Finite Group Databasebeta

← #356864

Subgroups — #356864

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4C2×C8×C4⋊C4C211yesmaximal, characteristic
5C4⋊C42C211yesmaximal, characteristic
6#10885C212yesmaximal
7#10880C212yesmaximal
8#10849C211yesmaximal, characteristic
9#10807C212yesmaximal
10C4×#340C212yesmaximal
11C4×#340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4×C8C2212yes—
15C22×C4×C8C2211yescharacteristic
16#2369—24no—
17#2369C418yescomplemented
18C2×#391—22no—
19C2×#389—21no—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4—21no—
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1990C418yescomplemented
30#1987C418yescomplemented
31C8×C4⋊C4—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#336C2212yes—
43#1436—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C23×Q8—21no—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8C2×C414yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42—21no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8C2×C414yes—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—24no—
80C2×C4×C8—24no—
81C2×C4×C8—42no—
82C2×C4×C8C2×C414yes—
83#391—216no—
84#389—28no—
85#389—24no—
86#389C2×C418yes—
87#384—22no—
88#384C2×C414yes—
89C4×C4⋊C4—28no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—42no—
95C4×C4⋊C4—44no—
96C4×C4⋊C4C2×C418yes—
97C4×C4⋊C4C2×C414yes—
98C4×C4⋊C4C2×C414yes—
99C4×C4⋊C4C2×C414yes—
100C4×C4⋊C4D414yes—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95D412yes—
109#340—22no—
110#340—22no—
111#340—24no—
112#340—22no—
113#340—42no—
114#340C2×C418yes—
115#340C2×C418yes—
116#340C2×C418yes—
117#340C2×C418yes—
118#340Q814yescomplemented
119#336—24no—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8—81no—
124C22×Q8—81no—
125C22×Q8—81no—
126C22×Q8—81no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2411yescharacteristic
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8C22×C412yes—
155C22×C8C22×C414yes—
156C22×C8C22×C412yes—
157C22×C8C2×Q812yes—
158C22×C8#4112yes—
159C4⋊Q8—28no—
160C4⋊Q8—28no—
161C4⋊Q8—28no—
162C4⋊Q8C42116yescomplemented
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C22⋊C414yes—
202C2×C4⋊C4C22⋊C414yes—
203C2×C4⋊C4SD1614yes—
204C2×C4⋊C4Q1614yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C414yes—
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C22×C414yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C2×Q812yes—
214C2×C4⋊C4C2×Q812yes—
215C2×C4⋊C4#4112yes—
216C2×C4⋊C4#4112yes—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—28no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—82no—
231C2×C42—82no—
232C2×C42—82no—
233C2×C42C22×C414yes—
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C22×C412yes—
237C2×C42C2×D414yes—
238C2×C42C2×D412yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C4⋊C8—28no—
242C4×C8—28no—
243C4×C8—48no—
244#95—28no—
245#95—28no—
246#95—24no—
247#95—24no—
248#95—44no—
249#95—44no—
250#95—82no—
251#95C22⋊C418yes—
252C24C22×D411yescharacteristic
253C2×Q8—216no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8—84no—
258C2×Q8—84no—
259C2×Q8—84no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4C2×C22⋊C414yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C414yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C4×D412yes—
321C22×C4C4×D412yes—
322C22×C4C4×D412yes—
323C22×C4C4×D412yes—
324C22×C4C2×SD1614yes—
325C22×C4C2×SD1612yes—
326C22×C4C2×Q1614yes—
327C22×C4C2×Q1612yes—
328C22×C4#13512yes—
329C22×C4#13512yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C23×C411yescharacteristic
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×Q811yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343C22×C42−1+411yescharacteristic
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8C4×Q814yes—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—216no—
375C4⋊C4—28no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—48no—
389C4⋊C4—84no—
390C4⋊C4—88no—
391C4⋊C4#103116yes—
392C4⋊C4C2×C4218yes—
393C4⋊C4C4×Q814yes—
394C4⋊C4C4×Q814yes—
395C42—24no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—48no—
