Finite Group Databasebeta

← #7542204

Subgroups — #7542204

Order 512.

1303 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59870C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#32358C214yesmaximal
4#29208C211yesmaximal, characteristic
5#29206C211yesmaximal, characteristic
6#28836C212yesmaximal
7#26878C214yesmaximal
8#26506C212yesmaximal
9#26280C212yesmaximal
10#26089C212yesmaximal
11#25480C212yesmaximal
12#25063C212yesmaximal
13#23153C211yesmaximal, characteristic
14#23068C212yesmaximal
15#22728C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#475C211yesmaximal, characteristic
18C2×#477C2211yescharacteristic
19C2×#475C2211yescharacteristic
20C2×#446C2212yes—
21#2868C2211yescharacteristic
22#2859—21no—
23#2805—22no—
24#2582C2212yes—
25#2581—22no—
26#2571C2212yes—
27#2568—21no—
28#2567C2212yes—
29#2567C2212yes—
30#2552C2212yes—
31#2550—22no—
32#2511—22no—
33#2509C2214yes—
34#2507—22no—
35#2451—22no—
36#2369C2212yes—
37#2365C2212yes—
38#2355C2212yes—
39#2355C2212yes—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2212yes—
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C42×Q8C2211yescharacteristic
47#2099C2214yes—
48#2097C2214yes—
49#2086C2214yes—
50#2085C2214yes—
51#2084C2214yes—
52#2066C2214yes—
53#2058C2214yes—
54#2057C2214yes—
55#1990C2212yes—
56#1990C2212yes—
57#1988C2211yescharacteristic
58#1987C2211yescharacteristic
59#1984C2212yes—
60#1949C2214yes—
61#1948C2212yes—
62#1948C2212yes—
63#1948C2211yescharacteristic
64#1947C2212yes—
65#1947C2212yes—
66#1947C2214yes—
67#1947C2211yescharacteristic
68#1944C2214yes—
69#1939C2212yes—
70#1935C2214yes—
71#1928C2214yes—
72#1921C2212yes—
73#1920C2214yes—
74#1920C2214yes—
75#1919C2212yes—
76#1918C2214yes—
77#1917C2214yes—
78#1865C2212yes—
79#1863C2212yes—
80#1847C2214yes—
81#1845C2212yes—
82#1844C2212yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85#1776C2214yes—
86C4×#107C2212yes—
87C4×#106C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#103C2212yes—
92C4×#103C2212yes—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8D411yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#125—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2312yes—
108#477—24no—
109#475Q818yescomplemented
110#446—28no—
111C2×#107C2312yes—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#107C2312yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×#106C2312yes—
118C2×#106C2312yes—
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2314yes—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×#103C2312yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#97C2314yes—
129C2×C4×C8C2312yes—
130C2×C4×C8C2312yes—
131#400—24no—
132#400—24no—
133#400—24no—
134#398—24no—
135#398—24no—
136#395—24no—
137#395—24no—
138#391—22no—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—21no—
143#391—21no—
144#391—24no—
145#391D412yes—
146#391C2314yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—24no—
155#389—24no—
156#389—21no—
157#389—21no—
158#389D412yes—
159#389D412yes—
160#389C2312yes—
161#389C2312yes—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#387—24no—
166#387—24no—
167#387—24no—
168#387—24no—
169#384—22no—
170#384—21no—
171#384—21no—
172#384D411yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2314yes—
176#384C2312yes—
177#384C2312yes—
178#382—22no—
179#382—22no—
180#382C2314yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2311yescharacteristic
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C43—21no—
199C43C2311yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2312yes—
205#340C2312yes—
206#340C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215#336C2314yes—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8C2×D412yes—
220C22×Q8C2411yescharacteristic
221C22×C8C2412yes—
222C22×C8C2412yes—
223C22×C8C2412yes—
224C22×C8C2412yes—
225C4⋊Q8—24no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—42no—
230C4⋊Q8—42no—
231C4⋊Q8—82no—
