Finite Group Databasebeta

← #416655

Subgroups — #416655

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic
2C2×#1949C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic
4#13954C211yesmaximal, characteristic
5#11328C212yesmaximal
6#11327C212yesmaximal
7#11208C212yesmaximal
8#11208C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16#2571—24no—
17#2571—24no—
18C2×#400—22no—
19C2×#389—21no—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#382—21no—
25C2×#382C2211yescharacteristic
26C2×#382C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1949—24no—
31#1836—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—21no—
43#1444—21no—
44C4⋊#95—22no—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C42—21no—
62C22×C42—21no—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C2×#107—22no—
66C2×#107—42no—
67C2×#106—22no—
68C2×#106—42no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8Q814yes—
72C2×#97—22no—
73C2×#97—22no—
74C2×#97Q814yes—
75#400—216no—
76#389—28no—
77#389—24no—
78#389—24no—
79#389—24no—
80#389—24no—
81#389—42no—
82#389—42no—
83#384—22no—
84#384D414yes—
85#382—28no—
86#382—24no—
87#382—24no—
88#382—22no—
89#382—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C4×C4⋊C4Q814yes—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95D412yes—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#336—42no—
115#336—42no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C2×Q814yes—
142C22×C8C2×Q812yes—
143C22×C8C2×Q812yes—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—81no—
179C2×C4⋊C4—81no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×Q814yes—
188C2×C4⋊C4#4114yes—
189C2×C4⋊C4#4114yes—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—44no—
199C2×C42—81no—
200C2×C42—81no—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×Q812yes—
204C2×C42C2×Q812yes—
205C2×C42C2×Q812yes—
206C2×C42#4112yes—
207#107—44no—
208#106—44no—
209C4⋊C8—28no—
210#97—28no—
211#95—28no—
212#95—28no—
213#95—28no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—44no—
218#95—44no—
219#95—42no—
220#95—42no—
221C24C22×D411yescharacteristic
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4#12012yes—
277C22×C4#12012yes—
278C22×C4C22⋊Q812yes—
279C22×C4C22⋊Q812yes—
280C22×C4C4⋊Q812yes—
281C22×C4C4⋊Q812yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×Q811yescharacteristic
285C22×C4C22×Q811yescharacteristic
286C22×C4C22×Q811yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4112yes—
289C22×C4C2×#4112yes—
290C22×C4H(5,𝔽2)12yes—
291C22×C42−1+412yes—
292C2×C8—24no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—48no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—84no—
332C4⋊C4—88no—
333C4⋊C4—88no—
334C4⋊C4—88no—
335C4⋊C4#12518yes—
336C42—48no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—84no—
342C42C22⋊Q814yes—
343C42C22⋊Q814yes—
344C42C4⋊Q814yes—
345C42C4⋊Q814yes—
346C23C2×#12011yescharacteristic
347C23C2×C22⋊Q811yescharacteristic
348C23C2×C4⋊Q811yescharacteristic
349C23#54612yes—
350C23#54812yes—
351C23#57512yes—
352C23#57512yes—
353C23#57611yescharacteristic
354C23#57711yescharacteristic
355C23#57711yescharacteristic
356C23#57811yescharacteristic
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—28no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#39514yes—
416C2×C4#39514yes—
417C2×C4C2×C22⋊Q811yescharacteristic
418C2×C4C2×C22⋊Q811yescharacteristic
419C2×C4C2×#12512yes—
420C2×C4C2×C4⋊Q811yescharacteristic
421C2×C4C2×C4⋊Q811yescharacteristic
422C2×C4#53211yescharacteristic
423C2×C4#53211yescharacteristic
424C2×C4#53312yes—
425C2×C4#53511yescharacteristic
426C2×C4#53611yescharacteristic
427C2×C4#54412yes—
428C2×C4#54412yes—
429C2×C4#54712yes—
430C2×C4#55012yes—
431C2×C4#55212yes—
432C2×C4#55312yes—
433C2×C4#55412yes—
434C2×C4#55712yes—
435C8—42no—
436C8—42no—
437C8—44no—
438C22C2×#39511yescharacteristic
439C22#243111yescharacteristic
440C22#243211yescharacteristic
441C22#244212yes—
442C22#255012yes—
443C22#300511yescharacteristic
444C22#300511yescharacteristic
445C22#301211yescharacteristic
446C22#301311yescharacteristic
447C22#308012yes—
448C22#308111yescharacteristic
449C22#308211yescharacteristic
450C22#316311yescharacteristic
451C22#316411yescharacteristic
452C22#316612yes—
453C22#331112yes—
454C22#331312yes—
455C22#331912yes—
456C22#332112yes—
457C22#337012yes—
458C22#337112yes—
459C22#337612yes—
460C22#337712yes—
461C4—24no—
462C4—24no—
463C4—48no—
464C4—82no—
465C4—84no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4—161no—
470C4—161no—
471C4#244112yes—
472C4#255212yes—
473C4#256511yescharacteristic
474C4#256611yescharacteristic
475C4#266011yescharacteristic
476C4#285911yescharacteristic
477C2#1522811yescharacteristic
478C2#2355311yescharacteristic
479C2#2355711yescharacteristic
480C2#2557811yescharacteristic
481C2#2558011yescharacteristic
482C2#2565611yescharacteristic
483C2#2565811yescharacteristic
484C2#2612312yes—
485C2#2612512yes—
486C2#2880612yes—
487C2#2880812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#19492
C2×#18362
#139542
#113282
#113272
#112082
#108492
#108352
#107032