Finite Group Databasebeta

← #7195426

Subgroups — #7195426

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic
2#59823C211yesmaximal, characteristic
3#32000C214yesmaximal, complemented
4#31324C214yesmaximal, complemented
5#29277C211yesmaximal, characteristic, complemented
6#29259C211yesmaximal, characteristic, complemented
7#28598C212yesmaximal, complemented
8#26302C212yesmaximal, complemented
9#26236C212yesmaximal, complemented
10#26132C214yesmaximal, complemented
11#24098C214yesmaximal, complemented
12#23835C211yesmaximal, characteristic, complemented
13D4×#107C212yesmaximal, complemented
14#22260C211yesmaximal, characteristic
15#22221C211yesmaximal, characteristic, complemented
16C2×#554C2212yes—
17C2×#487C2211yescharacteristic
18C2×#484C2214yes—
19C2×C8×D4C2212yes—
20C22×#107C2214yes—
21#2804—22no—
22#2732—22no—
23#2580—22no—
24#2580—22no—
25#2571—21no—
26#2563C2212yes—
27#2563C2212yes—
28#2552—21no—
29#2542C2212yes—
30#2538—22no—
31#2538C2212yes—
32#2487C2214yes—
33#2454—22no—
34#2417C2214yes—
35#2366—21no—
36#2366—21no—
37D4×C4⋊C4C2212yes—
38D4×C4⋊C4C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2100C2212yes—
46#2095C2212yes—
47#2083C2212yes—
48#2083C2212yes—
49#2076C2214yes—
50#2076C2214yes—
51#2057C2212yes—
52#2057C2212yes—
53#2057C2212yes—
54#2036C2214yes—
55#2036C2214yes—
56#2036C2214yes—
57#2036C2214yes—
58#2024C2212yes—
59#1990C2211yescharacteristic
60#1987C2212yes—
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1954C2212yes—
64#1947C2212yes—
65#1947C2212yes—
66#1928C2214yes—
67#1921C2214yes—
68#1892C2214yes—
69#1892C2214yes—
70#1872C2214yes—
71#1869C2214yes—
72#1869C2214yes—
73#1863C2214yes—
74#1863C2214yes—
75#1852C2214yes—
76#1852C2214yes—
77#1852C2214yes—
78#1845C2211yescharacteristic
79#1831C2211yescharacteristic
80#1824C2214yes—
81#1808C2214yes—
82C4×#107C2212yes—
83C4×#107C2212yes—
84#1744C2211yescharacteristic
85C23×C8C2314yes—
86#554—48no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127D411yescharacteristic
92C2×#125—22no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8C2314yes—
95C2×C22⋊Q8C2314yes—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8C2312yes—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109#487—24no—
110#484—48no—
111C8×D4—28no—
112C2×#107—48no—
113C2×#107C2312yes—
114C2×#107C2314yes—
115C2×#107C2312yes—
116C2×#107C2312yes—
117C2×#107C2312yes—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×C4⋊C8C2312yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130#400—24no—
131#399—24no—
132#399—24no—
133#398—24no—
134#396—24no—
135#396—24no—
136#393—24no—
137#393—24no—
138#391—21no—
139#391—21no—
140#391—21no—
141#391C2312yes—
142#390—22no—
143#390—22no—
144#390—22no—
145#390—22no—
146#390D412yes—
147#390D412yes—
148#390D412yes—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#386—24no—
162#386—24no—
163#385—24no—
164#385—24no—
165#385—24no—
166#385—24no—
167#384—22no—
168#384D411yescharacteristic
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#382—22no—
174#382—22no—
175#380—24no—
176#380C2314yes—
177#380C2314yes—
178#380C2314yes—
179#380C2314yes—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C43C2311yescharacteristic
189#340C2312yes—
190#340C2312yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2312yes—
196#340C2314yes—
197#340C2312yes—
198#336C2312yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8C2412yes—
206C22×D4C2×D412yes—
207C22×D4C2×D412yes—
208C22×D4C2×D412yes—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4C2414yes—
212C22×C8—28no—
213C22×C8—44no—
214C22×C8—44no—
215C22×C8C2414yes—
216C22×C8C2412yes—
217C22×C8C2412yes—
218C4⋊Q8—24no—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—42no—
