Finite Group Databasebeta

← #7538160

Subgroups — #7538160

Order 512.

1353 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59870C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#32660C214yesmaximal
4#28835C212yesmaximal
5#28687C211yesmaximal, characteristic
6#28542C212yesmaximal
7#28511C211yesmaximal, characteristic
8#26528C212yesmaximal
9#26093C212yesmaximal
10#25454C212yesmaximal
11#25070C214yesmaximal
12#24817C212yesmaximal
13#23250C211yesmaximal, characteristic
14#23102C212yesmaximal
15#22971C212yesmaximal
16C4×#475C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×#498C2212yes—
19C2×#477C2212yes—
20C2×#477C2212yes—
21C2×#475C2211yescharacteristic
22C2×#462C2211yescharacteristic
23C2×C4×Q16C2212yes—
24#2868C2211yescharacteristic
25#2859—21no—
26#2805—22no—
27#2582C2212yes—
28#2581C2214yes—
29#2571C2212yes—
30#2568C2212yes—
31#2567—21no—
32#2567C2212yes—
33#2552C2212yes—
34#2550—22no—
35#2511—22no—
36#2509—22no—
37#2507—22no—
38#2451—22no—
39#2369C2212yes—
40#2365C2212yes—
41#2355—21no—
42#2355C2212yes—
43Q8×C4⋊C4C2212yes—
44Q8×C4⋊C4C2212yes—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C42×Q8C2211yescharacteristic
50#2097C2214yes—
51#2096C2214yes—
52#2095C2214yes—
53#2085C2214yes—
54#2084C2214yes—
55#2083C2214yes—
56#2068C2214yes—
57#2066C2214yes—
58#1990C2212yes—
59#1988C2211yescharacteristic
60#1985C2214yes—
61#1981C2211yescharacteristic
62#1971C2214yes—
63#1956C2212yes—
64#1949C2214yes—
65#1948C2212yes—
66#1948C2212yes—
67#1948C2211yescharacteristic
68#1947C2211yescharacteristic
69#1943C2214yes—
70#1939C2212yes—
71#1939C2212yes—
72#1928C2212yes—
73#1927C2214yes—
74#1919C2212yes—
75#1918C2214yes—
76#1878C2214yes—
77#1878C2212yes—
78#1865C2212yes—
79#1865C2214yes—
80#1847C2214yes—
81#1845C2212yes—
82#1803C2211yescharacteristic
83#1803C2214yes—
84#1803C2211yescharacteristic
85C4×#106C2212yes—
86#1770C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93C22×Q16C2312yes—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8D411yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#125—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—21no—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2314yes—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2312yes—
109#498—44no—
110#477—28no—
111#477—44no—
112#475—24no—
113#462Q818yescomplemented
114C4×Q16—44no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#107C2312yes—
118C2×#106C2312yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×#106C2312yes—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×#103C2312yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#97C2314yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2312yes—
138#400—24no—
139#400—24no—
140#400—24no—
141#398—24no—
142#398—24no—
143#395—24no—
144#395—24no—
145#391—22no—
146#391—22no—
147#391—22no—
148#391—21no—
149#391—24no—
150#391D412yes—
151#391C2312yes—
152#391C2314yes—
153#391C2314yes—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—24no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—24no—
167#389D412yes—
168#389D412yes—
169#389C2312yes—
170#389C2314yes—
171#389C2314yes—
172#387—24no—
173#387—24no—
174#387—24no—
175#387—24no—
176#384—22no—
177#384—21no—
178#384—21no—
179#384D411yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2314yes—
183#384C2312yes—
184#384C2312yes—
185#382—22no—
186#382—22no—
187#382C2314yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
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⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C43—21no—
206C43C2311yescharacteristic
207#340C2312yes—
208#340C2314yes—
209#340C2312yes—
210#340C2312yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#336C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8C2×D412yes—
222C22×Q8C2412yes—
223C22×Q8C2411yescharacteristic
224C2×Q16—42no—
