Finite Group Databasebeta

← #7545520

Subgroups — #7545520

Order 512.

1311 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59870C211yesmaximal, characteristic
2#36598C211yesmaximal, characteristic
3#32553C214yesmaximal
4#29219C211yesmaximal, characteristic
5#28717C211yesmaximal, characteristic
6#27730C212yesmaximal
7#27730C212yesmaximal
8#26509C212yesmaximal
9#26113C212yesmaximal
10#25323C212yesmaximal
11#25000C214yesmaximal
12#24683C212yesmaximal
13#23247C211yesmaximal, characteristic
14#23214C212yesmaximal
15#22771C212yesmaximal
16C4×#475C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×#487C2212yes—
19C2×#475C2211yescharacteristic
20C2×#462C2211yescharacteristic
21C2×C4×Q16C2212yes—
22#2868C2211yescharacteristic
23#2862—21no—
24#2805—22no—
25#2582C2212yes—
26#2581C2214yes—
27#2571C2212yes—
28#2571C2212yes—
29#2570—22no—
30#2568C2212yes—
31#2568C2212yes—
32#2567—21no—
33#2511—22no—
34#2511—22no—
35#2509—22no—
36#2446—22no—
37#2365C2212yes—
38#2365C2212yes—
39#2355—21no—
40#2355C2212yes—
41#2355C2212yes—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C42×Q8C2211yescharacteristic
48#2101C2214yes—
49#2101C2214yes—
50#2099C2214yes—
51#2088C2214yes—
52#2074C2214yes—
53#2074C2214yes—
54#2072C2214yes—
55#2056C2214yes—
56#1990C2212yes—
57#1990C2212yes—
58#1988C2212yes—
59#1988C2212yes—
60#1988C2211yescharacteristic
61#1984C2214yes—
62#1981C2211yescharacteristic
63#1949C2214yes—
64#1948C2211yescharacteristic
65#1947C2211yescharacteristic
66#1939C2212yes—
67#1939C2212yes—
68#1933C2214yes—
69#1933C2214yes—
70#1928C2212yes—
71#1927C2214yes—
72#1921C2214yes—
73#1919C2212yes—
74#1878C2212yes—
75#1878C2214yes—
76#1865C2214yes—
77#1865C2212yes—
78#1835C2214yes—
79#1834C2212yes—
80#1805C2214yes—
81#1803C2212yes—
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85C4×#106C2212yes—
86C8×C4⋊C4C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#103C2212yes—
92C4×#103C2214yes—
93C22×Q16C2312yes—
94C2×C4⋊Q8—21no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8D411yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4×Q8—22no—
101C2×C4×Q8—21no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2314yes—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2312yes—
108#487—28no—
109#475—24no—
110#462—24no—
111C4×Q16—28no—
112C2×#107C2312yes—
113C2×#106C2312yes—
114C2×#106C2312yes—
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2314yes—
120C2×#103C2312yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×C4×C8C2312yes—
129C2×C4×C8C2312yes—
130C2×C4×C8C2314yes—
131#400—24no—
132#400—24no—
133#400—24no—
134#400—24no—
135#400—24no—
136#398—24no—
137#395—24no—
138#391—22no—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—21no—
146#391D412yes—
147#391C2312yes—
148#391C2314yes—
149#391C2314yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—24no—
158#389D412yes—
159#389D412yes—
160#389C2312yes—
161#389C2314yes—
162#389C2314yes—
163#387—24no—
164#387—24no—
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#384C2312yes—
176#384C2312yes—
177#382—22no—
178#382—22no—
179#382—22no—
180#382C2314yes—
181#382C2314yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2311yescharacteristic
194C4×C4⋊C4C2314yes—
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C43—21no—
200C43C2311yescharacteristic
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2312yes—
205#340C2312yes—
206#340C2312yes—
207#340C2314yes—
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2314yes—
212#340C2312yes—
213#336C2312yes—
214#336C2314yes—
215#336C2314yes—
216#336C2312yes—
217C22×Q8—22no—
218C22×Q8—21no—
219C22×Q8C2×D412yes—
220C22×Q8C2412yes—
