Finite Group Databasebeta

← #7195422

Subgroups — #7195422

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic
2#59823C211yesmaximal, characteristic
3#31998C214yesmaximal, complemented
4#31320C214yesmaximal, complemented
5#29276C211yesmaximal, characteristic, complemented
6#29255C211yesmaximal, characteristic, complemented
7#28600C212yesmaximal, complemented
8#26298C212yesmaximal, complemented
9#26234C212yesmaximal, complemented
10#26128C214yesmaximal, complemented
11#24097C214yesmaximal, complemented
12#23837C211yesmaximal, characteristic, complemented
13D4×#106C212yesmaximal, complemented
14#22260C211yesmaximal, characteristic
15#22218C211yesmaximal, characteristic, complemented
16C2×#559—21no—
17C2×#554C2212yes—
18C2×#487C2211yescharacteristic
19C2×#483C2214yes—
20C2×C8×D4C2212yes—
21C22×#106C2214yes—
22#2732—22no—
23#2724—22no—
24#2582—21no—
25#2552—21no—
26#2547C2212yes—
27#2547C2212yes—
28#2542C2212yes—
29#2538—22no—
30#2538C2212yes—
31#2486C2214yes—
32#2454—22no—
33#2447—22no—
34#2447—22no—
35#2417C2214yes—
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#2096C2212yes—
47#2084C2212yes—
48#2084C2212yes—
49#2077C2214yes—
50#2077C2214yes—
51#2058C2212yes—
52#2058C2212yes—
53#2058C2212yes—
54#2035C2214yes—
55#2035C2214yes—
56#2035C2214yes—
57#2035C2214yes—
58#2024C2212yes—
59#1990C2211yescharacteristic
60#1988C2212yes—
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1954C2212yes—
64#1948C2212yes—
65#1948C2212yes—
66#1927C2214yes—
67#1920C2214yes—
68#1892C2214yes—
69#1892C2214yes—
70#1872C2214yes—
71#1869C2214yes—
72#1869C2214yes—
73#1865C2214yes—
74#1865C2214yes—
75#1853C2214yes—
76#1853C2214yes—
77#1853C2214yes—
78#1844C2211yescharacteristic
79#1831C2211yescharacteristic
80#1824C2214yes—
81#1807C2214yes—
82C4×#106C2212yes—
83C4×#106C2212yes—
84#1744C2211yescharacteristic
85C23×C8C2314yes—
86#559—48no—
87#554—48no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127D411yescharacteristic
93C2×#125—22no—
94C2×#124—22no—
95C2×#124—22no—
96C2×#123—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8C2314yes—
99C2×C22⋊Q8C2314yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8C2312yes—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113#487—24no—
114#483—48no—
115C8×D4—28no—
116C2×#106—48no—
117C2×#106C2312yes—
118C2×#106C2314yes—
119C2×#106C2312yes—
120C2×#106C2312yes—
121C2×#106C2312yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134#399—24no—
135#398—24no—
136#398—24no—
137#397—24no—
138#397—24no—
139#397—24no—
140#396—24no—
141#396—24no—
142#391—22no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—21no—
147#391C2312yes—
148#390—22no—
149#390—22no—
150#390D412yes—
151#390D412yes—
152#390D412yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#386—24no—
166#385—24no—
167#385—24no—
168#385—24no—
169#384—22no—
170#384D411yescharacteristic
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#381—24no—
176#380—24no—
177#380C2314yes—
178#380C2314yes—
179#380C2314yes—
180#380C2314yes—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C43C2311yescharacteristic
190#340C2312yes—
191#340C2312yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2312yes—
197#340C2314yes—
198#340C2312yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2312yes—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8—21no—
206C22×Q8C2412yes—
207C22×D4C2×D412yes—
208C22×D4C2×D412yes—
209C22×D4C2×D412yes—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4C2414yes—
213C22×C8—28no—
214C22×C8—44no—
215C22×C8—44no—
216C22×C8C2414yes—
217C22×C8C2412yes—
218C22×C8C2412yes—
