Finite Group Databasebeta

← #380116

Subgroups — #380116

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1894C211yesmaximal, characteristic
2C2×#1892C211yesmaximal, characteristic
3C2×#1729C211yesmaximal, characteristic
4#14264C211yesmaximal, characteristic
5#14258C211yesmaximal, characteristic
6#11316C212yesmaximal, complemented
7#11292C212yesmaximal, complemented
8#11190C212yesmaximal, complemented
9#11067C212yesmaximal, complemented
10#10722C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C2×#391C2212yes—
20C2×#389C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#1894C418yescomplemented
25#1892—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33#1729—24no—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1444—21no—
37#1442—21no—
38#1441—21no—
39#1441—21no—
40#1440—22no—
41#1439—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×Q8C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—42no—
49C22×M16—42no—
50C22×M16C2×C414yes—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—42no—
54C23×C8—42no—
55C23×C8D414yes—
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2×C414yes—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—42no—
68C2×#117—42no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D412yes—
83C22×C42C2312yes—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103C2×C418yes—
91#391—42no—
92#391—42no—
93#389—42no—
94#389—42no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380C2×C414yes—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111#340—22no—
112#340—22no—
113#340—42no—
114#340—42no—
115#340—42no—
116#340—42no—
117#340C2×C418yes—
118#336—24no—
119#336—24no—
120C25C2×D411yescharacteristic
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8C22×C412yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4C22×C412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4C2411yescharacteristic
151C2×M16—28no—
152C2×M16—48no—
153C2×M16—44no—
154C2×M16—44no—
155C2×M16—48no—
156C22×C8—28no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—44no—
172C22×C8—44no—
173C22×C8—48no—
174C22×C8—48no—
175C22×C8C22×C414yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22⋊Q8—28no—
179C22⋊Q8—28no—
180C22⋊Q8—44no—
181C22⋊Q8—44no—
182#117—44no—
183#117—44no—
184#117—48no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C414yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—81no—
231C2×C22⋊C4—81no—
232C2×C22⋊C4—81no—
233C2×C22⋊C4—81no—
234C2×C22⋊C4—82no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241#103—48no—
242#103—44no—
243#103—44no—
244#95—24no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—42no—
249#95—42no—
250#95—82no—
251#95—82no—
252#95—82no—
253#95—82no—
254C24—22no—
255C24—22no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—21no—
262C24C4×D412yes—
263C24#12112yes—
264C24#12412yes—
265C24C22×D411yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4#11714yes—
351C22×C4C4×D414yes—
352C22×C4C4×D412yes—
353C22×C4#12112yes—
354C22×C4#12314yes—
355C22×C4#12412yes—
356C22×C4#12614yes—
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4H(5,𝔽2)12yes—
364C22×C4H(5,𝔽2)12yes—
365C22×C4H(5,𝔽2)12yes—
366C22×C42−1+412yes—
367M16—416no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—416no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—48no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—88no—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C22⋊C4—84no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C23—24no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—22no—
423C23—24no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23#38814yes—
440C23C2×C4×D411yescharacteristic
441C23C2×#12111yescharacteristic
442C23C2×#12411yescharacteristic
443C23#53412yes—
444C23#53712yes—
445C23#57611yescharacteristic
446C23#57711yescharacteristic
447C23#57712yes—
448C23#57711yescharacteristic
449C23#57811yescharacteristic
450C23#57812yes—
451Q8—44no—
452Q8—42no—
453Q8—42no—
454Q8—81no—
455Q8—81no—
456Q8—82no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
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—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4#38814yes—
518C2×C4C2×#11711yescharacteristic
519C2×C4C2×C4×D411yescharacteristic
520C2×C4#51812yes—
521C2×C4#52012yes—
522C2×C4C2×#12311yescharacteristic
523C2×C4C2×#12611yescharacteristic
524C2×C4#53411yescharacteristic
525C2×C4#53411yescharacteristic
526C2×C4#53511yescharacteristic
527C2×C4#53611yescharacteristic
528C2×C4#53812yes—
529C2×C4#53812yes—
530C2×C4#53812yes—
531C2×C4#53812yes—
532C2×C4#53912yes—
533C2×C4#53912yes—
534C2×C4#53912yes—
535C2×C4#53912yes—
536C8—44no—
537C8—42no—
538C8—42no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22C2×#38811yescharacteristic
551C22#232512yes—
552C22#261711yescharacteristic
553C22#261811yescharacteristic
554C22#263912yes—
555C22#294611yescharacteristic
556C22#294712yes—
557C22#294711yescharacteristic
558C22#305711yescharacteristic
559C22#305911yescharacteristic
560C22#306011yescharacteristic
561C22#306111yescharacteristic
562C22#313611yescharacteristic
563C22#313712yes—
564C22#313711yescharacteristic
565C22#319812yes—
566C22#320112yes—
567C22#320112yes—
568C22#320112yes—
569C22#326012yes—
570C22#326112yes—
571C22#326212yes—
572C22#326312yes—
573C4—24no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—84no—
579C4—161no—
580C4—161no—
581C4—162no—
582C4#231812yes—
583C4#232111yescharacteristic
584C4#232211yescharacteristic
585C4#262211yescharacteristic
586C4#262311yescharacteristic
587C4#264312yes—
588C2—44no—
589C2#1425911yescharacteristic
590C2#2109011yescharacteristic
591C2#2109211yescharacteristic
592C2#2219812yes—
593C2#2220012yes—
594C2#2973711yescharacteristic
595C2#2974011yescharacteristic
596C2#2977911yescharacteristic
597C2#2978111yescharacteristic
598C2#3019512yes—
599C2#3019612yes—
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. #11316C2semidirect
7. #11292C2semidirect
8. #11190C2semidirect
9. #11067C2semidirect
24. #1894C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18942
C2×#18922
C2×#17292
#142642
#142582
#113162
#112922
#111902
#110672
#107222
#107212