Finite Group Databasebeta

← #7604555

Subgroups — #7604555

Order 512.

1464 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#59871C211yesmaximal, characteristic
2#59832C211yesmaximal, characteristic
3#32566C214yesmaximal
4#29287C211yesmaximal, characteristic
5#28791C212yesmaximal
6#28283C211yesmaximal, characteristic
7#27815C212yesmaximal
8#26510C212yesmaximal
9#26462C212yesmaximal
10#26042C212yesmaximal
11#25472C214yesmaximal
12#25453C212yesmaximal
13#23214C212yesmaximal
14#22964C212yesmaximal
15C4×#498C211yesmaximal, characteristic
16C8×C4⋊Q8C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×Q82—21no—
19C2×#498C2211yescharacteristic
20C2×#487C2211yescharacteristic
21C2×#475C2212yes—
22C2×#475C2214yes—
23C2×C8×Q8C2212yes—
24#2806—22no—
25#2734—22no—
26#2584—22no—
27#2582—21no—
28#2582C2214yes—
29#2567C2212yes—
30#2552—21no—
31#2552C2212yes—
32#2551C2212yes—
33#2551C2212yes—
34#2546C2214yes—
35#2507—22no—
36#2451—22no—
37#2451—22no—
38#2369—21no—
39#2365C2212yes—
40#2365C2212yes—
41#2365C2212yes—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48#2100C2214yes—
49#2088C2214yes—
50#2088C2214yes—
51#2084C2214yes—
52#2066C2214yes—
53#2066C2214yes—
54#2058C2214yes—
55#2024C2214yes—
56#1990C2214yes—
57#1990C2211yescharacteristic
58#1990C2211yescharacteristic
59#1990C2211yescharacteristic
60#1988C2212yes—
61#1987C2212yes—
62#1984C2212yes—
63#1981C2212yes—
64#1956C2211yescharacteristic
65#1948C2212yes—
66#1948C2212yes—
67#1948C2214yes—
68#1948C2212yes—
69#1941C2212yes—
70#1941C2214yes—
71#1941C2212yes—
72#1939C2212yes—
73#1939C2212yes—
74#1933C2212yes—
75#1933C2212yes—
76#1933C2214yes—
77#1927C2214yes—
78#1920C2214yes—
79#1865C2214yes—
80#1865C2214yes—
81#1844C2211yescharacteristic
82#1842C2211yescharacteristic
83#1834C2211yescharacteristic
84#1803C2212yes—
85C4×#106C2212yes—
86C4×#106C2212yes—
87C8×C4⋊C4C2212yes—
88C8×C4⋊C4C2212yes—
89C4×#103C2212yes—
90C4×#103C2214yes—
91C4×#103C2212yes—
92C42×C8C2211yescharacteristic
93Q82—48no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8D411yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2314yes—
103C2×#125—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2312yes—
112#498—24no—
113#487Q818yescomplemented
114#475—28no—
115#475—48no—
116C8×Q8—44no—
117C2×#106C2312yes—
118C2×#106C2314yes—
119C2×#106C2312yes—
120C2×#106C2312yes—
121C2×#106C2314yes—
122C2×#106C2312yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×#103C2312yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2314yes—
139C2×C4×C8C2311yescharacteristic
140#398—24no—
141#398—24no—
142#398—24no—
143#398—24no—
144#398—24no—
145#395—24no—
146#395—24no—
147#395—24no—
148#391—24no—
149#391—22no—
150#391—22no—
151#391—24no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391D412yes—
156#391C2312yes—
157#391C2314yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—24no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—24no—
171#389D412yes—
172#389D412yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#389C2314yes—
178#389C2314yes—
179#387—24no—
180#387—24no—
181#387—24no—
182#384—22no—
183#384D411yescharacteristic
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2314yes—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C43C2311yescharacteristic
203#340C2312yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2312yes—
208#340C2314yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#336C2314yes—
216#336C2314yes—
217C22×Q8—21no—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—24no—
221C22×Q8C2×D412yes—
222C22×Q8C2412yes—
223C22×C8C2411yescharacteristic
224C22×C8C2414yes—
225C22×C8C2411yescharacteristic
226C22×C8C2411yescharacteristic
227C22×C8C2411yescharacteristic
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—44no—
