Finite Group Databasebeta

← #416974

Subgroups — #416974

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1993C211yesmaximal, characteristic
2C2×#1988C211yesmaximal, characteristic
3C2×#1940C211yesmaximal, characteristic
4C2×#1776C211yesmaximal, characteristic
5#14161C211yesmaximal, characteristic
6#13955C211yesmaximal, characteristic
7#11316C212yesmaximal
8#11315C212yesmaximal
9#11232C212yesmaximal
10#11231C212yesmaximal
11#10850C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1993—24no—
23#1988—24no—
24#1940—24no—
25#1776—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1446—21no—
36#1446—21no—
37#1442—22no—
38C4⋊#95—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43#1434—21no—
44#1434—21no—
45C23×Q8—21no—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8Q814yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C42—21no—
64C22×C42—21no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×#107—22no—
68C2×#107—22no—
69C2×#107—22no—
70C2×#106—22no—
71C2×#106—22no—
72C2×#106—22no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×#97—22no—
80C2×#97—22no—
81C2×#97D414yes—
82#382—22no—
83#382—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D414yes—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—22no—
110#340—24no—
111#340—24no—
112#336—24no—
113C22×Q8—24no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—82no—
117C22×Q8—82no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C22×C8#4112yes—
149C22×C8#4112yes—
150C4⋊Q8—28no—
151C4⋊Q8—28no—
152C4⋊Q8—44no—
153C4⋊Q8—44no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×Q814yes—
194C2×C4⋊C4#4112yes—
195C2×C4⋊C4#4112yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42—44no—
204C2×C42—42no—
205C2×C42—82no—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×Q812yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212#107—28no—
213#106—28no—
214C4⋊C8—48no—
215C4⋊C8—44no—
216#97—28no—
217#95—24no—
218#95—22no—
219#95—22no—
220#95—44no—
221#95—42no—
222#95—42no—
223#95—42no—
224#95—44no—
225#95—82no—
226#95—82no—
227#95—82no—
228#95—82no—
229#95—82no—
230#95—82no—
231C24C22×D411yescharacteristic
232C2×Q8—48no—
233C2×Q8—88no—
234C2×Q8—84no—
235C2×Q8—84no—
236C2×Q8—88no—
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—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4#12112yes—
289C22×C4#12112yes—
290C22×C4#12112yes—
291C22×C4#12112yes—
292C22×C4C22⋊Q812yes—
293C22×C4C22⋊Q812yes—
294C22×C4#13714yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×Q811yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C42−1+411yescharacteristic
306C22×C42−1+411yescharacteristic
307C22×C42−1+411yescharacteristic
308C22×C42−1+411yescharacteristic
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8#12414yes—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—48no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—84no—
361C42—84no—
362C42#12114yes—
363C42C22⋊Q814yes—
364C42C22⋊Q814yes—
365C42#12614yes—
366C23C2×#12111yescharacteristic
367C23C2×#12111yescharacteristic
368C23C2×C22⋊Q811yescharacteristic
369C23#54711yescharacteristic
370C23#54711yescharacteristic
371C23#54811yescharacteristic
372C23#54811yescharacteristic
373C23C2×#13712yes—
374C23#57512yes—
375C23#57612yes—
376C23#57712yes—
377Q8—816no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
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—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4#39614yes—
435C2×C4#39614yes—
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4C2×C22⋊Q811yescharacteristic
439C2×C4#52511yescharacteristic
440C2×C4#52511yescharacteristic
441C2×C4C2×#12411yescharacteristic
442C2×C4C2×#12611yescharacteristic
443C2×C4#52911yescharacteristic
444C2×C4#52911yescharacteristic
445C2×C4#53211yescharacteristic
446C2×C4#53611yescharacteristic
447C2×C4#53611yescharacteristic
448C2×C4#53611yescharacteristic
449C2×C4#53611yescharacteristic
450C2×C4#54111yescharacteristic
451C2×C4#54111yescharacteristic
452C2×C4#55211yescharacteristic
453C2×C4#55211yescharacteristic
454C2×C4#55611yescharacteristic
455C2×C4#55611yescharacteristic
456C2×C4#55611yescharacteristic
457C2×C4#55611yescharacteristic
458C2×C4#55611yescharacteristic
459C2×C4#55611yescharacteristic
460C2×C4#55711yescharacteristic
461C2×C4#55711yescharacteristic
462C2×C4#55711yescharacteristic
463C2×C4#55711yescharacteristic
464C8—24no—
465C8—42no—
466C8—82no—
467C22C2×#39611yescharacteristic
468C22#241511yescharacteristic
469C22#241511yescharacteristic
470C22#245311yescharacteristic
471C22#245311yescharacteristic
472C22#254511yescharacteristic
473C22#254711yescharacteristic
474C22#303911yescharacteristic
475C22#304011yescharacteristic
476C22#304111yescharacteristic
477C22#304311yescharacteristic
478C22#305612yes—
479C22#306212yes—
480C22#307812yes—
481C22#308112yes—
482C22#334412yes—
483C22#334612yes—
484C22#334712yes—
485C22#334812yes—
486C22#336912yes—
487C22#337012yes—
488C22#337512yes—
489C22#337612yes—
490C4—24no—
491C4—24no—
492C4—44no—
493C4—82no—
494C4—84no—
495C4—82no—
496C4—82no—
497C4—82no—
498C4—82no—
499C4—162no—
500C4#251411yescharacteristic
501C4#251411yescharacteristic
502C4#256511yescharacteristic
503C4#256811yescharacteristic
504C4#258111yescharacteristic
505C4#258111yescharacteristic
506C4#262111yescharacteristic
507C4#285011yescharacteristic
508C2#1523711yescharacteristic
509C2#2361411yescharacteristic
510C2#2361911yescharacteristic
511C2#2469311yescharacteristic
512C2#2469511yescharacteristic
513C2#2469611yescharacteristic
514C2#2469811yescharacteristic
515C2#2651912yes—
516C2#2652012yes—
517C2#2865112yes—
518C2#2871512yes—
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×#19932
C2×#19882
C2×#19402
C2×#17762
#141612
#139552
#113162
#113152
#112322
#112312
#108502