219C4⋊Q8—24no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—42no—
225#127SD1618yes—
226#125—48no—
227#124—48no—
228#124—48no—
229#123—416no—
230C22⋊Q8—28no—
231C22⋊Q8—44no—
232C22⋊Q8—416no—
233C22⋊Q8—44no—
234C22⋊Q8—44no—
235C4×Q8—44no—
236C4×Q8—44no—
237C4×Q8—44no—
238C4×Q8—82no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—44no—
242C4×D4—44no—
243C4×D4—48no—
244C4×D4—48no—
245C4×D4—48no—
246C4×D4—84no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4C2×D414yes—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302#106—24no—
303#106—28no—
304#106—24no—
305#106—44no—
306#106—44no—
307#106—42no—
308#106—42no—
309C4⋊C8—24no—
310#103—28no—
311#103—44no—
312#103—44no—
313#103—44no—
314C22⋊C8—28no—
315C22⋊C8—44no—
316C22⋊C8—44no—
317C4×C8—22no—
318C4×C8#4114yes—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331C24C22×D414yes—
332C2×Q8—24no—
333C2×Q8—22no—
334C2×Q8—24no—
335C2×Q8—22no—
336C2×Q8—24no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4C2×SD1618yes—
352C2×D4C2×SD1618yes—
353C2×D4C2×SD1618yes—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D414yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2511yescharacteristic
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—24no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—416no—
391C2×C8C2×#4114yes—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—81no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—84no—
424C4⋊C4—82no—
425C4⋊C4—81no—
426C4⋊C4—84no—
427C4⋊C4H(5,𝔽2)14yes—
428C22⋊C4—28no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—41no—
441C42—41no—
442C42—44no—
443C42—44no—
444C42—44no—
445C42—44no—
446C42C2×SD1614yes—
447C42C2×#4112yes—
448C422−1+412yes—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23C22×SD1618yes—
456C23C23×D411yescharacteristic
457Q8—44no—
458Q8—44no—
459Q8—84no—
460Q8—84no—
461Q8—84no—
462D4—216no—
463D4—48no—
464D4—48no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
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—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—168no—
509C2×C4—168no—
510C2×C4#54712yes—
511C2×C4#54712yes—
512C2×C4C22×SD1612yes—
513C2×C4C22×SD1612yes—
514C2×C4C22×SD1612yes—
515C2×C4#57712yes—
516C2×C4C22×#4111yescharacteristic
517C2×C4C2×H(5,𝔽2)12yes—
518C2×C4C2×2−1+411yescharacteristic
519C2×C4#58512yes—
520C8—24no—
521C8—44no—
522C8—42no—
523C22—28no—
524C22—28no—
525C22—28no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22C2×#54711yescharacteristic
531C22#348411yescharacteristic
532C22#348511yescharacteristic
533C22C23×SD1611yescharacteristic
534C22C2×#57711yescharacteristic
535C22#358611yescharacteristic
536C22#358611yescharacteristic
537C4—22no—
538C4—22no—
539C4—22no—
540C4—44no—
541C4—41no—
542C4—41no—
543C4—44no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4#333514yes—
552C4#350612yes—
553C4#355412yes—
554C2—48no—
555C2#4535411yescharacteristic
556C2C2×#333511yescharacteristic
557C2#6129211yescharacteristic
558C2#6156211yescharacteristic
559C2#6168011yescharacteristic
560C2#6169511yescharacteristic
561C2#6180911yescharacteristic
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. #31998C2semidirect
4. #31320C2semidirect
5. #29276C2semidirect
6. #29255C2semidirect
7. #28600C2semidirect
8. #26298C2semidirect
9. #26234C2semidirect
10. #26128C2semidirect
11. #24097C2semidirect
12. #23837C2semidirect
13. D4×#106C2semidirect
15. #22218C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#598232
#319982
#313202
#292762
#292552
#286002
#262982
#262342
#261282
#240972
#238372
D4×#1062
#222602
#222182