234C4⋊Q8—44no—
235C4⋊Q8—416no—
236C4⋊Q8—48no—
237C4⋊Q8—48no—
238C4⋊Q8SD1618yes—
239C4⋊Q8C2×Q814yes—
240C4⋊Q8C2×Q814yes—
241C4⋊Q8#4114yes—
242#125—84no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8—44no—
248C4×Q8—44no—
249C4×Q8—48no—
250C4×Q8—48no—
251C4×Q8—44no—
252C4×Q8—44no—
253C4×Q8—82no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2414yes—
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305#106—24no—
306#106—28no—
307#106—24no—
308#106—24no—
309#106—42no—
310#106—44no—
311C4⋊C8—28no—
312C4⋊C8—24no—
313C4⋊C8—24no—
314C4⋊C8—42no—
315#103—24no—
316#103—28no—
317#103—42no—
318#103—44no—
319#103C2×Q818yes—
320#103C2×Q818yes—
321C4×C8—22no—
322C4×C8—22no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—41no—
326C4×C8—44no—
327C4×C8C2×Q814yes—
328#95—24no—
329#95—22no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95C2×D414yes—
343#95C2×D414yes—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—48no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—81no—
360C2×Q8—81no—
361C2×Q8—84no—
362C2×Q8—84no—
363C2×Q8C2×SD1618yes—
364C2×Q8C22×Q814yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D414yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D414yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2511yescharacteristic
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—24no—
393C2×C8—21no—
394C2×C8C22×Q812yes—
395C2×C8C22×Q812yes—
396C2×C8H(5,𝔽2)12yes—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—81no—
433C4⋊C4—82no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4C2×SD1618yes—
438C4⋊C4C2×SD1618yes—
439C4⋊C4C22×Q814yes—
440C4⋊C4C2×#4114yes—
441C42—22no—
442C42—22no—
443C42—21no—
444C42—24no—
445C42—21no—
446C42—24no—
447C42—41no—
448C42—44no—
449C42—48no—
450C42—44no—
451C42—44no—
452C42—44no—
453C42—84no—
454C42C2×SD1614yes—
455C42C22×Q812yes—
456C42C2×#4112yes—
457C422−1+412yes—
458C422−1+412yes—
459C23C23×D411yescharacteristic
460Q8—24no—
461Q8—24no—
462Q8—48no—
463Q8—48no—
464Q8—48no—
465Q8—42no—
466Q8—42no—
467Q8—84no—
468Q8—168no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—24no—
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—42no—
507C2×C4#54812yes—
508C2×C4#54812yes—
509C2×C4D4×Q812yes—
510C2×C4D4×Q812yes—
511C2×C4C22×SD1618yes—
512C2×C4C22×SD1612yes—
513C2×C4C22×SD1612yes—
514C2×C4C22×SD1612yes—
515C2×C4#57712yes—
516C2×C4#57812yes—
517C2×C4C23×Q811yescharacteristic
518C2×C4C22×#4111yescharacteristic
519C2×C4C2×H(5,𝔽2)11yescharacteristic
520C2×C4C2×2−1+411yescharacteristic
521C2×C4C2×2−1+411yescharacteristic
522C2×C4#58511yescharacteristic
523C8—22no—
524C8—22no—
525C8—22no—
526C8—44no—
527C8—41no—
528C22C2×#54811yescharacteristic
529C22C2×D4×Q811yescharacteristic
530C22#349611yescharacteristic
531C22C23×SD1611yescharacteristic
532C22C2×#57711yescharacteristic
533C22C2×#57811yescharacteristic
534C22#358411yescharacteristic
535C4—22no—
536C4—28no—
537C4—22no—
538C4—22no—
539C4—44no—
540C4—44no—
541C4—41no—
542C4—44no—
543C4—44no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4#335914yes—
550C4Q8×SD1614yes—
551C4Q8×#4111yescharacteristic
552C4#350311yescharacteristic
553C4#352211yescharacteristic
554C4#355511yescharacteristic
555C2#5965511yescharacteristic
556C2C2×#335911yescharacteristic
557C2#6132711yescharacteristic
558C2C2×Q8×SD1611yescharacteristic
559C2#6137211yescharacteristic
560C2#6224311yescharacteristic
561C2#6234311yescharacteristic
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
113. #487Q8semidirect

Maximal subgroups

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

subgroupindex
#598712
#598322
#325662
#292872
#287912
#282832
#278152
#265102
#264622
#260422
#254722
#254532
#232142
#229642
C4×#4982
C8×C4⋊Q82
C4×#4872