Finite Group Databasebeta

← #366905

Subgroups — #366905

Order 512.

1320 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#21146C211yesmaximal, characteristic
5C2×#1801C211yesmaximal, characteristic
6C4⋊C4×M16C211yesmaximal, characteristic
7C2×C4×#101C211yesmaximal, characteristic
8#11354C212yesmaximal
9#11045C212yesmaximal
10#10963C212yesmaximal
11#10857C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#423C2212yes—
16C2×#423C2211yescharacteristic
17C22×#101C2211yescharacteristic
18C22×#101C2212yes—
19C22×#101C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2212yes—
22C2×C4×M16C2211yescharacteristic
23#2567—22no—
24#2567—22no—
25#2567—21no—
26#2567—21no—
27#2567—22no—
28C2×#395C414yes—
29C2×#391C2212yes—
30C2×#389C414yes—
31C2×#384C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1997—24no—
34#1997—24no—
35#1940—22no—
36#1939—22no—
37#1801—24no—
38#1771—22no—
39#1770—21no—
40C8×C4⋊C4—21no—
41C4×#101—24no—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#343C2212yes—
45C2×#343C2212yes—
46C2×#343C2211yescharacteristic
47C2×#343C2212yes—
48C2×#343C2211yescharacteristic
49#1728C2212yes—
50#1728C2212yes—
51C23×Q8C2×C412yes—
52C23×Q8C2311yescharacteristic
53C22×M16C2312yes—
54C22×M16C2312yes—
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C22×C4⋊C4C2×C412yes—
76C22×C4⋊C4C2×C412yes—
77C22×C4⋊C4C2×C412yes—
78C22×C4⋊C4C2×C418yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C42C2×C412yes—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#423—22no—
85#423—24no—
86#423—22no—
87#423—22no—
88#423—42no—
89#423D414yes—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—21no—
95C2×#101—22no—
96C2×#101—22no—
97C2×#101—22no—
98C2×#101—22no—
99C2×#101—22no—
100C2×#101—24no—
101C2×#101—24no—
102C2×#101—42no—
103C2×#101Q814yescomplemented
104C4×M16—22no—
105C4×M16—24no—
106C4×M16—22no—
107C4×M16—22no—
108C4×M16—42no—
109C4×M16D414yes—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—22no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—22no—
116#395—24no—
117#395—28no—
118#395—24no—
119#391—22no—
120#391—22no—
121#391—24no—
122#389—24no—
123#389—28no—
124#389—24no—
125#384—22no—
126#384—22no—
127#384—24no—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D412yes—
133C2×#95C2×C418yes—
134C2×#95C2×C412yes—
135#343—24no—
136#343—24no—
137#343—24no—
138#343—24no—
139#343—24no—
140#343—22no—
141#343—22no—
142#336—24no—
143#336—22no—
144#336—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—84no—
162C22×Q8C2×Q812yes—
163C22×Q8#4112yes—
164C23×C4C22×C412yes—
165C23×C4C22×C414yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C22×C414yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C2411yescharacteristic
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—22no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—22no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—22no—
193C2×M16—21no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16C2×D412yes—
201C2×M16C2×D414yes—
202C2×M16C2×D412yes—
203C2×M16C2×Q812yes—
204C2×M16C2×Q812yes—
205C2×M16#4112yes—
206C2×M16#4112yes—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—22no—
211C22×C8—21no—
212C22×C8—22no—
213C4⋊Q8—42no—
214C4⋊Q8—42no—
215C4⋊Q8—42no—
216C4⋊Q8—42no—
217C4⋊Q8—42no—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224C4×Q8—42no—
225C4×Q8—42no—
226C4×Q8—42no—
227C4×Q8—42no—
228C4×Q8—42no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—88no—
253C2×C4⋊C4C22⋊C412yes—
254C2×C4⋊C4C22⋊C414yes—
255C2×C4⋊C4C22⋊C412yes—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D414yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42#4112yes—
274C2×C42#4112yes—
275C4⋊C8—24no—
276C4⋊C8—24no—
277C4⋊C8—48no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280#101—28no—
281#101—28no—
282#101—48no—
283#97—24no—
284#97—44no—
285#97—42no—
286C4×C8—24no—
287C4×C8—44no—
288C4×C8—42no—
289#95—28no—
290#95—28no—
291#95C22⋊C412yes—
292#95C22⋊C414yes—
293#95C22⋊C412yes—
294C24C23×C411yescharacteristic
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—84no—
