Finite Group Databasebeta

← #416986

Subgroups — #416986

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1993C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1835C211yesmaximal, characteristic
4C2×#1768C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic
6#14277C211yesmaximal, characteristic
7#11316C212yesmaximal
8#11315C212yesmaximal
9#11228C212yesmaximal
10#11227C212yesmaximal
11#10843C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#389C2212yes—
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#376C2211yescharacteristic
22#1993—24no—
23#1990—24no—
24#1835—24no—
25#1768—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1446—21no—
37#1446—21no—
38#1445—21no—
39#1445—21no—
40#1444—21no—
41#1444—21no—
42#1442—21no—
43#1442—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8Q814yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×#97—22no—
76C2×#97—22no—
77C2×#97—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8D414yes—
81#389—24no—
82#389—24no—
83#384—22no—
84#384—22no—
85#382—22no—
86#382—22no—
87#382—22no—
88#382D414yes—
89#376—22no—
90#376—22no—
91C2×#95—21no—
92C2×#95—21no—
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
104C2×#95D411yescharacteristic
105C2×#95D411yescharacteristic
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#336—22no—
115#336—22no—
116C22×Q8—24no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—82no—
120C22×Q8—82no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8#4112yes—
159C22×C8#4112yes—
160C4⋊Q8—28no—
161C4⋊Q8—28no—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
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—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×Q814yes—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×Q812yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42#4112yes—
215C2×C42#4112yes—
216C4⋊C8—28no—
217C4⋊C8—28no—
218#97—44no—
219C4×C8—28no—
220#95—24no—
221#95—24no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—42no—
227#95—42no—
228#95—44no—
229#95—82no—
230#95—82no—
231#95—82no—
232#95—82no—
233#95—82no—
234#95—82no—
235C24C22×D411yescharacteristic
236C2×Q8—48no—
237C2×Q8—88no—
238C2×Q8—88no—
239C2×Q8—84no—
240C2×Q8—84no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
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—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—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4#12112yes—
292C22×C4#12112yes—
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4C22⋊Q812yes—
296C22×C4C22⋊Q812yes—
297C22×C4#12412yes—
298C22×C4#12412yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×Q811yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C42−1+411yescharacteristic
310C22×C42−1+411yescharacteristic
311C22×C42−1+411yescharacteristic
312C22×C42−1+411yescharacteristic
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—48no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C4⋊C4—88no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42#12114yes—
370C42C22⋊Q814yes—
371C42C22⋊Q814yes—
372C42#12614yes—
373C23C2×#12111yescharacteristic
374C23C2×#12111yescharacteristic
375C23C2×C22⋊Q811yescharacteristic
376C23C2×#12411yescharacteristic
377C23#53211yescharacteristic
378C23#53611yescharacteristic
379C23#53611yescharacteristic
380C23#57612yes—
381C23#57712yes—
382C23#57712yes—
383C23#57812yes—
384Q8—816no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—41no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
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—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4#39614yes—
442C2×C4#39614yes—
443C2×C4C2×#12111yescharacteristic
444C2×C4C2×C22⋊Q811yescharacteristic
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4#52511yescharacteristic
447C2×C4#52511yescharacteristic
448C2×C4C2×#12611yescharacteristic
449C2×C4#52911yescharacteristic
450C2×C4#52911yescharacteristic
451C2×C4#53611yescharacteristic
452C2×C4#53611yescharacteristic
453C2×C4#54111yescharacteristic
454C2×C4#54111yescharacteristic
455C2×C4#54711yescharacteristic
456C2×C4#54711yescharacteristic
457C2×C4#54811yescharacteristic
458C2×C4#54811yescharacteristic
459C2×C4#55211yescharacteristic
460C2×C4#55211yescharacteristic
461C2×C4#55611yescharacteristic
462C2×C4#55611yescharacteristic
463C2×C4#55611yescharacteristic
464C2×C4#55611yescharacteristic
465C2×C4#55611yescharacteristic
466C2×C4#55611yescharacteristic
467C2×C4#55711yescharacteristic
468C2×C4#55711yescharacteristic
469C2×C4#55711yescharacteristic
470C2×C4#55711yescharacteristic
471C8—42no—
472C8—42no—
473C8—42no—
474C8—42no—
475C22C2×#39611yescharacteristic
476C22#254511yescharacteristic
477C22#254711yescharacteristic
478C22#256511yescharacteristic
479C22#256811yescharacteristic
480C22#262111yescharacteristic
481C22#285011yescharacteristic
482C22#304512yes—
483C22#304612yes—
484C22#305711yescharacteristic
485C22#305711yescharacteristic
486C22#305811yescharacteristic
487C22#305911yescharacteristic
488C22#308112yes—
489C22#308212yes—
490C22#313612yes—
491C22#313712yes—
492C22#317812yes—
493C22#317912yes—
494C22#324312yes—
495C22#324312yes—
496C22#324412yes—
497C22#324412yes—
498C4—24no—
499C4—24no—
500C4—42no—
501C4—42no—
502C4—84no—
503C4—84no—
504C4—81no—
505C4—81no—
506C4—162no—
507C4—162no—
508C4#241511yescharacteristic
509C4#241511yescharacteristic
510C4#245311yescharacteristic
511C4#245311yescharacteristic
512C4#251411yescharacteristic
513C4#251411yescharacteristic
514C4#258111yescharacteristic
515C4#258111yescharacteristic
516C2#1523711yescharacteristic
517C2#2361211yescharacteristic
518C2#2361311yescharacteristic
519C2#2866812yes—
520C2#2873012yes—
521C2#2918912yes—
522C2#2922912yes—
523C2#2988011yescharacteristic
524C2#2988111yescharacteristic
525C2#3304511yescharacteristic
526C2#3304611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ 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×#19902
C2×#18352
C2×#17682
#142822
#142772
#113162
#113152
#112282
#112272
#108432