221C22×Q8C2411yescharacteristic
222C2×Q16—24no—
223C2×Q16—24no—
224C2×Q16—42no—
225C22×C8C2412yes—
226C22×C8C2412yes—
227C22×C8C2412yes—
228C22×C8C2412yes—
229C4⋊Q8—24no—
230C4⋊Q8—22no—
231C4⋊Q8—24no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—24no—
235C4⋊Q8—44no—
236C4⋊Q8—82no—
237C4⋊Q8#4114yes—
238C4×Q8—22no—
239C4×Q8—24no—
240C4×Q8—22no—
241C4×Q8—24no—
242C4×Q8—22no—
243C4×Q8—42no—
244C4×Q8—42no—
245C4×Q8—41no—
246C4×Q8—41no—
247C4×Q8—44no—
248C4×Q8—44no—
249C4×Q8—44no—
250C4×Q8—84no—
251C4×Q8#4114yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2411yescharacteristic
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2411yescharacteristic
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2411yescharacteristic
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2414yes—
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303#107—24no—
304#106—24no—
305#106—42no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—44no—
311#103—24no—
312#103—24no—
313#103—24no—
314#103—28no—
315#103—28no—
316#103—24no—
317#103—42no—
318#103—42no—
319C4×C8—24no—
320C4×C8—24no—
321C4×C8—44no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95C2×D414yes—
341#95C2×D414yes—
342C2×Q8—24no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—44no—
357C2×Q8—42no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—42no—
361C2×Q8C2×#4114yes—
362C2×Q8C2×#4112yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4C22×D414yes—
373C22×C4C22×D414yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2511yescharacteristic
381Q16—48no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4C2×#4112yes—
434C42—24no—
435C42—22no—
436C42—22no—
437C42—24no—
438C42—24no—
439C42—21no—
440C42—21no—
441C42—41no—
442C42—42no—
443C42—42no—
444C42—44no—
445C42—42no—
446C42—44no—
447C42—42no—
448C42—42no—
449C42—84no—
450C42—84no—
451C42#13514yes—
452C42C2×#4112yes—
453C42C2×#4112yes—
454C422−1+412yes—
455C422−1+412yes—
456C23C23×D411yescharacteristic
457Q8—22no—
458Q8—24no—
459Q8—22no—
460Q8—24no—
461Q8—42no—
462Q8—41no—
463Q8—41no—
464Q8—42no—
465Q8—84no—
466Q8—84no—
467Q8—88no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4#54812yes—
511C2×C4#54812yes—
512C2×C4#54812yes—
513C2×C4#54812yes—
514C2×C4C2×#13512yes—
515C2×C4C2×#13512yes—
516C2×C4C2×#13512yes—
517C2×C4#57712yes—
518C2×C4#57812yes—
519C2×C4C22×#4111yescharacteristic
520C2×C4C22×#4111yescharacteristic
521C2×C4C2×2−1+411yescharacteristic
522C2×C4C2×2−1+411yescharacteristic
523C2×C4#58511yescharacteristic
524C2×C4#58511yescharacteristic
525C8—42no—
526C8—42no—
527C8—42no—
528C8—42no—
529C22C2×#54811yescharacteristic
530C22C2×#54811yescharacteristic
531C22#348511yescharacteristic
532C22C22×#13511yescharacteristic
533C22C2×#57711yescharacteristic
534C22C2×#57811yescharacteristic
535C22#358611yescharacteristic
536C4—22no—
537C4—22no—
538C4—22no—
539C4—44no—
540C4—44no—
541C4—41no—
542C4—42no—
543C4—42no—
544C4—44no—
545C4—82no—
546C4—81no—
547C4—82no—
548C4—84no—
549C4—81no—
550C4—82no—
551C4#336414yes—
552C4#336514yes—
553C4#347511yescharacteristic
554C4#350711yescharacteristic
555C4#351511yescharacteristic
556C4#355511yescharacteristic
557C2#5907911yescharacteristic
558C2C2×#336411yescharacteristic
559C2C2×#336511yescharacteristic
560C2#6134211yescharacteristic
561C2#6134611yescharacteristic
562C2#6170111yescharacteristic
563C2#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

Maximal subgroups

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

subgroupindex
#598702
#365982
#325532
#292192
#287172
#277302
#265092
#261132
#253232
#250002
#246832
#232472
#232142
#227712
C4×#4752
C4×#4622