213C2×SD16—28no—
214C2×SD16—28no—
215C2×SD16—44no—
216C22×C8C2414yes—
217C22×C8C2412yes—
218C22×C8C2412yes—
219C4⋊Q8—24no—
220C4⋊Q8—28no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—44no—
225C4⋊Q8—48no—
226C4⋊Q8—416no—
227C4⋊Q8—84no—
228C4⋊Q8SD1618yescomplemented
229C4⋊Q8#4114yes—
230#127—24no—
231#125—48no—
232C22⋊Q8—416no—
233C22⋊Q8—48no—
234C22⋊Q8—84no—
235C4×Q8—28no—
236C4×Q8—44no—
237C4×Q8—44no—
238C4×Q8—44no—
239C4×Q8—48no—
240C4×Q8—44no—
241C4×Q8—82no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—44no—
245C4×D4—44no—
246C4×D4—48no—
247C4×D4—44no—
248C4×D4—82no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2414yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301#106—28no—
302#106—44no—
303C4⋊C8—24no—
304C4⋊C8—44no—
305#103—28no—
306#103—28no—
307#103—44no—
308#102—24no—
309#102—24no—
310#102—24no—
311#102—24no—
312#102—44no—
313C4×C8—22no—
314C4×C8—22no—
315C4×C8—44no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95C2×D414yes—
327#95C2×D414yes—
328C24—22no—
329C2×Q8—24no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—48no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—81no—
340C2×Q8—81no—
341C2×Q8—84no—
342C2×Q8—84no—
343C2×Q8C2×SD1618yes—
344C2×Q8C2×#4114yes—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—81no—
355C2×D4—81no—
356C2×D42−1+414yes—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—48no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D414yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D414yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2511yescharacteristic
379SD16—416no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—48no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—81no—
420C4⋊C4—82no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4C2×SD1618yes—
424C4⋊C4C2×SD1618yes—
425C4⋊C4H(5,𝔽2)14yes—
426C22⋊C4—44no—
427C22⋊C4—48no—
428C22⋊C4—44no—
429C22⋊C4—84no—
430C22⋊C4—82no—
431C22⋊C4—84no—
432C42—22no—
433C42—24no—
434C42—24no—
435C42—22no—
436C42—22no—
437C42—41no—
438C42—41no—
439C42—48no—
440C42—44no—
441C42—44no—
442C42—44no—
443C42C2×SD1614yes—
444C42C2×#4112yes—
445C422−1+412yes—
446C23—24no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23C23×D411yescharacteristic
454Q8—28no—
455Q8—48no—
456Q8—44no—
457Q8—84no—
458Q8—84no—
459Q8—168no—
460D4—24no—
461D4—24no—
462D4—48no—
463D4—42no—
464D4—42no—
465D4—84no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—24no—
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—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—88no—
505C2×C4—168no—
506C2×C4#54812yes—
507C2×C4#54812yes—
508C2×C4C22×SD1612yes—
509C2×C4C22×SD1612yes—
510C2×C4C22×SD1618yes—
511C2×C4C22×SD1612yes—
512C2×C4#57612yes—
513C2×C4C22×#4111yescharacteristic
514C2×C4C2×H(5,𝔽2)12yes—
515C2×C4C2×2−1+412yes—
516C2×C4C2×2−1+411yescharacteristic
517C8—42no—
518C8—44no—
519C8—42no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22C2×#54811yescharacteristic
528C22#348411yescharacteristic
529C22#348611yescharacteristic
530C22C23×SD1611yescharacteristic
531C22C2×#57611yescharacteristic
532C22#358611yescharacteristic
533C22#358611yescharacteristic
534C4—28no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—44no—
539C4—41no—
540C4—41no—
541C4—44no—
542C4—44no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4#336214yes—
549C4#352012yes—
550C4#355212yes—
551C2—84no—
552C2#4533711yescharacteristic
553C2C2×#336211yescharacteristic
554C2#6133511yescharacteristic
555C2#6156211yescharacteristic
556C2#6168411yescharacteristic
557C2#6181811yescharacteristic
558C2#6184911yescharacteristic
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. #59832C2semidirect
3. #32546C2semidirect
4. #31456C2semidirect
5. #29276C2semidirect
7. #27813C2semidirect
9. #26044C2semidirect
10. #25320C2semidirect
11. #24329C2semidirect
13. #22996C2semidirect
14. #22307C2semidirect
15. #22218C2semidirect
228. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#598322
#598232
#325462
#314562
#292762
#281572
#278132
#268972
#260442
#253202
#243292
#238152
#229962
#223072
#222182