Finite Group Databasebeta

← #7598820

Subgroups — #7598820

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59875C211yesmaximal, characteristic
2#59832C211yesmaximal, characteristic
3#31813C214yesmaximal
4#29287C211yesmaximal, characteristic
5#29081C214yesmaximal
6#28255C211yesmaximal, characteristic
7#27823C212yesmaximal
8#26022C212yesmaximal
9#25936C212yesmaximal
10#25323C212yesmaximal
11#25302C212yesmaximal
12#24997C212yesmaximal
13#23102C212yesmaximal
14#22525C212yesmaximal
15#21712C211yesmaximal, characteristic
16C4×#487C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#487C2211yescharacteristic
20C2×#487C2211yescharacteristic
21C2×#462C2212yes—
22C2×#462C2212yes—
23C2×#446C2212yes—
24#2810—22no—
25#2734—22no—
26#2582C2212yes—
27#2568—21no—
28#2568C2212yes—
29#2568C2212yes—
30#2551—21no—
31#2551C2212yes—
32#2546—22no—
33#2514—22no—
34#2514—22no—
35#2514C2214yes—
36#2451—22no—
37#2451C2214yes—
38#2365—21no—
39#2365C2212yes—
40#2355—21no—
41#2355C2212yes—
42#2355C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48#2101C2214yes—
49#2088C2214yes—
50#2074C2214yes—
51#2066C2214yes—
52#2056C2214yes—
53#2056C2214yes—
54#2024C2214yes—
55#2024C2214yes—
56#1993C2214yes—
57#1990C2211yescharacteristic
58#1990C2211yescharacteristic
59#1988C2214yes—
60#1988C2212yes—
61#1987C2212yes—
62#1984C2212yes—
63#1971C2214yes—
64#1956C2211yescharacteristic
65#1956C2211yescharacteristic
66#1948C2212yes—
67#1948C2212yes—
68#1939C2212yes—
69#1939C2212yes—
70#1935C2214yes—
71#1933C2212yes—
72#1933C2212yes—
73#1933C2212yes—
74#1933C2212yes—
75#1927C2214yes—
76#1919C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1842C2211yescharacteristic
80#1834C2211yescharacteristic
81#1834C2211yescharacteristic
82#1805C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1770C2212yes—
86#1770C2212yes—
87#1763C2214yes—
88C4×#103C2212yes—
89C4×#103C2212yes—
90C4×#103C2212yes—
91C4×#103C2212yes—
92C42×C8C2211yescharacteristic
93C22×Q16C2314yes—
94Q82—28no—
95Q82—44no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2314yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4×Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2312yes—
112#487—24no—
113#487—24no—
114#462—28no—
115#462—44no—
116#446—44no—
117C2×#106C2314yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2314yes—
122C2×#103C2312yes—
123C2×#103C2314yes—
124C2×#103C2312yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#97C2314yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139#401—24no—
140#400—24no—
141#400—24no—
142#400—24no—
143#398—24no—
144#398—24no—
145#395—24no—
146#395—24no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391—24no—
155#391D412yes—
156#391C2312yes—
157#391C2312yes—
158#391C2314yes—
159#391C2314yes—
160#391C2314yes—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389D412yes—
171#389D412yes—
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#387—24no—
176#387—24no—
177#387—24no—
178#387—24no—
179#387—24no—
180#384D411yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#383—24no—
186#382—22no—
187#382—22no—
188#382C2314yes—
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⋊C4C2312yes—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C43C2311yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2312yes—
207#340C2314yes—
208#340C2312yes—
209#340C2312yes—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2314yes—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—24no—
221C22×Q8C2×D412yes—
222C22×Q8C2412yes—
223C22×Q8C2412yes—
224C2×Q16—28no—
225C2×Q16—44no—
226C2×Q16—44no—
227C22×C8C2414yes—
228C22×C8C2411yescharacteristic
229C22×C8C2411yescharacteristic