224#127Q1618yes—
225#125—48no—
226C22⋊Q8—28no—
227C22⋊Q8—44no—
228C22⋊Q8—416no—
229C22⋊Q8—44no—
230C22⋊Q8—44no—
231C4×Q8—44no—
232C4×Q8—82no—
233C4×Q8—82no—
234C4×Q8—82no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—48no—
240C4×D4—84no—
241C4×D4—84no—
242C4×D4—84no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4C2×D414yes—
285C2×C22⋊C4C2×D414yes—
286C2×C22⋊C4C2×D414yes—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298#107—24no—
299#107—28no—
300#107—24no—
301#107—44no—
302#107—44no—
303#107—42no—
304#107—42no—
305C4⋊C8—24no—
306#103—28no—
307#103—44no—
308#103—44no—
309#103—44no—
310C22⋊C8—28no—
311C22⋊C8—44no—
312C22⋊C8—44no—
313C4×C8—22no—
314C4×C8#4114yes—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327C24C22×D414yes—
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—81no—
336C2×Q8—81no—
337C2×Q8—81no—
338C2×Q8—81no—
339C2×Q8—81no—
340C2×Q8—81no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4C2×Q1618yes—
348C2×D4C2×Q1618yes—
349C2×D4C2×Q1618yes—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—48no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D414yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2511yescharacteristic
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—416no—
387C2×C8C2×#4114yes—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—81no—
415C4⋊C4—84no—
416C4⋊C4—81no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—81no—
420C4⋊C4—82no—
421C4⋊C4—81no—
422C4⋊C4—84no—
423C4⋊C4H(5,𝔽2)14yes—
424C22⋊C4—28no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—48no—
428C22⋊C4—84no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—41no—
437C42—41no—
438C42—44no—
439C42—44no—
440C42—44no—
441C42—44no—
442C42C2×Q1614yes—
443C42C2×#4112yes—
444C422−1+412yes—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23C22×Q1618yes—
452C23C23×D411yescharacteristic
453Q8—44no—
454Q8—44no—
455Q8—84no—
456Q8—84no—
457Q8—84no—
458D4—216no—
459D4—48no—
460D4—48no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—24no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
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—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—168no—
505C2×C4—168no—
506C2×C4#54712yes—
507C2×C4#54712yes—
508C2×C4C22×Q1612yes—
509C2×C4C22×Q1612yes—
510C2×C4C22×Q1612yes—
511C2×C4#57612yes—
512C2×C4C22×#4111yescharacteristic
513C2×C4C2×H(5,𝔽2)12yes—
514C2×C4C2×2−1+411yescharacteristic
515C2×C4#58512yes—
516C8—24no—
517C8—44no—
518C8—42no—
519C22—28no—
520C22—28no—
521C22—28no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22C2×#54711yescharacteristic
527C22#348411yescharacteristic
528C22#348511yescharacteristic
529C22C23×Q1611yescharacteristic
530C22C2×#57611yescharacteristic
531C22#358511yescharacteristic
532C22#358611yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—44no—
537C4—41no—
538C4—41no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4#333814yes—
548C4#350612yes—
549C4#355412yes—
550C2—48no—
551C2#4535411yescharacteristic
552C2C2×#333811yescharacteristic
553C2#6129111yescharacteristic
554C2#6156311yescharacteristic
555C2#6167911yescharacteristic
556C2#6169811yescharacteristic
557C2#6180711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32000C2semidirect
4. #31324C2semidirect
5. #29277C2semidirect
6. #29259C2semidirect
7. #28598C2semidirect
8. #26302C2semidirect
9. #26236C2semidirect
10. #26132C2semidirect
11. #24098C2semidirect
12. #23835C2semidirect
13. D4×#107C2semidirect
15. #22221C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#598232
#320002
#313242
#292772
#292592
#285982
#263022
#262362
#261322
#240982
#238352
D4×#1072
#222602
#222212