Finite Group Databasebeta

← #7187371

Subgroups — #7187371

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59828C211yesmaximal, characteristic, complemented
2#59823C211yesmaximal, characteristic
3#32546C214yesmaximal, complemented
4#31636C214yesmaximal, complemented
5#29255C211yesmaximal, characteristic, complemented
6#28160C211yesmaximal, characteristic
7#27749C212yesmaximal, complemented
8#26166C214yesmaximal, complemented
9#26108C212yesmaximal, complemented
10#25368C212yesmaximal
11#24187C214yesmaximal, complemented
12#23816C211yesmaximal, characteristic, complemented
13#22997C212yesmaximal
14#22307C211yesmaximal, characteristic, complemented
15#22268C211yesmaximal, characteristic
16C2×#557C2212yes—
17C2×#554—21no—
18C2×#492C2211yescharacteristic
19C2×#460C2212yes—
20C2×#461C2214yes—
21C2×C4×SD16C2214yes—
22#2732—22no—
23#2726—22no—
24#2582C2212yes—
25#2552—21no—
26#2552C2212yes—
27#2546C2212yes—
28#2542C2212yes—
29#2538—21no—
30#2538C2214yes—
31#2506—22no—
32#2454—22no—
33#2453—22no—
34#2446C2214yes—
35#2417—22no—
36#2366C2212yes—
37#2365—21no—
38Q8×C4⋊C4C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2100C2212yes—
46#2099C2214yes—
47#2096C2212yes—
48#2088C2214yes—
49#2072C2212yes—
50#2072C2214yes—
51#2071C2212yes—
52#2055C2214yes—
53#2043C2212yes—
54#2043C2214yes—
55#2024C2212yes—
56#2024C2212yes—
57#2013C2212yes—
58#2013C2214yes—
59#1984C2214yes—
60#1981C2212yes—
61#1957C2211yescharacteristic
62#1956C2211yescharacteristic
63#1956C2214yes—
64#1948C2212yes—
65#1939C2212yes—
66#1934C2214yes—
67#1933C2214yes—
68#1916C2214yes—
69#1880C2214yes—
70#1880C2214yes—
71#1877C2214yes—
72#1877C2214yes—
73#1877C2214yes—
74#1865C2214yes—
75#1865C2214yes—
76#1864C2214yes—
77#1834C2211yescharacteristic
78#1831C2211yescharacteristic
79#1803C2212yes—
80#1803C2214yes—
81C8×C4⋊C4C2212yes—
82C4×#103C2212yes—
83C4×#103C2212yes—
84#1744C2211yescharacteristic
85C22×SD16C2314yes—
86C22×SD16C2314yes—
87#557—48no—
88#554—48no—
89C2×C4⋊Q8D411yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2314yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#125—22no—
96C2×#125—22no—
97C2×#125—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—21no—
103C2×C4×Q8C2312yes—
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#492—24no—
114#461—48no—
115#460—28no—
116C4×SD16—48no—
117C2×#106C2314yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2314yes—
120C2×#103C2312yes—
121C2×#103C2312yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2312yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2314yes—
133#400—24no—
134#399—24no—
135#398—24no—
136#398—24no—
137#398—24no—
138#398—24no—
139#396—24no—
140#396—24no—
141#391—22no—
142#391—21no—
143#391D412yes—
144#391C2312yes—
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#389—21no—
161#389—24no—
162#389D412yes—
163#389D412yes—
164#389C2312yes—
165#389C2314yes—
166#389C2314yes—
167#387—24no—
168#387—24no—
169#386—24no—
170#385—24no—
171#384—22no—
172#384D411yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2314yes—
178#380—22no—
179#380—24no—
180C4×C4⋊C4—22no—
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#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2312yes—
195#340C2312yes—
196#340C2312yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202#336C2312yes—
203C22×Q8—21no—
204C22×Q8C2×D412yes—
205C22×Q8C2412yes—
206C22×Q8C2412yes—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C2412yes—
210C23×C4—22no—
211C23×C4—24no—
212C2×SD16—28no—
213C2×SD16—28no—
214C2×SD16—28no—
215C2×SD16—44no—
216C2×SD16—44no—
217C2×SD16—44no—
218C22×C8C2412yes—
219C22×C8C2414yes—
220C22×C8C2412yes—
221C4⋊Q8—24no—
222C4⋊Q8—28no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C4⋊Q8—42no—
