Finite Group Databasebeta

← #7597120

Subgroups — #7597120

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59832C211yesmaximal, characteristic
2#59828C211yesmaximal, characteristic
3#31868C214yesmaximal
4#29287C211yesmaximal, characteristic
5#28947C214yesmaximal
6#28791C212yesmaximal
7#28283C211yesmaximal, characteristic
8#26528C212yesmaximal
9#26444C212yesmaximal
10#25837C212yesmaximal
11#25472C212yesmaximal
12#25060C212yesmaximal
13#23214C212yesmaximal
14Q8×#103C212yesmaximal
15C4×#498C211yesmaximal, characteristic
16C8×C4⋊Q8C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#557—21no—
20C2×#498C2211yescharacteristic
21C2×#487C2211yescharacteristic
22C2×#475C2212yes—
23C2×C8×Q8C2212yes—
24#2734—22no—
25#2726—22no—
26#2582—21no—
27#2582C2212yes—
28#2552C2212yes—
29#2551—21no—
30#2551C2212yes—
31#2546—22no—
32#2546C2212yes—
33#2506C2214yes—
34#2453—22no—
35#2451—22no—
36#2451C2214yes—
37#2446—22no—
38#2365—21no—
39#2365C2212yes—
40#2365C2212yes—
41Q8×C4⋊C4C2212yes—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47#2096C2214yes—
48#2088C2214yes—
49#2088C2214yes—
50#2084C2214yes—
51#2066C2214yes—
52#2066C2214yes—
53#2058C2214yes—
54#2058C2214yes—
55#1990C2214yes—
56#1990C2211yescharacteristic
57#1990C2211yescharacteristic
58#1990C2211yescharacteristic
59#1988C2212yes—
60#1987C2214yes—
61#1987C2212yes—
62#1984C2212yes—
63#1981C2212yes—
64#1956C2211yescharacteristic
65#1948C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1941C2212yes—
69#1941C2214yes—
70#1941C2212yes—
71#1939C2212yes—
72#1939C2212yes—
73#1933C2212yes—
74#1933C2214yes—
75#1933C2212yes—
76#1927C2214yes—
77#1920C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1844C2211yescharacteristic
81#1842C2211yescharacteristic
82#1834C2211yescharacteristic
83#1803C2212yes—
84#1803C2214yes—
85C4×#106C2212yes—
86C4×#106C2212yes—
87C8×C4⋊C4C2212yes—
88C8×C4⋊C4C2212yes—
89C4×#103C2214yes—
90C4×#103C2212yes—
91C4×#103C2212yes—
92C42×C8C2211yescharacteristic
93Q82—28no—
94Q82—44no—
95#557—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8D411yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2314yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#125—22no—
107C2×#125—24no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2312yes—
117#498—24no—
118#487Q818yescomplemented
119#475—44no—
120C8×Q8—44no—
121C2×#106C2312yes—
122C2×#106C2312yes—
123C2×#106C2312yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2314yes—
143C2×C4×C8C2311yescharacteristic
144#400—24no—
145#398—24no—
146#398—24no—
147#398—24no—
148#398—24no—
149#398—24no—
150#395—24no—
151#395—24no—
152#391—22no—
153#391—22no—
154#391—22no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391—24no—
159#391D412yes—
160#391C2312yes—
161#391C2314yes—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—24no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389D412yes—
174#389D412yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2314yes—
179#389C2314yes—
180#389C2314yes—
181#387—24no—
182#387—24no—
183#387—24no—
184#387—24no—
185#384D411yescharacteristic
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2314yes—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C43C2311yescharacteristic
204#340C2312yes—
205#340C2312yes—
206#340C2314yes—
207#340C2314yes—
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2314yes—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—21no—
221C22×Q8—24no—
222C22×Q8C2×D412yes—
223C22×Q8C2412yes—
224C22×C8C2414yes—
225C22×C8C2411yescharacteristic
226C22×C8C2411yescharacteristic
227C22×C8C2411yescharacteristic
228C22×C8C2411yescharacteristic
229C4⋊Q8—28no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—44no—
236C4⋊Q8—48no—
237C4⋊Q8—416no—
238C4⋊Q8—48no—
239C4⋊Q8—84no—
240C4⋊Q8SD1618yes—