230C22×C8C2411yescharacteristic
231C22×C8C2411yescharacteristic
232C4⋊Q8—28no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8—44no—
241C4⋊Q8—48no—
242C4⋊Q8—416no—
243C4⋊Q8—84no—
244C4⋊Q8#4114yes—
245C4⋊Q8#4114yes—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—28no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—24no—
253C4×Q8—44no—
254C4×Q8—48no—
255C4×Q8—44no—
256C4×Q8—82no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2414yes—
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304#106—44no—
305C4⋊C8—24no—
306C4⋊C8—24no—
307C4⋊C8—42no—
308C4⋊C8—44no—
309#103—28no—
310#103—24no—
311#103—24no—
312#103—24no—
313#103—24no—
314#103—24no—
315#103—24no—
316#103—42no—
317#103—42no—
318#103—44no—
319#97—44no—
320C4×C8—22no—
321C4×C8—22no—
322C4×C8—22no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—41no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—22no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95C2×D414yes—
345#95C2×D414yes—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—48no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—81no—
366C2×Q8—81no—
367C2×Q8—84no—
368C2×Q8—84no—
369C2×Q8C2×#4114yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D414yes—
380C22×C4C22×D414yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2511yescharacteristic
387Q16—88no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—24no—
395C2×C8—21no—
396C2×C8—24no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—24no—
402C2×C82−1+412yes—
403C4⋊C4—24no—
404C4⋊C4—28no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—42no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—82no—
440C4⋊C4—84no—
441C4⋊C4C2×#4114yes—
442C42—22no—
443C42—22no—
444C42—24no—
445C42—24no—
446C42—21no—
447C42—21no—
448C42—41no—
449C42—48no—
450C42—44no—
451C42—44no—
452C42—44no—
453C42—44no—
454C42—84no—
455C42#13714yes—
456C42C2×#4112yes—
457C42C2×#4112yes—
458C42H(5,𝔽2)12yes—
459C422−1+412yes—
460C23C23×D411yescharacteristic
461Q8—24no—
462Q8—24no—
463Q8—48no—
464Q8—42no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470Q8—84no—
471Q8—84no—
472Q8—168no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—24no—
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—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4#54612yes—
513C2×C4#54612yes—
514C2×C4#54812yes—
515C2×C4#54812yes—
516C2×C4C2×#13712yes—
517C2×C4C2×#13712yes—
518C2×C4C2×#13712yes—
519C2×C4#57712yes—
520C2×C4#57812yes—
521C2×C4C22×#4111yescharacteristic
522C2×C4C22×#4111yescharacteristic
523C2×C4C2×H(5,𝔽2)11yescharacteristic
524C2×C4C2×2−1+411yescharacteristic
525C2×C4C2×2−1+411yescharacteristic
526C2×C4#58511yescharacteristic
527C8—22no—
528C8—41no—
529C8—41no—
530C8—44no—
531C8—41no—
532C22C2×#54611yescharacteristic
533C22C2×#54811yescharacteristic
534C22#349711yescharacteristic
535C22C22×#13711yescharacteristic
536C22C2×#57711yescharacteristic
537C22C2×#57811yescharacteristic
538C22#358411yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—44no—
543C4—44no—
544C4—44no—
545C4—41no—
546C4—44no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4#332814yes—
554C4#336914yes—
555C4#348211yescharacteristic
556C4#351411yescharacteristic
557C4#351811yescharacteristic
558C4#355211yescharacteristic
559C2#5942111yescharacteristic
560C2C2×#332811yescharacteristic
561C2#6121211yescharacteristic
562C2C2×#336911yescharacteristic
563C2#6132811yescharacteristic
564C2#6236611yescharacteristic
565C2#6238211yescharacteristic
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
#598322
#318132
#292872
#290812
#282552
#278232
#260222
#259362
#253232
#253022
#249972
#231022
#225252
#217122
C4×#4872