Finite Group Databasebeta

← #370097

Subgroups — #370097

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#1949C212yesmaximal, complemented
3C2×#1854C211yesmaximal, characteristic
4C2×#1851C211yesmaximal, characteristic
5#13881C211yesmaximal, characteristic
6#13801C211yesmaximal, characteristic
7#11325C212yesmaximal, complemented
8#11232C212yesmaximal, complemented
9#10908C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428—28no—
16C2×#428—24no—
17C22×#107C2212yes—
18C22×#107C2211yescharacteristic
19C22×#106C2212yes—
20C22×#106C2211yescharacteristic
21C22×C4⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C2×#382C2212yes—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1949—28no—
29#1854—24no—
30#1851C418yescomplemented
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1442—21no—
39#1442—21no—
40#1437—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43#1434—21no—
44C22⋊#95—22no—
45C4×#95—21no—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—24no—
51C22×M16—24no—
52C22×M16C2×C414yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C2×#117—22no—
57C2×#117—24no—
58C2×#117—42no—
59C2×#117D414yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42—22no—
77C22×C42C2312yes—
78#428—232no—
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—22no—
83C2×#107—24no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—22no—
87C2×#106—42no—
88C2×#106D418yes—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8C2×C418yescomplemented
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8C2×C414yes—
95#382—24no—
96#382—42no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380Q814yes—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—42no—
103C4×C22⋊C4C2×C414yes—
104C4×C22⋊C4Q814yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120#340—24no—
121#340—42no—
122#336C2×C418yescomplemented
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—21no—
133C23×C4—21no—
134C23×C4—44no—
135C23×C4—42no—
136C23×C4—44no—
137C23×C4—81no—
138C23×C4—81no—
139C23×C4C22×C412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×Q812yes—
145C23×C4C2×Q812yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C2×M16—216no—
150C2×M16—28no—
151C2×M16—28no—
152C2×M16—28no—
153C2×M16—216no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C22×C412yes—
167C22×C8C22×C412yes—
168C22×C8C22×C414yes—
169C22×C8C2×D414yes—
170C22×C8#4114yes—
171#117—28no—
172#117—44no—
173#117—48no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—28no—
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—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×Q814yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—81no—
215C2×C22⋊C4—81no—
216C2×C22⋊C4C4⋊C418yes—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42—44no—
225C2×C42—82no—
226C2×C42C22×C414yes—
227C2×C42C2×Q814yes—
228#107—216no—
229#107—44no—
230#106—216no—
231#106—44no—
232C4⋊C8—216no—
233C22⋊C8—28no—
234#95—24no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—42no—
240#95—82no—
241#95—82no—
242#95—82no—
243#95C4⋊C414yes—
244#95C4⋊C414yes—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24C2×C4⋊C412yes—
254C24#12112yes—
255C24C22⋊Q812yes—
256C24C22×D411yescharacteristic
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
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—24no—
289C22×C4—24no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—48no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4C2×C4⋊C414yes—
325C22×C4C2×C4⋊C414yes—
326C22×C4C2×C4⋊C412yes—
327C22×C4C4×D414yes—
328C22×C4C4×Q814yes—
329C22×C4#12112yes—
330C22×C4C22⋊Q812yes—
331C22×C4#12514yes—
332C22×C4C4⋊Q814yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C22×Q811yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)12yes—
342C22×C42−1+412yes—
343M16—232no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8C4×D418yes—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—48no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4#12014yes—
382C22⋊C4—48no—
383C22⋊C4—48no—
384C22⋊C4—84no—
385C22⋊C4—84no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—48no—
391C42—88no—
392C23—24no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—24no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—24no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23#38914yes—
415C23C22×C4⋊C411yescharacteristic
416C23C2×#12111yescharacteristic
417C23C2×C22⋊Q811yescharacteristic
418C23D4212yes—
419C23#54712yes—
420C23#57512yes—
421C23#57512yes—
422C23#57611yescharacteristic
423C23#57711yescharacteristic
424C23#57711yescharacteristic
425C23#57811yescharacteristic
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—816no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4#38914yes—
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C2×C4×D412yes—
488C2×C4C2×C4×Q811yescharacteristic
489C2×C4#52012yes—
490C2×C4C2×#12011yescharacteristic
491C2×C4#52511yescharacteristic
492C2×C4#52512yes—
493C2×C4C2×#12511yescharacteristic
494C2×C4C2×C4⋊Q811yescharacteristic
495C2×C4#53411yescharacteristic
496C2×C4#53611yescharacteristic
497C2×C4#53912yes—
498C2×C4#54612yes—
499C2×C4D4×Q812yes—
500C2×C4D4×Q812yes—
501C2×C4#55212yes—
502C2×C4#55412yes—
503C2×C4#55412yes—
504C8—28no—
505C8—42no—
506C8—42no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22C2×#38911yescharacteristic
519C22D4×C4⋊C412yes—
520C22#241011yescharacteristic
521C22#241311yescharacteristic
522C22#244912yes—
523C22#291112yes—
524C22#291211yescharacteristic
525C22#291311yescharacteristic
526C22#304011yescharacteristic
527C22#304011yescharacteristic
528C22#304511yescharacteristic
529C22#304611yescharacteristic
530C22#308012yes—
531C22#308111yescharacteristic
532C22#308211yescharacteristic
533C22#326812yes—
534C22#326912yes—
535C22#327512yes—
536C22#327512yes—
537C22#334712yes—
538C22#334912yes—
539C22#335112yes—
540C22#335112yes—
541C4—24no—
542C4—44no—
543C4—44no—
544C4—44no—
545C4—82no—
546C4—84no—
547C4—84no—
548C4—161no—
549C4—161no—
550C4C4×#12011yescharacteristic
551C4#232211yescharacteristic
552C4#236012yes—
553C4#243411yescharacteristic
554C4#247312yes—
555C4#258311yescharacteristic
556C2—44no—
557C2#1404311yescharacteristic
558C2#2117211yescharacteristic
559C2#2117511yescharacteristic
560C2#2266012yes—
561C2#2266612yes—
562C2#2437111yescharacteristic
563C2#2437711yescharacteristic
564C2#2459311yescharacteristic
565C2#2459911yescharacteristic
566C2#2636112yes—
567C2#2636712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1949C2semidirect
7. #11325C2semidirect
8. #11232C2semidirect
9. #10908C2semidirect
30. #1851C4semidirect
91. C2×C4⋊C8C2×C4semidirect
122. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#19492
C2×#18542
C2×#18512
#138812
#138012
#113252
#112322
#109082
#107302
#107222