326C2×Q8—84no—
327C2×Q8—88no—
328C2×Q8—88no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—28no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—48no—
349C22×C4C2×C22⋊C414yes—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C414yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C4×D412yes—
356C22×C4C4×D412yes—
357C22×C4C4×D412yes—
358C22×C4C4×D412yes—
359C22×C4C4×Q814yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×Q811yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C42−1+411yescharacteristic
372C22×C42−1+411yescharacteristic
373C22×C42−1+411yescharacteristic
374M16—22no—
375M16—22no—
376M16—22no—
377M16—22no—
378M16—22no—
379M16—22no—
380M16—22no—
381M16—24no—
382M16—24no—
383M16—22no—
384M16—22no—
385M16—42no—
386M16—41no—
387M16—41no—
388M16—42no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—42no—
394M16—42no—
395M16—42no—
396M16—42no—
397M16C22⋊Q814yes—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—42no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—42no—
412C2×C8—41no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—88no—
431C4⋊C4—88no—
432C4⋊C4#10118yescomplemented
433C42—22no—
434C42—22no—
435C42—24no—
436C42—22no—
437C42—24no—
438C42—22no—
439C42—44no—
440C42—44no—
441C42#12112yes—
442C42#12112yes—
443C42#12112yes—
444C42#12112yes—
445C42#12412yes—
446C42#12412yes—
447C42#12712yes—
448C42#12712yes—
449C23—22no—
450C23—22no—
451C23C22×C22⋊C411yescharacteristic
452C23C2×C4×D411yescharacteristic
453C23C2×C4×D411yescharacteristic
454C23C2×C4×Q811yescharacteristic
455C23C4×#4111yescharacteristic
456C23#51911yescharacteristic
457C23#52011yescharacteristic
458Q8—42no—
459Q8—42no—
460Q8—42no—
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465Q8—42no—
466Q8—88no—
467Q8—88no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4#39012yes—
498C2×C4#39012yes—
499C2×C4#39012yes—
500C2×C4#39012yes—
501C2×C4C2×#10118yes—
502C2×C4C2×#10114yes—
503C2×C4C2×#12111yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×C22⋊Q811yescharacteristic
506C2×C4#52511yescharacteristic
507C2×C4C2×#12411yescharacteristic
508C2×C4#52911yescharacteristic
509C2×C4C2×#12711yescharacteristic
510C2×C4#53311yescharacteristic
511C2×C4#53611yescharacteristic
512C2×C4#53711yescharacteristic
513C2×C4#54111yescharacteristic
514C2×C4#54211yescharacteristic
515C2×C4#54611yescharacteristic
516C2×C4#54611yescharacteristic
517C2×C4#54611yescharacteristic
518C2×C4#54711yescharacteristic
519C2×C4#54711yescharacteristic
520C2×C4#54711yescharacteristic
521C2×C4#54811yescharacteristic
522C2×C4#54811yescharacteristic
523C2×C4#54811yescharacteristic
524C2×C4D4×Q811yescharacteristic
525C2×C4D4×Q811yescharacteristic
526C2×C4D4×Q811yescharacteristic
527C2×C4#55211yescharacteristic
528C2×C4#55211yescharacteristic
529C2×C4#55411yescharacteristic
530C2×C4#55611yescharacteristic
531C8—24no—
532C8—42no—
533C8—44no—
534C8—42no—
535C8—42no—
536C8—82no—
537C22—22no—
538C22—22no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22C2×#39011yescharacteristic
546C22C4×C22⋊Q811yescharacteristic
547C22#232311yescharacteristic
548C22#233911yescharacteristic
549C22Q8×C22⋊C411yescharacteristic
550C22#236011yescharacteristic
551C22#236211yescharacteristic
552C22C22×#10112yes—
553C22#288812yes—
554C4—28no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—44no—
559C4—44no—
560C4—82no—
561C4—84no—
562C4#196614yes—
563C4#241511yescharacteristic
564C4#241711yescharacteristic
565C4#242711yescharacteristic
566C4#242911yescharacteristic
567C4#248911yescharacteristic
568C4#249211yescharacteristic
569C4#254011yescharacteristic
570C4#254211yescharacteristic
571C2—21no—
572C2—21no—
573C2—21no—
574C2—21no—
575C2#1399711yescharacteristic
576C2C2×#196611yescharacteristic
577C2#2121911yescharacteristic
578C2#2248112yes—
579C2Q8×#10112yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
103. C2×#101Q8semidirect
432. C4⋊C4#101semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#19972
#211462
C2×#18012
C4⋊C4×M162
C2×C4×#1012
#113542
#110452
#109632
#108572