Finite Group Databasebeta

← #7541642

Subgroups — #7541642

Order 512.

1305 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#59875C211yesmaximal, characteristic
2#36598C211yesmaximal, characteristic
3#32439C214yesmaximal
4#29208C211yesmaximal, characteristic
5#29141C211yesmaximal, characteristic
6#27730C212yesmaximal
7#27127C214yesmaximal
8#26319C212yesmaximal
9#26093C212yesmaximal
10#25294C212yesmaximal
11#24995C212yesmaximal
12#24683C212yesmaximal
13#23149C211yesmaximal, characteristic
14#22808C212yesmaximal
15#22739C212yesmaximal
16C4×#475C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×#475C2211yescharacteristic
19C2×#462C2211yescharacteristic
20C2×C8×Q8C2212yes—
21C2×C4×Q16C2212yes—
22#2868C2211yescharacteristic
23#2862—21no—
24#2810—22no—
25#2582C2212yes—
26#2571C2212yes—
27#2570—22no—
28#2568—21no—
29#2568C2212yes—
30#2568C2212yes—
31#2568C2212yes—
32#2514—22no—
33#2514—22no—
34#2514C2214yes—
35#2511—22no—
36#2446—22no—
37#2365C2212yes—
38#2355—21no—
39#2355C2212yes—
40#2355C2212yes—
41#2355C2212yes—
42#2355C2212yes—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C42×Q8C2211yescharacteristic
48#2101C2214yes—
49#2097C2214yes—
50#2074C2214yes—
51#2074C2214yes—
52#2068C2214yes—
53#2057C2214yes—
54#2056C2214yes—
55#2024C2214yes—
56#1990C2212yes—
57#1988C2211yescharacteristic
58#1984C2212yes—
59#1981C2211yescharacteristic
60#1969C2214yes—
61#1956C2212yes—
62#1949C2214yes—
63#1949C2214yes—
64#1948C2211yescharacteristic
65#1947C2212yes—
66#1947C2212yes—
67#1947C2211yescharacteristic
68#1939C2212yes—
69#1933C2214yes—
70#1928C2214yes—
71#1928C2214yes—
72#1928C2212yes—
73#1927C2214yes—
74#1919C2212yes—
75#1918C2214yes—
76#1916C2214yes—
77#1878C2212yes—
78#1865C2212yes—
79#1845C2212yes—
80#1835C2214yes—
81#1834C2212yes—
82#1803C2211yescharacteristic
83#1803C2212yes—
84#1803C2212yes—
85#1803C2211yescharacteristic
86C4×#107C2214yes—
87C4×#106C2212yes—
88C8×C4⋊C4C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
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×C4×Q8—22no—
101C2×C4×Q8—21no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2312yes—
107#475—24no—
108#462—24no—
109C8×Q8—28no—
110C4×Q16—28no—
111C2×#107C2312yes—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#106C2312yes—
115C2×#106C2312yes—
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2314yes—
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×#103C2312yes—
122C2×#103C2312yes—
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×C4×C8C2312yes—
130C2×C4×C8C2314yes—
131C2×C4×C8C2312yes—
132#401—24no—
133#400—24no—
134#400—24no—
135#400—24no—
136#400—24no—
137#400—24no—
138#398—24no—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391—22no—
148#391—21no—
149#391D412yes—
150#391C2312yes—
151#391C2314yes—
152#391C2314yes—
153#391C2314yes—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389D412yes—
160#389D412yes—
161#389C2312yes—
162#389C2314yes—
163#387—24no—
164#387—24no—
165#387—24no—
166#387—24no—
167#387—24no—
168#387—24no—
169#384—21no—
170#384—21no—
171#384D411yescharacteristic
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2312yes—
175#384C2312yes—
176#383—24no—
177#382—22no—
178#382—22no—
179#382—22no—
180#382—22no—
181#382C2314yes—
182#382C2314yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C43—21no—
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2312yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
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—22no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—42no—
235C4⋊Q8—42no—
236C4⋊Q8—44no—
237C4⋊Q8—82no—
238C4×Q8—24no—
239C4×Q8—22no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—24no—
245C4×Q8—41no—
246C4×Q8—41no—
247C4×Q8—44no—
248C4×Q8—84no—
249C4×Q8#4114yes—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2411yescharacteristic
276C2×C4⋊C4C2411yescharacteristic
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2411yescharacteristic
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2411yescharacteristic
296C2×C42C2414yes—
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299#107—24no—
300#107—44no—
301#107—44no—
302#106—24no—
303#106—42no—
304C4⋊C8—28no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8#4114yes—
309#103—24no—
310#103—24no—
311#103—24no—
312#103—24no—
313#103—42no—
314#103—42no—
315#103—44no—
316#103—44no—
317C4×C8—28no—
318C4×C8—24no—
319C4×C8—24no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—24no—
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—24no—
344C2×Q8—24no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—44no—
361C2×Q8—42no—
362C2×Q8C2×#4112yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4C22×D414yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D414yes—
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×C8C2×#4114yes—
394C4⋊C4—24no—
395C4⋊C4—28no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—42no—
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—82no—
430C4⋊C4—82no—
431C4⋊C4C2×#4112yes—
432C42—24no—
433C42—22no—
434C42—22no—
435C42—24no—
436C42—24no—
437C42—21no—
438C42—21no—
439C42—41no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—44no—
444C42—44no—
445C42—42no—
446C42—42no—
447C42—84no—
448C42—84no—
449C42#13514yes—
450C42C2×#4112yes—
451C42C2×#4112yes—
452C422−1+412yes—
453C422−1+412yes—
454C23C23×D411yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—48no—
458Q8—41no—
459Q8—41no—
460Q8—42no—
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—88no—
465Q8—84no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4#54712yes—
509C2×C4#54712yes—
510C2×C4#54812yes—
511C2×C4#54812yes—
512C2×C4C2×#13512yes—
513C2×C4C2×#13512yes—
514C2×C4C2×#13512yes—
515C2×C4#57812yes—
516C2×C4#57812yes—
517C2×C4C22×#4111yescharacteristic
518C2×C4C22×#4111yescharacteristic
519C2×C4C2×2−1+411yescharacteristic
520C2×C4C2×2−1+411yescharacteristic
521C2×C4#58511yescharacteristic
522C2×C4#58511yescharacteristic
523C8—24no—
524C8—42no—
525C8—42no—
526C8—42no—
527C22C2×#54711yescharacteristic
528C22C2×#54811yescharacteristic
529C22#348711yescharacteristic
530C22C22×#13511yescharacteristic
531C22C2×#57811yescharacteristic
532C22C2×#57811yescharacteristic
533C22#358511yescharacteristic
534C4—22no—
535C4—22no—
536C4—22no—
537C4—44no—
538C4—44no—
539C4—42no—
540C4—42no—
541C4—41no—
542C4—81no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—84no—
548C4—81no—
549C4#334214yes—
550C4#336414yes—
551C4#347511yescharacteristic
552C4#350711yescharacteristic
553C4#351811yescharacteristic
554C4#355511yescharacteristic
555C2#5906511yescharacteristic
556C2C2×#334211yescharacteristic
557C2#6128211yescharacteristic
558C2C2×#336411yescharacteristic
559C2#6134811yescharacteristic
560C2#6188511yescharacteristic
561C2#6193611yescharacteristic
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
#598752
#365982
#324392
#292082
#291412
#277302
#271272
#263192
#260932
#252942
#249952
#246832
#231492
#228082
#227392
C4×#4752
C4×#4622