Finite Group Databasebeta

← #380302

Subgroups — #380302

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1825C211yesmaximal, characteristic
4C2×#1825C211yesmaximal, characteristic
5#14264C211yesmaximal, characteristic
6#11328C212yesmaximal, complemented
7#11328C212yesmaximal, complemented
8#11293C212yesmaximal, complemented
9#11292C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2317—24no—
19#2317—24no—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#385—22no—
24C2×#384—21no—
25C2×#382—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28#1852—24no—
29#1825—24no—
30#1825C418yescomplemented
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—21no—
43#1442—21no—
44#1439—22no—
45C4⋊#95C2212yes—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—42no—
57C2×#117—42no—
58C2×#117C2×C414yes—
59C2×#117D414yes—
60C2×#117Q814yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D414yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D412yes—
81C22×C42—21no—
82C22×C42C2312yes—
83C2×#107—24no—
84C2×#107—42no—
85C2×#107—42no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8C2×C414yes—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—24no—
96#389—42no—
97#389—42no—
98#385—216no—
99#384—28no—
100#382—28no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380C2×C414yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—24no—
117#340—42no—
118#340—42no—
119#340C2×C418yescomplemented
120#340C2×C418yescomplemented
121#336—42no—
122#336—42no—
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—44no—
136C23×C4—82no—
137C23×C4—81no—
138C23×C4—81no—
139C23×C4C22×C412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×Q812yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C22×C414yes—
166#117—28no—
167#117—28no—
168#117—28no—
169#117—48no—
170#117—44no—
171#117—44no—
172#117C4⋊C4116yescomplemented
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—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—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C22×C414yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×Q812yes—
222C2×C4⋊C4C2×Q812yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—81no—
232C2×C22⋊C4—81no—
233C2×C22⋊C4—82no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243#107—48no—
244C22⋊C8—28no—
245C22⋊C8—44no—
246#95—28no—
247#95—28no—
248#95—24no—
249#95—44no—
250#95—42no—
251#95—82no—
252#95—82no—
253C24—22no—
254C24—22no—
255C24—21no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24C4×D412yes—
262C24#12012yes—
263C24C22⋊Q812yes—
264C24C22×D411yescharacteristic
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—48no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4—88no—
335C22×C4—88no—
336C22×C4C2×C4⋊C414yes—
337C22×C4C4×D414yes—
338C22×C4C4×D412yes—
339C22×C4#12012yes—
340C22×C4C22⋊Q814yes—
341C22×C4C22⋊Q812yes—
342C22×C4#12314yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4H(5,𝔽2)12yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+412yes—
352C22×C42−1+412yes—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—48no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—88no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C4⋊C4C2×C4⋊C414yes—
397C4⋊C4C2×C4⋊C414yes—
398C22⋊C4—216no—
399C22⋊C4—48no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C42—216no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—42no—
407C23—22no—
408C23—24no—
409C23—22no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23#38514yes—
430C23C2×C4×D411yescharacteristic
431C23C2×#12011yescharacteristic
432C23C2×C22⋊Q811yescharacteristic
433C23#53412yes—
434C23#53612yes—
435C23#57612yes—
436C23#57611yescharacteristic
437C23#57711yescharacteristic
438C23#57711yescharacteristic
439C23#57812yes—
440C23#57811yescharacteristic
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—816no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4#38514yes—
503C2×C4C22×C4⋊C411yescharacteristic
504C2×C4C2×C4×D411yescharacteristic
505C2×C4#52012yes—
506C2×C4#52012yes—
507C2×C4C2×C22⋊Q811yescharacteristic
508C2×C4C2×#12311yescharacteristic
509C2×C4#53412yes—
510C2×C4#53511yescharacteristic
511C2×C4#53511yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#53611yescharacteristic
514C2×C4#53612yes—
515C2×C4#53912yes—
516C2×C4#53912yes—
517C2×C4#53912yes—
518C2×C4#53912yes—
519C2×C4#54412yes—
520C2×C4#54412yes—
521C8—44no—
522C8—81no—
523C8—81no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22C2×#38511yescharacteristic
536C22#232212yes—
537C22#261311yescharacteristic
538C22#261411yescharacteristic
539C22#262712yes—
540C22#294611yescharacteristic
541C22#294711yescharacteristic
542C22#295112yes—
543C22#300811yescharacteristic
544C22#301111yescharacteristic
545C22#301211yescharacteristic
546C22#301311yescharacteristic
547C22#308311yescharacteristic
548C22#308312yes—
549C22#308411yescharacteristic
550C22#319812yes—
551C22#319912yes—
552C22#320112yes—
553C22#320212yes—
554C22#324112yes—
555C22#324212yes—
556C22#324312yes—
557C22#324312yes—
558C4—24no—
559C4—44no—
560C4—44no—
561C4—84no—
562C4—82no—
563C4—84no—
564C4—82no—
565C4—161no—
566C4—161no—
567C4#231711yescharacteristic
568C4#231711yescharacteristic
569C4#232212yes—
570C4#262311yescharacteristic
571C4#262311yescharacteristic
572C4#262712yes—
573C2—44no—
574C2#1426311yescharacteristic
575C2#2087811yescharacteristic
576C2#2088111yescharacteristic
577C2#2208812yes—
578C2#2209412yes—
579C2#2960211yescharacteristic
580C2#2960511yescharacteristic
581C2#2964611yescharacteristic
582C2#2964911yescharacteristic
583C2#2997612yes—
584C2#2997912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11328C2semidirect
7. #11328C2semidirect
8. #11293C2semidirect
9. #11292C2semidirect
30. #1825C4semidirect
119. #340C2×C4semidirect
120. #340C2×C4semidirect
172. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#18522
C2×#18252
#142642
#113282
#112932
#112922
#107302
#107132