Finite Group Databasebeta

← #409727

Subgroups — #409727

Order 512.

1040 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14714C211yesmaximal, characteristic
2#14699C211yesmaximal, characteristic
3#11384C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic
7#11315C212yesmaximal
8#11292C212yesmaximal
9#10881C212yesmaximal
10#10821C212yesmaximal
11#10693C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#97C2212yes—
14C2×#391—21no—
15C2×#391—21no—
16C2×#389C2211yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#387—21no—
19C2×#384C2211yescharacteristic
20C2×C43C2211yescharacteristic
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32#1436—21no—
33#1436—21no—
34C4⋊#95—21no—
35C4⋊#95—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42#1434—21no—
43C23×Q8—21no—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—42no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#97—24no—
69C2×#97—24no—
70C2×#97—42no—
71#391—44no—
72#391—44no—
73#389—22no—
74#389—42no—
75#389Q814yescomplemented
76#387—44no—
77#384—22no—
78#384—22no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C43—42no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—42no—
101#340—42no—
102#336—24no—
103#336—22no—
104#336—22no—
105C22×Q8—24no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—42no—
111C22×Q8—42no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8#4114yes—
138C22×C8#4114yes—
139C4⋊Q8—48no—
140C4⋊Q8—48no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
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⋊C4C2×Q812yes—
182C2×C4⋊C4C2×Q812yes—
183C2×C4⋊C4#4114yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42C2×Q812yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208#97—48no—
209#95—24no—
210#95—42no—
211#95—42no—
212#95—42no—
213#95—42no—
214#95—44no—
215#95—44no—
216#95—42no—
217#95—42no—
218#95—42no—
219#95—44no—
220#95—44no—
221#95—42no—
222#95—42no—
223C24C22×D411yescharacteristic
224C2×Q8—48no—
225C2×Q8—48no—
226C2×Q8—48no—
227C2×Q8—84no—
228C2×Q8—84no—
229C2×Q8—84no—
230C2×Q8—84no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
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—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4C22⋊Q812yes—
306C22×C4C22⋊Q812yes—
307C22×C4#12412yes—
308C22×C4#12412yes—
309C22×C4AGL(1,ℤ/8)12yes—
310C22×C4AGL(1,ℤ/8)12yes—
311C22×C4AGL(1,ℤ/8)12yes—
312C22×C4#13712yes—
313C22×C4#13712yes—
314C22×C4#13712yes—
315C22×C4C22×Q811yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4112yes—
321C22×C4C2×#4112yes—
322C22×C4H(5,𝔽2)12yes—
323C22×C42−1+411yescharacteristic
324C22×C42−1+411yescharacteristic
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—44no—
365C42—44no—
366C42—44no—
367C42—44no—
368C42—42no—
369C42—42no—
370C23C2×C22⋊Q811yescharacteristic
371C23#52511yescharacteristic
372C23C2×#12411yescharacteristic
373C23#53211yescharacteristic
374C23#53211yescharacteristic
375C23#53411yescharacteristic
376C23#53611yescharacteristic
377C23C2×AGL(1,ℤ/8)11yescharacteristic
378C23C2×AGL(1,ℤ/8)11yescharacteristic
379C23C2×AGL(1,ℤ/8)11yescharacteristic
380C23C2×#13711yescharacteristic
381C23C2×#13711yescharacteristic
382C23C2×#13711yescharacteristic
383C23#57511yescharacteristic
384C23#57511yescharacteristic
385Q8—816no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
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—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4#48914yes—
469C2×C4#49114yes—
470C2×C4#52912yes—
471C2×C4#52912yes—
472C2×C4#52912yes—
473C2×C4#53312yes—
474C2×C4#53912yes—
475C2×C4#54212yes—
476C2×C4#55112yes—
477C2×C4#55112yes—
478C2×C4#55112yes—
479C2×C4#55212yes—
480C2×C4#55412yes—
481C2×C4#55612yes—
482C2×C4#55611yescharacteristic
483C2×C4#55611yescharacteristic
484C2×C4#55711yescharacteristic
485C2×C4#55711yescharacteristic
486C8—44no—
487C8—44no—
488C22#239611yescharacteristic
489C22#256111yescharacteristic
490C22#256411yescharacteristic
491C22#256811yescharacteristic
492C22#260411yescharacteristic
493C22#266311yescharacteristic
494C22#286111yescharacteristic
495C22#307711yescharacteristic
496C22#307811yescharacteristic
497C22#307911yescharacteristic
498C22#307911yescharacteristic
499C22#310911yescharacteristic
500C22#311011yescharacteristic
501C22#311711yescharacteristic
502C22#311711yescharacteristic
503C22C2×#48911yescharacteristic
504C22C2×#49111yescharacteristic
505C22#313811yescharacteristic
506C22#314011yescharacteristic
507C22#317111yescharacteristic
508C22#317111yescharacteristic
509C22#318011yescharacteristic
510C22#318011yescharacteristic
511C22#318211yescharacteristic
512C22#318211yescharacteristic
513C22#318511yescharacteristic
514C22#318511yescharacteristic
515C22#319211yescharacteristic
516C22#319311yescharacteristic
517C22#319611yescharacteristic
518C22#319611yescharacteristic
519C22#323511yescharacteristic
520C22#323711yescharacteristic
521C22#323811yescharacteristic
522C22#323811yescharacteristic
523C4—24no—
524C4—24no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—84no—
530C4—82no—
531C4—82no—
532C4—162no—
533C4—161no—
534C4—161no—
535C4#252012yes—
536C4#252312yes—
537C4#252712yes—
538C4#258812yes—
539C2#1505311yescharacteristic
540C2#2378011yescharacteristic
541C2#2378211yescharacteristic
542C2#2910911yescharacteristic
543C2#2911211yescharacteristic
544C2#2916211yescharacteristic
545C2#2916611yescharacteristic
546C2#2922411yescharacteristic
547C2#2922611yescharacteristic
548C2#2951911yescharacteristic
549C2#2952111yescharacteristic
550C2#3036411yescharacteristic
551C2#3036711yescharacteristic
552C2#3328011yescharacteristic
553C2#3328711yescharacteristic
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
75. #389Q8semidirect

Maximal subgroups

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

subgroupindex
#147142
#146992
#113842
#113712
#113152
#112922
#108812
#108212
#106932