Finite Group Databasebeta

← #380794

Subgroups — #380794

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C212yesmaximal
2C2×#1798C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#13795C211yesmaximal, characteristic
5#11325C212yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#11232C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#10730C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#380C2212yes—
21#1854—28no—
22#1798C418yescomplemented
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30#1441—21no—
31#1441—21no—
32#1439—21no—
33#1439—21no—
34C4⋊#95—21no—
35C4⋊#95—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C22⋊#95—21no—
40C22⋊#95—21no—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#117—22no—
46C2×#117—41no—
47C2×#117—41no—
48C2×#117—41no—
49C2×#117—41no—
50C2×#117—41no—
51C2×#117—41no—
52C2×#117—41no—
53C2×#117—41no—
54C2×#117C2×C414yes—
55C2×#117D414yes—
56C2×#117Q814yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—41no—
62C22×C4⋊C4—41no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4Q814yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—42no—
81C2×#107—42no—
82C2×#106—24no—
83C2×#106—42no—
84C2×#106—42no—
85C2×C4⋊C8—41no—
86C2×C4⋊C8—41no—
87C2×C4⋊C8—41no—
88C2×C4⋊C8—41no—
89C2×C4⋊C8C2×C414yescomplemented
90C2×C4⋊C8C2×C414yescomplemented
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8C2×C418yes—
94#384—41no—
95#384—41no—
96#382—41no—
97#382—41no—
98#380—24no—
99#380—24no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109#340—24no—
110#340—24no—
111#340—42no—
112#340—42no—
113#340—42no—
114#340—42no—
115#336C2×C418yescomplemented
116C25C2×D411yescharacteristic
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—82no—
133C23×C4—82no—
134C23×C4C22×C412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×Q812yes—
142C23×C4C2×Q812yes—
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8C22×C414yes—
159C22×C8C22×C414yes—
160#117—44no—
161#117—44no—
162#117—44no—
163#117—44no—
164#117—44no—
165#117—44no—
166#117—44no—
167#117—44no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
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—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C4⋊C414yes—
206C2×C4⋊C4C4⋊C414yes—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×Q814yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4C4⋊C414yes—
220C2×C22⋊C4C4⋊C414yes—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—24no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C22×C412yes—
233C2×C42C22×C412yes—
234C2×C42C2×Q812yes—
235C2×C42C2×Q812yes—
236#107—48no—
237#106—48no—
238C4⋊C8—44no—
239C4⋊C8—44no—
240C22⋊C8—216no—
241#95—42no—
242#95—42no—
243#95—44no—
244#95—82no—
245#95—82no—
246#95—82no—
247#95—82no—
248#95—82no—
249#95—82no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24C2×C4⋊C412yes—
259C24#12112yes—
260C24C22⋊Q812yes—
261C24C22×D411yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
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—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—48no—
309C22×C4—42no—
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—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—88no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4—88no—
339C22×C4C2×C4⋊C412yes—
340C22×C4C2×C4⋊C412yes—
341C22×C4C2×C4⋊C412yes—
342C22×C4C2×C4⋊C412yes—
343C22×C4C2×C4⋊C412yes—
344C22×C4C4×D414yes—
345C22×C4C4×Q814yes—
346C22×C4#12112yes—
347C22×C4C22⋊Q812yes—
348C22×C4#12514yes—
349C22×C4C4⋊Q814yes—
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C22×Q811yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C42−1+411yescharacteristic
363C22×C42−1+411yescharacteristic
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—48no—
402C22⋊C4—88no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—48no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23#38914yes—
435C23C22×C4⋊C411yescharacteristic
436C23C2×#12111yescharacteristic
437C23C2×C22⋊Q811yescharacteristic
438C23#53411yescharacteristic
439C23#53411yescharacteristic
440C23#53511yescharacteristic
441C23#53611yescharacteristic
442C23#57612yes—
443C23#57712yes—
444C23#57712yes—
445C23#57812yes—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—816no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4#38914yes—
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×C4×Q811yescharacteristic
512C2×C4#51811yescharacteristic
513C2×C4#51811yescharacteristic
514C2×C4#52011yescharacteristic
515C2×C4#52011yescharacteristic
516C2×C4C2×#12511yescharacteristic
517C2×C4C2×C4⋊Q811yescharacteristic
518C2×C4#53411yescharacteristic
519C2×C4#53411yescharacteristic
520C2×C4#53711yescharacteristic
521C2×C4#53711yescharacteristic
522C2×C4#53811yescharacteristic
523C2×C4#53811yescharacteristic
524C2×C4#53811yescharacteristic
525C2×C4#53811yescharacteristic
526C2×C4#53911yescharacteristic
527C2×C4#53911yescharacteristic
528C2×C4#53911yescharacteristic
529C2×C4#53911yescharacteristic
530C2×C4#54311yescharacteristic
531C2×C4#54311yescharacteristic
532C2×C4#54311yescharacteristic
533C2×C4#54311yescharacteristic
534C2×C4#54411yescharacteristic
535C2×C4#54411yescharacteristic
536C2×C4#54411yescharacteristic
537C2×C4#54411yescharacteristic
538C8—44no—
539C8—44no—
540C22—28no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—44no—
545C22—44no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22C2×#38911yescharacteristic
552C22#231511yescharacteristic
553C22#231711yescharacteristic
554C22#261711yescharacteristic
555C22#261811yescharacteristic
556C22#262611yescharacteristic
557C22#262711yescharacteristic
558C22#291212yes—
559C22#291312yes—
560C22#304512yes—
561C22#304612yes—
562C22#308112yes—
563C22#308212yes—
564C22#319911yescharacteristic
565C22#320011yescharacteristic
566C22#320011yescharacteristic
567C22#320211yescharacteristic
568C22#320312yes—
569C22#320412yes—
570C22#322412yes—
571C22#322512yes—
572C22#324112yes—
573C22#324212yes—
574C4—24no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—84no—
580C4—84no—
581C4—162no—
582C4—162no—
583C4#232511yescharacteristic
584C4#232511yescharacteristic
585C4#232911yescharacteristic
586C4#232911yescharacteristic
587C4#264611yescharacteristic
588C4#264611yescharacteristic
589C4#265911yescharacteristic
590C4#265911yescharacteristic
591C2—44no—
592C2#1427511yescharacteristic
593C2#2117512yes—
594C2#2194611yescharacteristic
595C2#2194711yescharacteristic
596C2#2199212yes—
597C2#2972912yes—
598C2#2976511yescharacteristic
599C2#2976611yescharacteristic
600C2#2995911yescharacteristic
601C2#2996011yescharacteristic
602C2#2997812yes—
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
5. #11325C2semidirect
6. #11267C2semidirect
7. #11232C2semidirect
8. #11231C2semidirect
22. #1798C4semidirect
89. C2×C4⋊C8C2×C4semidirect
90. C2×C4⋊C8C2×C4semidirect
115. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
C2×#17982
#143402
#137952
#113252
#112672
#112322
#112312
#107302