Finite Group Databasebeta

← #417024

Subgroups — #417024

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4#14160C211yesmaximal, characteristic
5#11293C212yesmaximal
6#11292C212yesmaximal
7#11267C212yesmaximal
8#11267C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×#125—21no—
13C22×#125C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16#2552—24no—
17#2552—24no—
18C2×#398—22no—
19C2×#389—21no—
20C2×#389C2212yes—
21C2×#384C2212yes—
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1992—24no—
27#1989—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1441—21no—
41#1441—21no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C2×#125—28no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—48no—
55C2×#125Q814yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×#97—22no—
79C2×#97—22no—
80C2×#97—22no—
81#398—216no—
82#389—28no—
83#389—24no—
84#389—24no—
85#384—24no—
86#384—24no—
87#384—22no—
88#384D414yes—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D412yes—
103#340—24no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#336—24no—
114#336—24no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8#4114yes—
143#125—28no—
144#125—28no—
145#125—432no—
146#125—44no—
147#125—44no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×Q812yes—
202C2×C4⋊C4C2×Q812yes—
203C2×C4⋊C4#4114yes—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—44no—
212C2×C42—42no—
213C2×C42—81no—
214C2×C42—81no—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×Q812yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C4⋊C8—44no—
222#97—44no—
223#95—28no—
224#95—24no—
225#95—24no—
226#95—42no—
227#95—44no—
228#95—44no—
229#95—82no—
230#95—82no—
231C24C22×D411yescharacteristic
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
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—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4#12012yes—
287C22×C4#12012yes—
288C22×C4C22⋊Q812yes—
289C22×C4C22⋊Q812yes—
290C22×C4#12412yes—
291C22×C4#12412yes—
292C22×C4AGL(1,ℤ/8)14yes—
293C22×C4#13714yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×Q811yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C42−1+412yes—
303C22×C42−1+412yes—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C4⋊C4—28no—
322C4⋊C4—28no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—48no—
336C4⋊C4—416no—
337C4⋊C4—416no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—84no—
351C4⋊C4—88no—
352C42—48no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—84no—
358C42C22⋊Q814yes—
359C42C22⋊Q814yes—
360C42#12414yes—
361C42#12414yes—
362C23C2×#12011yescharacteristic
363C23C2×C22⋊Q811yescharacteristic
364C23C2×#12411yescharacteristic
365C23#54712yes—
366C23#54812yes—
367C23C2×AGL(1,ℤ/8)12yes—
368C23C2×#13712yes—
369C23#57611yescharacteristic
370C23#57711yescharacteristic
371C23#57711yescharacteristic
372C23#57811yescharacteristic
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
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—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4#39314yes—
434C2×C4#39314yes—
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4C2×C22⋊Q811yescharacteristic
437C2×C4C2×#12411yescharacteristic
438C2×C4C2×#12411yescharacteristic
439C2×C4#52912yes—
440C2×C4#52912yes—
441C2×C4#53211yescharacteristic
442C2×C4#53211yescharacteristic
443C2×C4#53611yescharacteristic
444C2×C4#53611yescharacteristic
445C2×C4#54112yes—
446C2×C4#54112yes—
447C2×C4#54712yes—
448C2×C4#54812yes—
449C2×C4#55212yes—
450C2×C4#55212yes—
451C2×C4#55712yes—
452C2×C4#55712yes—
453C8—44no—
454C8—81no—
455C8—81no—
456C22C2×#39311yescharacteristic
457C22#243211yescharacteristic
458C22#243211yescharacteristic
459C22#245312yes—
460C22#248912yes—
461C22#300311yescharacteristic
462C22#300411yescharacteristic
463C22#301211yescharacteristic
464C22#301311yescharacteristic
465C22#307912yes—
466C22#308311yescharacteristic
467C22#308411yescharacteristic
468C22#313611yescharacteristic
469C22#313711yescharacteristic
470C22#314112yes—
471C22#334312yes—
472C22#334412yes—
473C22#334512yes—
474C22#334612yes—
475C22#336612yes—
476C22#336812yes—
477C22#337212yes—
478C22#337412yes—
479C4—24no—
480C4—24no—
481C4—44no—
482C4—44no—
483C4—82no—
484C4—84no—
485C4—84no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4#245312yes—
490C4#248912yes—
491C4#256511yescharacteristic
492C4#256511yescharacteristic
493C4#264211yescharacteristic
494C4#286111yescharacteristic
495C2#1523811yescharacteristic
496C2#2345211yescharacteristic
497C2#2345811yescharacteristic
498C2#2565511yescharacteristic
499C2#2565711yescharacteristic
500C2#2570911yescharacteristic
501C2#2571211yescharacteristic
502C2#2652212yes—
503C2#2652412yes—
504C2#2773912yes—
505C2#2774112yes—
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×#25522
C2×#19922
C2×#19892
#141602
#112932
#112922
#112672
#108492
#108352
#107032