Finite Group Databasebeta

← #467315

Subgroups — #467315

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2058C211yesmaximal, characteristic
2C2×#1831C211yesmaximal, characteristic
3#14714C211yesmaximal, characteristic
4#14699C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11378C212yesmaximal
7#11316C212yesmaximal
8#11267C211yesmaximal, characteristic
9#11231C212yesmaximal
10#11227C212yesmaximal
11#11207C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C2×#391—21no—
16C2×#391—21no—
17C2×#389C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#387—21no—
20C2×#384C2211yescharacteristic
21C2×C43C2211yescharacteristic
22#2058—24no—
23#1831—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1436—21no—
35#1436—21no—
36C4⋊#95—21no—
37C4⋊#95—21no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44#1434—21no—
45C23×Q8—21no—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—42no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—42no—
76#391—44no—
77#391—44no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389Q814yescomplemented
82#387—44no—
83#384—42no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C43—22no—
99C43—22no—
100#340—24no—
101#340—24no—
102#340—42no—
103#340—42no—
104#340—42no—
105#340—42no—
106#340—42no—
107#336—24no—
108#336—22no—
109#336D414yes—
110C22×Q8—24no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—42no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8C2×D414yes—
144C4⋊Q8—48no—
145C4⋊Q8—48no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C2×Q812yes—
189C2×C4⋊C4C2×Q812yes—
190C2×C4⋊C4#4114yes—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—41no—
211C2×C42—41no—
212C2×C42SD1614yes—
213C2×C42C2×D412yes—
214C2×C42C2×Q812yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217#106—48no—
218C4⋊C8—48no—
219#95—24no—
220#95—42no—
221#95—42no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—42no—
226#95—42no—
227#95—42no—
228#95—42no—
229#95—42no—
230#95—42no—
231#95—44no—
232#95—42no—
233C24C22×D411yescharacteristic
234C2×Q8—48no—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—84no—
238C2×Q8—84no—
239C2×Q8—84no—
240C2×Q8—84no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4#12112yes—
314C22×C4#12112yes—
315C22×C4C22⋊Q812yes—
316C22×C4C22⋊Q812yes—
317C22×C4C2×SD1612yes—
318C22×C4C2×SD1612yes—
319C22×C4C2×SD1612yes—
320C22×C4#13512yes—
321C22×C4AGL(1,ℤ/8)12yes—
322C22×C4AGL(1,ℤ/8)12yes—
323C22×C4#13712yes—
324C22×C4#13712yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×Q811yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4112yes—
329C22×C4C2×#4111yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C42−1+412yes—
333C22×C42−1+412yes—
334C22×C42−1+411yescharacteristic
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—48no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—48no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C42—22no—
373C42—24no—
374C42—22no—
375C42—24no—
376C42—22no—
377C42—22no—
378C42—28no—
379C42—44no—
380C42—44no—
381C42—42no—
382C42—42no—
383C42—44no—
384C23C2×#12111yescharacteristic
385C23C2×C22⋊Q811yescharacteristic
386C23#52511yescharacteristic
387C23#53511yescharacteristic
388C23#53611yescharacteristic
389C23#53611yescharacteristic
390C23#53711yescharacteristic
391C23C22×SD1611yescharacteristic
392C23C2×#13511yescharacteristic
393C23C2×AGL(1,ℤ/8)11yescharacteristic
394C23C2×AGL(1,ℤ/8)11yescharacteristic
395C23C2×#13711yescharacteristic
396C23C2×#13711yescharacteristic
397C23#57511yescharacteristic
398C23#57511yescharacteristic
399Q8—816no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4#46514yes—
482C2×C4#46814yes—
483C2×C4#47614yes—
484C2×C4#53612yes—
485C2×C4#54112yes—
486C2×C4#54112yes—
487C2×C4#54212yes—
488C2×C4#54212yes—
489C2×C4#54412yes—
490C2×C4#54611yescharacteristic
491C2×C4#54612yes—
492C2×C4D4×Q811yescharacteristic
493C2×C4#55112yes—
494C2×C4#55211yescharacteristic
495C2×C4#55411yescharacteristic
496C2×C4#55612yes—
497C2×C4#55712yes—
498C2×C4#56212yes—
499C2×C4#56312yes—
500C8—44no—
501C8—82no—
502C22#241111yescharacteristic
503C22#241711yescharacteristic
504C22#258111yescharacteristic
505C22#262111yescharacteristic
506C22#262811yescharacteristic
507C22#263711yescharacteristic
508C22#282011yescharacteristic
509C22C2×#46511yescharacteristic
510C22C2×#46811yescharacteristic
511C22#305511yescharacteristic
512C22#305611yescharacteristic
513C22C2×#47611yescharacteristic
514C22#307611yescharacteristic
515C22#307811yescharacteristic
516C22#307911yescharacteristic
517C22#310311yescharacteristic
518C22#310811yescharacteristic
519C22#311011yescharacteristic
520C22#311411yescharacteristic
521C22#320611yescharacteristic
522C22#321111yescharacteristic
523C22#321511yescharacteristic
524C22#321711yescharacteristic
525C22#322711yescharacteristic
526C22#322811yescharacteristic
527C22#322911yescharacteristic
528C22#323111yescharacteristic
529C22#323511yescharacteristic
530C22#323611yescharacteristic
531C22#323911yescharacteristic
532C22#324011yescharacteristic
533C22#324611yescharacteristic
534C22#324711yescharacteristic
535C22#325111yescharacteristic
536C22#325511yescharacteristic
537C4—24no—
538C4—24no—
539C4—24no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—82no—
544C4—82no—
545C4—84no—
546C4—162no—
547C4—161no—
548C4—161no—
549C4#270012yes—
550C4#271312yes—
551C4#275512yes—
552C4#276212yes—
553C2#1656511yescharacteristic
554C2#2450311yescharacteristic
555C2#2450711yescharacteristic
556C2#2483611yescharacteristic
557C2#2484511yescharacteristic
558C2#2927611yescharacteristic
559C2#2928011yescharacteristic
560C2#2987711yescharacteristic
561C2#2988411yescharacteristic
562C2#2998111yescharacteristic
563C2#2998511yescharacteristic
564C2#3014911yescharacteristic
565C2#3015211yescharacteristic
566C2#3282011yescharacteristic
567C2#3282311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
81. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20582
C2×#18312
#147142
#146992
#113842
#113782
#113162
#112672
#112312
#112272
#112072