Finite Group Databasebeta

← #374513

Subgroups — #374513

Order 512.

1249 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#21332C211yesmaximal, characteristic
4#21306C211yesmaximal, characteristic
5C4×#419C211yesmaximal, characteristic
6#11184C214yesmaximal
7#10903C214yesmaximal
8#10855C211yesmaximal, characteristic
9#10836C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3078—22no—
13#3078—22no—
14#3078—22no—
15#3078—22no—
16C2×#479C414yescomplemented
17C2×#477C412yescomplemented
18C2×#475C412yescomplemented
19C2×#428C2212yes—
20C2×#423C2211yescharacteristic
21C2×#419C2212yes—
22C2×#419C2212yes—
23C2×C4×M16C2211yescharacteristic
24#2356—22no—
25#2356—22no—
26#2355—22no—
27#2355—22no—
28#2313—22no—
29#2310C2212yes—
30#2285C2211yescharacteristic
31#2285C2211yescharacteristic
32#2283—22no—
33#2274C2212yes—
34C4×#117C2211yescharacteristic
35#1996—22no—
36#1996—22no—
37#1986—22no—
38#1986—22no—
39#1763—22no—
40C4×#103—22no—
41C2×#344C2214yes—
42#1740C2214yes—
43#1736C2212yes—
44#1733C2214yes—
45C2×H(3,ℤ/4)C2214yes—
46C23×Q8C2311yescharacteristic
47C22×M16C2312yes—
48C22×M16C2312yes—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125C2×C412yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8C2×C412yes—
71C2×C4×Q8C2×C412yes—
72C2×C4×Q8D412yes—
73C2×C4×Q8D412yes—
74C2×C4×Q8D412yes—
75C2×C4×Q8D412yes—
76C2×#117C2312yes—
77C2×#117C2312yes—
78C2×#117C2312yes—
79C2×#117C2312yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#479—28no—
85#479—28no—
86#477—24no—
87#477—24no—
88#475—24no—
89#475—24no—
90#428—24no—
91#428—24no—
92C2×#107C2×C414yes—
93C2×#106C2×C414yes—
94#423—21no—
95#423—21no—
96#423—21no—
97#423—21no—
98C2×C4⋊C8C2×C412yes—
99C2×C4⋊C8C2×C412yes—
100#419—24no—
101#419—24no—
102#419—24no—
103#419—24no—
104C2×#103—24no—
105C2×#103C2×C414yes—
106C2×#103C2×C414yes—
107C4×M16—21no—
108C4×M16—21no—
109C4×M16—21no—
110C4×M16—21no—
111C2×#97—21no—
112C2×C4×C8—21no—
113#391—24no—
114#391—24no—
115#387—28no—
116#387—28no—
117#384—21no—
118#384—21no—
119#383—24no—
120#382—24no—
121#382—21no—
122#382—21no—
123#381—24no—
124#380—21no—
125#380—21no—
126#380D412yes—
127#380D412yes—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4D412yes—
136C4×C4⋊C4D412yes—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140#376—24no—
141#376—21no—
142C43—21no—
143#344—28no—
144#342—28no—
145#340—24no—
146#339—28no—
147H(3,ℤ/4)—28no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8C22×C412yes—
158C22×Q8C2×D412yes—
159C22×Q8C2×D412yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2411yescharacteristic
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—24no—
168C2×M16—24no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16#4114yes—
175C22×C8—22no—
176C22×C8C22×C414yes—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185#125—22no—
186#125—22no—
187#125—22no—
188#125—22no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8C22⋊C414yes—
218C4×Q8C22⋊C414yes—
219C4×Q8C22⋊C414yes—
220C4×Q8C22⋊C414yes—
221#117—24no—
222#117—24no—
223#117—22no—
224#117—24no—
225#117—24no—
226#117—22no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117#4114yes—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C412yes—
255C2×C4⋊C4C22×C412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4C2×D414yes—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—28no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D414yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42#4111yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42#4111yescharacteristic
286C2×C42#4111yescharacteristic
287#107—28no—