400C42—48no—
401C42—88no—
402C42—88no—
403C42C2×C4214yes—
404C42#11714yes—
405C42C4×D414yes—
406C42C4×D414yes—
407C23C22×C22⋊C411yescharacteristic
408C23C2×C4×D411yescharacteristic
409C23C2×C4×D411yescharacteristic
410C23#54611yescharacteristic
411C23#54711yescharacteristic
412C23#54811yescharacteristic
413C23D4×Q811yescharacteristic
414C23C22×SD1611yescharacteristic
415C23C22×Q1611yescharacteristic
416C23C2×#13511yescharacteristic
417C23C2×#13511yescharacteristic
418C23#57611yescharacteristic
419C23#57611yescharacteristic
420C23#57711yescharacteristic
421C23#57711yescharacteristic
422Q8—816no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4C4×C22⋊C414yes—
485C2×C4C4×C22⋊C414yes—
486C2×C4C2×#10318yes—
487C2×C4C2×#10314yes—
488C2×C4#41614yes—
489C2×C4C4×SD1614yes—
490C2×C4C4×SD1614yes—
491C2×C4C4×Q1614yes—
492C2×C4C4×Q1614yes—
493C2×C4C22×C4211yescharacteristic
494C2×C4C2×#11711yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×C4×Q811yescharacteristic
498C2×C4C2×C4×Q811yescharacteristic
499C2×C4C2×C4×Q811yescharacteristic
500C2×C4C4×#4111yescharacteristic
501C2×C4C4×#4111yescharacteristic
502C2×C4C4×#4111yescharacteristic
503C2×C4#51911yescharacteristic
504C2×C4#51911yescharacteristic
505C2×C4#51911yescharacteristic
506C2×C4#52011yescharacteristic
507C2×C4#52011yescharacteristic
508C2×C4#52011yescharacteristic
509C2×C4#54611yescharacteristic
510C2×C4#54711yescharacteristic
511C2×C4#54811yescharacteristic
512C2×C4D4×Q811yescharacteristic
513C2×C4#55211yescharacteristic
514C2×C4#55211yescharacteristic
515C2×C4#55411yescharacteristic
516C2×C4#55611yescharacteristic
517C2×C4#55611yescharacteristic
518C2×C4#55611yescharacteristic
519C2×C4#55711yescharacteristic
520C2×C4#55911yescharacteristic
521C8—24no—
522C8—42no—
523C8—44no—
524C22C2×C4×C22⋊C411yescharacteristic
525C22#233911yescharacteristic
526C22Q8×C22⋊C411yescharacteristic
527C22#236011yescharacteristic
528C22#236211yescharacteristic
529C22#236511yescharacteristic
530C22#236711yescharacteristic
531C22C22×#10311yescharacteristic
532C22C2×#41611yescharacteristic
533C22#289711yescharacteristic
534C22#289711yescharacteristic
535C22C2×C4×SD1611yescharacteristic
536C22C2×C4×SD1611yescharacteristic
537C22C2×C4×Q1611yescharacteristic
538C22C2×C4×Q1611yescharacteristic
539C22#294611yescharacteristic
540C22#294611yescharacteristic
541C22#294811yescharacteristic
542C22#294911yescharacteristic
543C22#329511yescharacteristic
544C22#329911yescharacteristic
545C22#330311yescharacteristic
546C22#330511yescharacteristic
547C22#333511yescharacteristic
548C22#333811yescharacteristic
549C22#334011yescharacteristic
550C22#334111yescharacteristic
551C22#336011yescharacteristic
552C22#336111yescharacteristic
553C22#336211yescharacteristic
554C22#336311yescharacteristic
555C22Q8×SD1611yescharacteristic
556C22#338011yescharacteristic
557C22#338111yescharacteristic
558C22Q8×Q1611yescharacteristic
559C4—24no—
560C4—28no—
561C4—44no—
562C4—48no—
563C4—44no—
564C4—44no—
565C4—82no—
566C4—84no—
567C4—82no—
568C4C4×#10318yes—
569C4C42×Q811yescharacteristic
570C4#227611yescharacteristic
571C4#235511yescharacteristic
572C4#235611yescharacteristic
573C4#236011yescharacteristic
574C4#236211yescharacteristic
575C4#236511yescharacteristic
576C4#236711yescharacteristic
577C2#1381711yescharacteristic
578C2C2×C4×#10311yescharacteristic
579C2#2041611yescharacteristic
580C2#2248711yescharacteristic
581C2#2248911yescharacteristic
582C2#2251911yescharacteristic
583C2Q8×#10311yescharacteristic
584C2#2296311yescharacteristic
585C2#2296511yescharacteristic
586C2#2306111yescharacteristic
587C2#2306311yescharacteristic
588C2#2315611yescharacteristic
589C2#2315811yescharacteristic
590C2#2321511yescharacteristic
591C2#2321711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #2369C4semidirect
29. #1990C4semidirect
30. #1987C4semidirect
118. #340Q8semidirect
162. C4⋊Q8C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#19902
C2×#19872
C2×C8×C4⋊C42
C4⋊C422
#108852
#108802
#108492
#108072
C4×#3402