Finite Group Databasebeta

← #382615

Subgroups — #382615

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1825C212yesmaximal
3C2×#1798C211yesmaximal, characteristic
4#14340C211yesmaximal, characteristic
5#11228C212yesmaximal, complemented
6#11227C212yesmaximal, complemented
7#11208C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#10730C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16#2317—24no—
17#2317C418yescomplemented
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#385—21no—
21C2×#385—21no—
22C2×#384—21no—
23C2×#384C2211yescharacteristic
24C2×#382—21no—
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27#1825—28no—
28#1798—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1441—21no—
39#1441—21no—
40#1439—21no—
41#1439—21no—
42C4⋊#95C2212yes—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—41no—
51C2×#117—41no—
52C2×#117—41no—
53C2×#117—41no—
54C2×#117C2×C414yes—
55C2×#117C2×C414yes—
56C2×#117D414yes—
57C2×#117D414yes—
58C2×#117Q814yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—41no—
66C22×C4⋊C4—41no—
67C22×C4⋊C4—41no—
68C22×C4⋊C4—41no—
69C22×C4⋊C4C2×C414yes—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—41no—
86C2×C4⋊C8—41no—
87C2×C4⋊C8—41no—
88C2×C4⋊C8—41no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92#389—24no—
93#389—24no—
94#389C2×C418yes—
95#389C2×C418yes—
96#385—28no—
97#385—28no—
98#384—28no—
99#384—41no—
100#384—41no—
101#382—28no—
102#382—41no—
103#382—41no—
104#380—24no—
105#380—24no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#340—42no—
120#340—42no—
121#336—24no—
122C25C2×D411yescharacteristic
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C22×C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×Q812yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166#117—28no—
167#117—28no—
168#117—28no—
169#117—44no—
170#117—44no—
171#117—44no—
172#117—44no—
173#117C4⋊C4116yescomplemented
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
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—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
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⋊C414yes—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C22×C414yes—
219C2×C4⋊C4C2×D414yes—
220C2×C4⋊C4C2×Q814yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4C22⋊C414yes—
232C2×C22⋊C4C22⋊C414yes—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42C22×C412yes—
245C2×C42C22×C412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C4⋊C8—44no—
249C4⋊C8—44no—
250C22⋊C8—48no—
251#95—28no—
252#95—28no—
253#95—42no—
254#95—42no—
255#95—44no—
256#95—82no—
257#95—82no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24C2×C22⋊C412yes—
267C24#12112yes—
268C24C22⋊Q812yes—
269C24C22×D411yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
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—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
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—24no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—48no—
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—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—88no—
344C22×C4—88no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C4⋊C414yes—
353C22×C4#11714yes—
354C22×C4#12114yes—
355C22×C4#12112yes—
356C22×C4C22⋊Q814yes—
357C22×C4C22⋊Q812yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×Q811yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4H(5,𝔽2)11yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C42−1+411yescharacteristic
369C22×C42−1+411yescharacteristic
370C22×C42−1+411yescharacteristic
371C22×C42−1+411yescharacteristic
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
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—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—88no—
405C4⋊C4—88no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C4⋊C4C2×C4⋊C418yes—
409C22⋊C4—216no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—88no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—216no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—42no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23#38014yes—
445C23C22×C22⋊C411yescharacteristic
446C23C2×#12111yescharacteristic
447C23C2×C22⋊Q811yescharacteristic
448C23#53411yescharacteristic
449C23#53411yescharacteristic
450C23#53611yescharacteristic
451C23#53611yescharacteristic
452C23#57612yes—
453C23#57712yes—
454C23#57712yes—
455C23#57812yes—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—816no—
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—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#38014yes—
520C2×C4C22×C4⋊C411yescharacteristic
521C2×C4C2×#11711yescharacteristic
522C2×C4#51811yescharacteristic
523C2×C4#51811yescharacteristic
524C2×C4#51911yescharacteristic
525C2×C4#51911yescharacteristic
526C2×C4C2×#12111yescharacteristic
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4#53411yescharacteristic
529C2×C4#53411yescharacteristic
530C2×C4#53611yescharacteristic
531C2×C4#53611yescharacteristic
532C2×C4#53711yescharacteristic
533C2×C4#53711yescharacteristic
534C2×C4#53711yescharacteristic
535C2×C4#53711yescharacteristic
536C2×C4#53911yescharacteristic
537C2×C4#53911yescharacteristic
538C2×C4#53911yescharacteristic
539C2×C4#53911yescharacteristic
540C2×C4#53911yescharacteristic
541C2×C4#53911yescharacteristic
542C2×C4#53911yescharacteristic
543C2×C4#53911yescharacteristic
544C2×C4#54411yescharacteristic
545C2×C4#54411yescharacteristic
546C2×C4#54411yescharacteristic
547C2×C4#54411yescharacteristic
548C8—82no—
549C8—82no—
550C22—24no—
551C22—28no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22C2×#38011yescharacteristic
562C22#230811yescharacteristic
563C22#231011yescharacteristic
564C22#261811yescharacteristic
565C22#261811yescharacteristic
566C22#262711yescharacteristic
567C22#262711yescharacteristic
568C22#289712yes—
569C22#289812yes—
570C22#304512yes—
571C22#304612yes—
572C22#308312yes—
573C22#308412yes—
574C22#319812yes—
575C22#319911yescharacteristic
576C22#320011yescharacteristic
577C22#320011yescharacteristic
578C22#320112yes—
579C22#320211yescharacteristic
580C22#324112yes—
581C22#324212yes—
582C22#324312yes—
583C22#324412yes—
584C4—24no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—48no—
590C4—84no—
591C4—162no—
592C4—162no—
593C4#231311yescharacteristic
594C4#231311yescharacteristic
595C4#231611yescharacteristic
596C4#231611yescharacteristic
597C4#261811yescharacteristic
598C4#261811yescharacteristic
599C4#262711yescharacteristic
600C4#262711yescharacteristic
601C2—44no—
602C2#1433911yescharacteristic
603C2#2065212yes—
604C2#2177011yescharacteristic
605C2#2177111yescharacteristic
606C2#2184312yes—
607C2#2976511yescharacteristic
608C2#2976611yescharacteristic
609C2#2976912yes—
610C2#2997611yescharacteristic
611C2#2997711yescharacteristic
612C2#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
5. #11228C2semidirect
6. #11227C2semidirect
7. #11208C2semidirect
8. #11207C2semidirect
17. #2317C4semidirect
173. #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×#18252
C2×#17982
#143402
#112282
#112272
#112082
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