288#106—28no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293#103—216no—
294#103—28no—
295#103—28no—
296#97—41no—
297#97—41no—
298C4×C8—41no—
299C4×C8—41no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—41no—
308#95—41no—
309C24C22×D411yescharacteristic
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×Q8C2×C22⋊C414yes—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—81no—
349C22×C4—82no—
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12011yescharacteristic
356C22×C4#12011yescharacteristic
357C22×C4#12011yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4C2×#4112yes—
369C22×C4C2×#4111yescharacteristic
370C22×C42−1+412yes—
371C22×C42−1+412yes—
372M16—24no—
373M16—24no—
374M16—44no—
375M16—42no—
376M16—42no—
377M16—42no—
378M16—42no—
379C2×C8—24no—
380C2×C8—28no—
381C2×C8—24no—
382C2×C8—42no—
383C2×C8—42no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4C2×C22⋊C414yes—
413C22⋊C4—28no—
414C22⋊C4—24no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C42—24no—
420C42—24no—
421C42—24no—
422C42—24no—
423C42—28no—
424C42—24no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—82no—
431C42—88no—
432C42—82no—
433C42C2×C22⋊C411yescharacteristic
434C42C2×C22⋊C411yescharacteristic
435C42C2×C22⋊C411yescharacteristic
436C42C2×C22⋊C411yescharacteristic
437C42C4×D411yescharacteristic
438C42C4×D411yescharacteristic
439C42C4×D411yescharacteristic
440C42C4×D411yescharacteristic
441C42#12111yescharacteristic
442C42#12111yescharacteristic
443C42#12111yescharacteristic
444C42#12111yescharacteristic
445C42#12311yescharacteristic
446C42#12311yescharacteristic
447C42#12311yescharacteristic
448C42#12311yescharacteristic
449C23—41no—
450C23—41no—
451C23C2×C4×D411yescharacteristic
452C23C2×#12011yescharacteristic
453C23C2×#12111yescharacteristic
454C23#54712yes—
455C23#54812yes—
456Q8—24no—
457Q8—24no—
458Q8—44no—
459Q8—44no—
460Q8—42no—
461Q8—42no—
462Q8—82no—
463Q8—82no—
464Q8—82no—
465Q8—82no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4#38611yescharacteristic
501C2×C4#38611yescharacteristic
502C2×C4#38611yescharacteristic
503C2×C4#38611yescharacteristic
504C2×C4#38611yescharacteristic
505C2×C4#38611yescharacteristic
506C2×C4#38611yescharacteristic
507C2×C4#38611yescharacteristic
508C2×C4C22×C22⋊C411yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C4×#4112yes—
511C2×C4#51912yes—
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12311yescharacteristic
514C2×C4#52512yes—
515C2×C4#53211yescharacteristic
516C2×C4#53211yescharacteristic
517C2×C4#53612yes—
518C2×C4#53611yescharacteristic
519C2×C4#53611yescharacteristic
520C2×C4#54712yes—
521C2×C4#54812yes—
522C2×C4#55212yes—
523C2×C4#55212yes—
524C2×C4#55612yes—
525C2×C4#55612yes—
526C8—44no—
527C8—44no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—42no—
535C22—42no—
536C22C2×#38611yescharacteristic
537C22#236212yes—
538C22#241512yes—
539C22#243211yescharacteristic
540C22#243211yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—44no—
549C4—44no—
550C4—84no—
551C4—84no—
552C4—84no—
553C4—82no—
554C4—82no—
555C4#190512yes—
556C4#229711yescharacteristic
557C4#231011yescharacteristic
558C4#236212yes—
559C4#241512yes—
560C4#249911yescharacteristic
561C4#249911yescharacteristic
562C2—21no—
563C2—21no—
564C2—22no—
565C2—22no—
566C2#1413811yescharacteristic
567C2C2×#190511yescharacteristic
568C2#2099811yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. C2×#479C4semidirect
17. C2×#477C4semidirect
18. C2×#475C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#341872
#213322
#213062
C4×#4192
#111842
#109032
#108552
#108362