Finite Group Databasebeta

← #7604575

Subgroups — #7604575

Order 512.

1346 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59924C211yesmaximal, characteristic
2#59870C211yesmaximal, characteristic
3#32575C214yesmaximal
4#29287C211yesmaximal, characteristic
5#28799C212yesmaximal
6#28284C211yesmaximal, characteristic
7#27823C212yesmaximal
8#26511C212yesmaximal
9#26462C212yesmaximal
10#26042C212yesmaximal
11#25487C214yesmaximal
12#25454C212yesmaximal
13#23221C212yesmaximal
14#22971C212yesmaximal
15C4×#498C211yesmaximal, characteristic
16#21712C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×#498C2211yescharacteristic
19C2×#487C2211yescharacteristic
20C2×#477C2212yes—
21C2×#475C2214yes—
22C2×#446C2212yes—
23#2811—22no—
24#2805—22no—
25#2581C2214yes—
26#2571C2212yes—
27#2568C2212yes—
28#2567—22no—
29#2567—22no—
30#2567—21no—
31#2567C2214yes—
32#2551—21no—
33#2551C2212yes—
34#2551C2212yes—
35#2511—22no—
36#2509—22no—
37#2369—21no—
38#2369C2212yes—
39#2369C2212yes—
40#2365C2212yes—
41#2355—21no—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48#2101C2214yes—
49#2088C2214yes—
50#2088C2214yes—
51#2085C2214yes—
52#2068C2214yes—
53#2066C2214yes—
54#2058C2214yes—
55#2024C2214yes—
56#1993C2214yes—
57#1990C2211yescharacteristic
58#1990C2211yescharacteristic
59#1990C2211yescharacteristic
60#1988C2212yes—
61#1987C2212yes—
62#1984C2212yes—
63#1981C2212yes—
64#1956C2211yescharacteristic
65#1949C2214yes—
66#1948C2212yes—
67#1948C2212yes—
68#1947C2212yes—
69#1943C2214yes—
70#1941C2212yes—
71#1941C2212yes—
72#1939C2212yes—
73#1939C2212yes—
74#1935C2214yes—
75#1933C2212yes—
76#1933C2212yes—
77#1927C2214yes—
78#1918C2214yes—
79#1865C2214yes—
80#1865C2214yes—
81#1844C2211yescharacteristic
82#1842C2211yescharacteristic
83#1834C2211yescharacteristic
84#1803C2212yes—
85C4×#106C2212yes—
86C4×#106C2212yes—
87#1770C2212yes—
88#1770C2212yes—
89#1763C2214yes—
90C4×#103C2212yes—
91C4×#103C2212yes—
92C42×C8C2211yescharacteristic
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8D411yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2314yes—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2314yes—
106C2×C4×Q8C2312yes—
107#498—24no—
108#487Q818yescomplemented
109#477—28no—
110#475—48no—
111#446—44no—
112C2×#107C2314yes—
113C2×#106C2312yes—
114C2×#106C2312yes—
115C2×#106C2312yes—
116C2×#106C2314yes—
117C2×#106C2312yes—
118C2×C4⋊C8C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×#103C2312yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#97C2314yes—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135#400—24no—
136#400—24no—
137#400—24no—
138#395—24no—
139#395—24no—
140#395—24no—
141#395—24no—
142#395—24no—
143#391—24no—
144#391—22no—
145#391—22no—
146#391—21no—
147#391—21no—
148#391—21no—
149#391—24no—
150#391—24no—
151#391D412yes—
152#391C2312yes—
153#391C2314yes—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—24no—
166#389D412yes—
167#389D412yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2314yes—
172#389C2314yes—
173#389C2314yes—
174#387—24no—
175#387—24no—
176#387—24no—
177#384—24no—
178#384—22no—
179#384D411yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#382—22no—
185#382C2314yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C43C2311yescharacteristic
202#340C2312yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2312yes—
207#340C2314yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#336C2314yes—
215#336C2314yes—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8—21no—
219C22×Q8—24no—
220C22×Q8C2×D412yes—
221C22×Q8C2412yes—
222C22×C8C2411yescharacteristic
223C22×C8C2414yes—
224C22×C8C2411yescharacteristic