232C4⋊Q8—82no—
233C4⋊Q8C2×Q814yes—
234#125—84no—
235C4×Q8—24no—
236C4×Q8—22no—
237C4×Q8—24no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—41no—
241C4×Q8—41no—
242C4×Q8—44no—
243C4×Q8—82no—
244C4×Q8—84no—
245C4×Q8—84no—
246C4×Q8C2×Q814yes—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
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—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2411yescharacteristic
275C2×C4⋊C4C2411yescharacteristic
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2411yescharacteristic
281C2×C4⋊C4C2412yes—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2411yescharacteristic
297C2×C42C2414yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300#107—24no—
301#107—44no—
302#107—44no—
303#107—42no—
304#106—44no—
305#106—42no—
306#106—44no—
307#106C2×Q818yes—
308C4⋊C8—28no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8C2×Q814yes—
312C4⋊C8#4114yes—
313#103—24no—
314#103—42no—
315#103—42no—
316#103C2×Q818yes—
317#97—28no—
318C4×C8—24no—
319C4×C8—24no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—24no—
324#95—22no—
325#95—24no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95C2×D414yes—
337#95C2×D414yes—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—44no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—81no—
352C2×Q8—81no—
353C2×Q8—82no—
354C2×Q8—82no—
355C2×Q8C22×Q812yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4C22×D414yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D414yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2511yescharacteristic
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8C22×Q814yes—
385C2×C8C2×#4114yes—
386C4⋊C4—24no—
387C4⋊C4—28no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—81no—
418C4⋊C4—84no—
419C4⋊C4—81no—
420C4⋊C4—82no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4C22×Q814yes—
428C4⋊C4C22×Q812yes—
429C42—24no—
430C42—22no—
431C42—22no—
432C42—24no—
433C42—24no—
434C42—21no—
435C42—21no—
436C42—41no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—44no—
441C42—44no—
442C42—42no—
443C42—42no—
444C42—84no—
445C42—84no—
446C42#13714yes—
447C42C22×Q812yes—
448C42C2×#4112yes—
449C422−1+412yes—
450C422−1+412yes—
451C23C23×D411yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—48no—
455Q8—48no—
456Q8—41no—
457Q8—41no—
458Q8—84no—
459Q8—164no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4#54712yes—
503C2×C4#54712yes—
504C2×C4D4×Q812yes—
505C2×C4D4×Q812yes—
506C2×C4C2×#13712yes—
507C2×C4C2×#13712yes—
508C2×C4C2×#13712yes—
509C2×C4#57612yes—
510C2×C4#57712yes—
511C2×C4C23×Q811yescharacteristic
512C2×C4C22×#4111yescharacteristic
513C2×C4C2×2−1+411yescharacteristic
514C2×C4C2×2−1+411yescharacteristic
515C2×C4#58511yescharacteristic
516C2×C4#58511yescharacteristic
517C8—24no—
518C8—24no—
519C8—42no—
520C8—42no—
521C22C2×#54711yescharacteristic
522C22C2×D4×Q811yescharacteristic
523C22#348711yescharacteristic
524C22C22×#13711yescharacteristic
525C22C2×#57611yescharacteristic
526C22C2×#57711yescharacteristic
527C22#358611yescharacteristic
528C4—22no—
529C4—22no—
530C4—22no—
531C4—44no—
532C4—44no—
533C4—44no—
534C4—42no—
535C4—42no—
536C4—41no—
537C4—81no—
538C4—82no—
539C4—82no—
540C4—81no—
541C4—82no—
542C4—84no—
543C4#335614yes—
544C4#338514yes—
545C4Q8×#4111yescharacteristic
546C4#350211yescharacteristic
547C4#351511yescharacteristic
548C4#355511yescharacteristic
549C2#5906711yescharacteristic
550C2C2×#335611yescharacteristic
551C2#6127811yescharacteristic
552C2C2×#338511yescharacteristic
553C2#6138311yescharacteristic
554C2#6189311yescharacteristic
555C2#6193711yescharacteristic
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
109. #475Q8semidirect

Maximal subgroups

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

subgroupindex
#598702
#365632
#323582
#292082
#292062
#288362
#268782
#265062
#262802
#260892
#254802
#250632
#231532
#230682
#227282
C4×#4772
C4×#4752