226C4⋊Q8—44no—
227C4⋊Q8SD1618yescomplemented
228#127—22no—
229#127#4114yes—
230#125—416no—
231#125—48no—
232#125—48no—
233C22⋊Q8—416no—
234C22⋊Q8—48no—
235C22⋊Q8—84no—
236C4×Q8—28no—
237C4×Q8—24no—
238C4×Q8—24no—
239C4×Q8—44no—
240C4×Q8—44no—
241C4×Q8—44no—
242C4×Q8—48no—
243C4×Q8—44no—
244C4×D4—44no—
245C4×D4—44no—
246C4×D4—44no—
247C4×D4—48no—
248C4×D4—44no—
249C4×D4—82no—
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—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2414yes—
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303#106—44no—
304C4⋊C8—24no—
305C4⋊C8—44no—
306#103—24no—
307#103—24no—
308#103—44no—
309#103—44no—
310#103—42no—
311#103—44no—
312#103—42no—
313#103—44no—
314#102—28no—
315#102—44no—
316C4×C8—22no—
317C4×C8—22no—
318C4×C8—44no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95C2×D414yes—
330#95C2×D414yes—
331C24—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—24no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8C2×SD1618yes—
347C2×Q8H(5,𝔽2)14yes—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—81no—
356C2×D4—81no—
357C2×D4C2×#4114yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—48no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D414yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D414yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2511yescharacteristic
380SD16—416no—
381SD16—88no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—42no—
420C4⋊C4—81no—
421C4⋊C4—81no—
422C4⋊C4—84no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—84no—
426C4⋊C4C2×SD1618yes—
427C4⋊C4C2×SD1618yes—
428C22⋊C4—44no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—84no—
432C22⋊C4—82no—
433C22⋊C4—84no—
434C42—22no—
435C42—24no—
436C42—24no—
437C42—22no—
438C42—22no—
439C42—41no—
440C42—41no—
441C42—48no—
442C42—44no—
443C42—44no—
444C42—44no—
445C42C2×SD1614yes—
446C42C2×#4112yes—
447C422−1+412yes—
448C23—24no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23C23×D411yescharacteristic
456Q8—24no—
457Q8—24no—
458Q8—48no—
459Q8—42no—
460Q8—42no—
461Q8—44no—
462Q8—44no—
463Q8—84no—
464D4—28no—
465D4—48no—
466D4—44no—
467D4—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
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—42no—
505C2×C4—42no—
506C2×C4—88no—
507C2×C4—168no—
508C2×C4#54812yes—
509C2×C4#54812yes—
510C2×C4C22×SD1612yes—
511C2×C4C22×SD1612yes—
512C2×C4C22×SD1618yes—
513C2×C4C22×SD1612yes—
514C2×C4#57812yes—
515C2×C4C22×#4111yescharacteristic
516C2×C4C2×H(5,𝔽2)12yes—
517C2×C4C2×2−1+411yescharacteristic
518C2×C4#58512yes—
519C8—44no—
520C8—42no—
521C8—42no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22C2×#54811yescharacteristic
530C22#348411yescharacteristic
531C22#348711yescharacteristic
532C22C23×SD1611yescharacteristic
533C22C2×#57811yescharacteristic
534C22#358511yescharacteristic
535C22#358611yescharacteristic
536C4—28no—
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#335914yes—
551C4#351312yes—
552C4#355912yes—
553C2—84no—
554C2#4527111yescharacteristic
555C2C2×#335911yescharacteristic
556C2#6133311yescharacteristic
557C2#6156211yescharacteristic
558C2#6168711yescharacteristic
559C2#6188111yescharacteristic
560C2#6191511yescharacteristic
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. #59828C2semidirect
3. #32546C2semidirect
4. #31636C2semidirect
5. #29255C2semidirect
7. #27749C2semidirect
8. #26166C2semidirect
9. #26108C2semidirect
11. #24187C2semidirect
12. #23816C2semidirect
14. #22307C2semidirect
227. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#598282
#598232
#325462
#316362
#292552
#281602
#277492
#261662
#261082
#253682
#241872
#238162
#229972
#223072
#222682