225C22×C8C2411yescharacteristic
226C22×C8C2411yescharacteristic
227C4⋊Q8—24no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—44no—
234C4⋊Q8—44no—
235C4⋊Q8—84no—
236C4⋊Q8—84no—
237C4⋊Q8C2×Q814yes—
238C4⋊Q8C2×Q814yes—
239C4⋊Q8#4114yes—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—42no—
243C4×Q8—42no—
244C4×Q8—44no—
245C4×Q8—44no—
246C4×Q8—44no—
247C4×Q8—82no—
248C4×Q8—82no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2414yes—
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298#107—28no—
299#106—24no—
300#106—24no—
301#106—24no—
302#106—42no—
303#106—44no—
304C4⋊C8—28no—
305C4⋊C8—24no—
306C4⋊C8—24no—
307C4⋊C8—42no—
308#103—24no—
309#103—28no—
310#103—42no—
311#103—44no—
312#103C2×Q818yes—
313#103C2×Q818yes—
314#97—44no—
315C4×C8—22no—
316C4×C8—22no—
317C4×C8—22no—
318C4×C8—22no—
319C4×C8—41no—
320C4×C8C2×Q814yes—
321#95—24no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95C2×D414yes—
338#95C2×D414yes—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—81no—
353C2×Q8—81no—
354C2×Q8—81no—
355C2×Q8—81no—
356C2×Q8—84no—
357C2×Q8—84no—
358C2×Q8—84no—
359C2×Q8C22×Q814yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4C22×D414yes—
369C22×C4C22×D414yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C2511yescharacteristic
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—24no—
388C2×C8—21no—
389C2×C8C22×Q812yes—
390C2×C8C22×Q812yes—
391C2×C8H(5,𝔽2)12yes—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
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—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4C22×Q814yes—
433C4⋊C4C2×#4114yes—
434C42—22no—
435C42—22no—
436C42—21no—
437C42—24no—
438C42—21no—
439C42—24no—
440C42—41no—
441C42—44no—
442C42—44no—
443C42—44no—
444C42—44no—
445C42—84no—
446C42—84no—
447C42#13714yes—
448C42C22×Q812yes—
449C42C2×#4112yes—
450C422−1+412yes—
451C422−1+412yes—
452C23C23×D411yescharacteristic
453Q8—24no—
454Q8—24no—
455Q8—42no—
456Q8—42no—
457Q8—84no—
458Q8—84no—
459Q8—84no—
460Q8—84no—
461Q8—168no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
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#54812yes—
502C2×C4#54812yes—
503C2×C4D4×Q812yes—
504C2×C4D4×Q812yes—
505C2×C4C2×#13712yes—
506C2×C4C2×#13712yes—
507C2×C4C2×#13712yes—
508C2×C4#57712yes—
509C2×C4#57712yes—
510C2×C4C23×Q811yescharacteristic
511C2×C4C22×#4111yescharacteristic
512C2×C4C2×H(5,𝔽2)11yescharacteristic
513C2×C4C2×2−1+411yescharacteristic
514C2×C4C2×2−1+411yescharacteristic
515C2×C4#58511yescharacteristic
516C8—22no—
517C8—22no—
518C8—22no—
519C8—44no—
520C8—41no—
521C22C2×#54811yescharacteristic
522C22C2×D4×Q811yescharacteristic
523C22#349611yescharacteristic
524C22C22×#13711yescharacteristic
525C22C2×#57711yescharacteristic
526C22C2×#57711yescharacteristic
527C22#358411yescharacteristic
528C4—22no—
529C4—22no—
530C4—22no—
531C4—44no—
532C4—44no—
533C4—41no—
534C4—44no—
535C4—44no—
536C4—44no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4#336914yes—
543C4#338714yes—
544C4Q8×#4111yescharacteristic
545C4#350311yescharacteristic
546C4#352211yescharacteristic
547C4#355511yescharacteristic
548C2#5965511yescharacteristic
549C2C2×#336911yescharacteristic
550C2#6132611yescharacteristic
551C2C2×#338711yescharacteristic
552C2#6137111yescharacteristic
553C2#6226011yescharacteristic
554C2#6232611yescharacteristic
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
108. #487Q8semidirect

Maximal subgroups

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

subgroupindex
#599242
#598702
#325752
#292872
#287992
#282842
#278232
#265112
#264622
#260422
#254872
#254542
#232212
#229712
C4×#4982
#217122
C4×#4872