241C4⋊Q8C2×Q814yes—
242C4⋊Q8C2×Q814yes—
243C4⋊Q8#4114yes—
244#125—416no—
245#125—48no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—28no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—44no—
252C4×Q8—48no—
253C4×Q8—48no—
254C4×Q8—48no—
255C4×Q8—44no—
256C4×Q8—44no—
257C4×Q8—82no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2414yes—
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309#106—24no—
310#106—24no—
311#106—24no—
312#106—42no—
313#106—44no—
314#106—44no—
315C4⋊C8—24no—
316C4⋊C8—24no—
317C4⋊C8—42no—
318C4⋊C8—44no—
319#103—24no—
320#103—44no—
321#103—42no—
322#103—44no—
323#103C2×Q818yes—
324#103C2×Q818yes—
325C4×C8—22no—
326C4×C8—22no—
327C4×C8—22no—
328C4×C8—22no—
329C4×C8—41no—
330C4×C8—44no—
331C4×C8C2×Q814yes—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95C2×D414yes—
347#95C2×D414yes—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—48no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×Q8—84no—
366C2×Q8—84no—
367C2×Q8C2×SD1618yes—
368C2×Q8C22×Q814yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D414yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D414yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C2511yescharacteristic
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—24no—
394C2×C8—21no—
395C2×C8—24no—
396C2×C8—21no—
397C2×C8—24no—
398C2×C8C22×Q812yes—
399C2×C8C22×Q812yes—
400C2×C82−1+412yes—
401C4⋊C4—24no—
402C4⋊C4—28no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—84no—
438C4⋊C4—81no—
439C4⋊C4—82no—
440C4⋊C4—84no—
441C4⋊C4C2×SD1618yes—
442C4⋊C4C2×SD1618yes—
443C4⋊C4C22×Q814yes—
444C4⋊C4C2×#4114yes—
445C42—22no—
446C42—22no—
447C42—24no—
448C42—24no—
449C42—21no—
450C42—21no—
451C42—41no—
452C42—48no—
453C42—44no—
454C42—48no—
455C42—44no—
456C42—44no—
457C42—44no—
458C42C2×SD1614yes—
459C42C22×Q812yes—
460C42C2×#4112yes—
461C42H(5,𝔽2)12yes—
462C422−1+412yes—
463C23C23×D411yescharacteristic
464Q8—216no—
465Q8—24no—
466Q8—24no—
467Q8—48no—
468Q8—42no—
469Q8—42no—
470Q8—84no—
471Q8—84no—
472Q8—168no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—21no—
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—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4#54612yes—
512C2×C4#54612yes—
513C2×C4D4×Q812yes—
514C2×C4D4×Q812yes—
515C2×C4C22×SD1618yes—
516C2×C4C22×SD1612yes—
517C2×C4C22×SD1612yes—
518C2×C4C22×SD1612yes—
519C2×C4#57712yes—
520C2×C4#57812yes—
521C2×C4C23×Q811yescharacteristic
522C2×C4C22×#4111yescharacteristic
523C2×C4C2×H(5,𝔽2)11yescharacteristic
524C2×C4C2×2−1+411yescharacteristic
525C2×C4C2×2−1+411yescharacteristic
526C2×C4#58511yescharacteristic
527C8—22no—
528C8—22no—
529C8—22no—
530C8—41no—
531C8—44no—
532C22C2×#54611yescharacteristic
533C22C2×D4×Q811yescharacteristic
534C22#349711yescharacteristic
535C22C23×SD1611yescharacteristic
536C22C2×#57711yescharacteristic
537C22C2×#57811yescharacteristic
538C22#358411yescharacteristic
539C4—22no—
540C4—22no—
541C4—28no—
542C4—22no—
543C4—44no—
544C4—44no—
545C4—44no—
546C4—44no—
547C4—41no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4#329814yes—
554C4Q8×SD1614yes—
555C4Q8×#4111yescharacteristic
556C4#350311yescharacteristic
557C4#352311yescharacteristic
558C4#355411yescharacteristic
559C2#5941311yescharacteristic
560C2C2×#329811yescharacteristic
561C2#6121211yescharacteristic
562C2C2×Q8×SD1611yescharacteristic
563C2#6137211yescharacteristic
564C2#6235711yescharacteristic
565C2#6238211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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
118. #487Q8semidirect

Maximal subgroups

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

subgroupindex
#598322
#598282
#318682
#292872
#289472
#287912
#282832
#265282
#264442
#258372
#254722
#250602
#232142
Q8×#1032
C4×#4982
C8×C4⋊Q82
C4×#4872