225C2×Q16—42no—
226C2×Q16C2×Q818yes—
227C22×C8C2412yes—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C4⋊Q8—24no—
232C4⋊Q8—22no—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—82no—
238C4⋊Q8—82no—
239C4⋊Q8C2×Q814yes—
240#125—84no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—24no—
244C4×Q8—22no—
245C4×Q8—24no—
246C4×Q8—42no—
247C4×Q8—42no—
248C4×Q8—41no—
249C4×Q8—41no—
250C4×Q8—44no—
251C4×Q8—44no—
252C4×Q8—44no—
253C4×Q8—84no—
254C4×Q8C2×Q814yes—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2411yescharacteristic
285C2×C4⋊C4C2411yescharacteristic
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2411yescharacteristic
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2414yes—
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#107—28no—
309#107—42no—
310#107—44no—
311#106—24no—
312#106—42no—
313#106—44no—
314#106—44no—
315C4⋊C8—22no—
316C4⋊C8—22no—
317C4⋊C8—44no—
318C4⋊C8C2×Q814yes—
319C4⋊C8#4114yes—
320#103—24no—
321#103—28no—
322#103—42no—
323#103—42no—
324#103—42no—
325#103—42no—
326#103—44no—
327#103C2×Q818yes—
328#97—44no—
329C4×C8—24no—
330C4×C8—24no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95C2×D414yes—
348#95C2×D414yes—
349C2×Q8—24no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—44no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—82no—
367C2×Q8—82no—
368C2×Q8C22×Q814yes—
369C2×Q8C22×Q812yes—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4C22×D414yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D414yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2511yescharacteristic
388Q16—48no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8C22×Q814yes—
400C2×C8C2×#4114yes—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4C22×Q812yes—
443C42—24no—
444C42—22no—
445C42—22no—
446C42—24no—
447C42—24no—
448C42—21no—
449C42—21no—
450C42—41no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—44no—
455C42—44no—
456C42—42no—
457C42—42no—
458C42—84no—
459C42—84no—
460C42#13714yes—
461C42C22×Q812yes—
462C42C2×#4112yes—
463C42H(5,𝔽2)12yes—
464C422−1+412yes—
465C23C23×D411yescharacteristic
466Q8—22no—
467Q8—24no—
468Q8—22no—
469Q8—24no—
470Q8—48no—
471Q8—41no—
472Q8—42no—
473Q8—41no—
474Q8—42no—
475Q8—84no—
476Q8—164no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4D4×Q812yes—
520C2×C4D4×Q812yes—
521C2×C4#55012yes—
522C2×C4#55012yes—
523C2×C4C2×#13712yes—
524C2×C4C2×#13712yes—
525C2×C4C2×#13712yes—
526C2×C4#57712yes—
527C2×C4#57812yes—
528C2×C4C23×Q811yescharacteristic
529C2×C4C22×#4111yescharacteristic
530C2×C4C2×H(5,𝔽2)11yescharacteristic
531C2×C4C2×2−1+411yescharacteristic
532C2×C4#58511yescharacteristic
533C2×C4#58511yescharacteristic
534C8—24no—
535C8—24no—
536C8—42no—
537C8—42no—
538C22C2×D4×Q811yescharacteristic
539C22C2×#55011yescharacteristic
540C22#348511yescharacteristic
541C22C22×#13711yescharacteristic
542C22C2×#57711yescharacteristic
543C22C2×#57811yescharacteristic
544C22#358511yescharacteristic
545C4—22no—
546C4—22no—
547C4—22no—
548C4—44no—
549C4—44no—
550C4—41no—
551C4—42no—
552C4—44no—
553C4—42no—
554C4—82no—
555C4—81no—
556C4—82no—
557C4—81no—
558C4—82no—
559C4—84no—
560C4#338714yes—
561C4#339814yes—
562C4Q8×#4111yescharacteristic
563C4#350111yescharacteristic
564C4#351511yescharacteristic
565C4#354711yescharacteristic
566C2#5905211yescharacteristic
567C2C2×#338711yescharacteristic
568C2#6138411yescharacteristic
569C2C2×#339811yescharacteristic
570C2#6142111yescharacteristic
571C2#6171911yescharacteristic
572C2#6178911yescharacteristic
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. #462Q8semidirect

Maximal subgroups

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

subgroupindex
#598702
#365632
#326602
#288352
#286872
#285422
#285112
#265282
#260932
#254542
#250702
#248172
#232502
#231022
#229712
C4×#4752
C4×#4622