Finite Group Databasebeta

← #364840

Subgroups — #364840

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3C2×#1997C211yesmaximal, characteristic, complemented
4#20646C211yesmaximal, characteristic
5C4×#418C211yesmaximal, characteristic
6#11210C212yesmaximal, complemented
7#11054C212yesmaximal
8#10958C212yesmaximal
9#10891C212yesmaximal, complemented
10#10852C211yesmaximal, characteristic, complemented
11#10851C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—22no—
14C22×C4×D4C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#418C2211yescharacteristic
17C2×#418C2212yes—
18C2×#418C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2542—24no—
21#2542C418yescomplemented
22C2×#399—22no—
23#2369—22no—
24#2365—22no—
25#2345—22no—
26Q8×C22⋊C4—22no—
27#2331—22no—
28#2316C2212yes—
29C4×C4⋊Q8—22no—
30C2×#390C2212yes—
31#2285C2211yescharacteristic
32C2×#384C2212yes—
33#2274C2212yes—
34C4×#117C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1997—24no—
37#1804—22no—
38#1802—22no—
39#1762—22no—
40C4×#102—22no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43#1736C2211yescharacteristic
44#1736C2212yes—
45#1736C2211yescharacteristic
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C23×Q8D412yes—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2×C414yes—
63C2×C4⋊Q8D414yes—
64C2×C22⋊Q8—216no—
65C2×C22⋊Q8—416no—
66C2×C4×Q8—24no—
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4C2×C414yes—
75C2×#117C2311yescharacteristic
76C2×#117C2312yes—
77C2×#117C2311yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4D412yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—24no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87#423—22no—
88#423—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91#418—22no—
92#418—22no—
93#418—22no—
94#418—24no—
95#418—42no—
96#418Q814yescomplemented
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—24no—
100C4×M16—22no—
101C4×M16—22no—
102C2×#97—21no—
103C2×C4×C8—21no—
104#399—216no—
105#391—24no—
106#391—28no—
107#390—24no—
108#390C2×C418yes—
109#389—24no—
110#389—24no—
111#389—24no—
112#387—28no—
113#384—24no—
114#384C2×C418yes—
115#384D412yes—
116#382D412yes—
117#380—24no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4C2×C414yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D412yes—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—21no—
132#376—21no—
133C2×#95—21no—
134C43—21no—
135#343—24no—
136#343—24no—
137#340—24no—
138#340—22no—
139#340—22no—
140#339—24no—
141#339—24no—
142#339—42no—
143#339—42no—
144C25—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—48no—
149C22×Q8C22⋊C418yes—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4C22×C412yes—
162C22×D4C2×Q812yes—
163C22×D4#4112yes—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—24no—
167C23×C4—48no—
168C23×C4—41no—
169C23×C4—41no—
170C23×C4—42no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D414yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16C2×Q812yes—
192C2×M16#4112yes—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C4⋊Q8—24no—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201C4⋊Q8—24no—
202C4⋊Q8—24no—
203C4⋊Q8—24no—
204C22⋊Q8—464no—
205C4×Q8—88no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214#117—21no—
215#117—21no—
216#117—22no—
217#117—22no—
218#117—21no—
219#117—21no—
220#117—21no—
221#117—22no—
222#117—22no—
223#117—21no—
224#117C2×Q812yes—
225#117#4112yes—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—216no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—416no—
245C2×C4⋊C4C22⋊C418yes—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D412yes—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—416no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—42no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—28no—
272C2×C42C22×C414yes—
273C2×C42C22×C412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D412yes—
279C2×C42#4112yes—
280C2×C42#4112yes—
281C4⋊C8—42no—
282C4⋊C8—42no—
283#102—28no—
284#102—28no—
285#102—48no—
286#97—42no—
287C4×C8—42no—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—44no—
299C24—44no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24C22×D411yescharacteristic
305C2×Q8—28no—
306C2×Q8—28no—
307C2×Q8—416no—
308C2×Q8—48no—
309C2×Q8—84no—
310C2×Q8—88no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4C4×Q814yescomplemented
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—42no—
349C22×C4—416no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—48no—
354C22×C4—44no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—84no—
360C22×C4—88no—
361C22×C4—84no—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C418yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12011yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12312yes—
370C22×C4#12312yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C42−1+411yescharacteristic
383C22×C42−1+411yescharacteristic
384C22×C42−1+411yescharacteristic
385M16—22no—
386M16—22no—
387M16—22no—
388M16—22no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—42no—
394M16—42no—
395M16—42no—
396M16—42no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C4⋊C4—216no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—44no—
421C4⋊C4—432no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—44no—
425C4⋊C4—88no—
426C4⋊C4C22⋊Q814yescomplemented
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—432no—
430C22⋊C4—42no—
431C22⋊C4—41no—
432C22⋊C4—42no—
433C22⋊C4—41no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C42—22no—
437C42—22no—
438C42—24no—
439C42—24no—
440C42—44no—
441C42—42no—
442C42—42no—
443C42—41no—
444C42—41no—
445C42—42no—
446C42—42no—
447C42—44no—
448C42—88no—
449C42C4×D412yes—
450C42C4×D412yes—
451C42C4×D412yes—
452C42C4×D412yes—
453C42#12112yes—
454C42#12112yes—
455C42#12112yes—
456C42#12112yes—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—44no—
461C23—48no—
462C23—48no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—82no—
471C23—84no—
472C23—84no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23C22×C22⋊C411yescharacteristic
477C23C2×#12011yescharacteristic
478C23C2×#12311yescharacteristic
479C23#54811yescharacteristic
480C23#54811yescharacteristic
481C23D4×Q811yescharacteristic
482C23#55011yescharacteristic
483Q8—416no—
484Q8—88no—
485D4—24no—
486D4—24no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—42no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—48no—
509C2×C4—42no—
510C2×C4—48no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—88no—
520C2×C4—84no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—88no—
524C2×C4#38612yes—
525C2×C4#38612yes—
526C2×C4#38612yes—
527C2×C4#38612yes—
528C2×C4#45314yes—
529C2×C4C2×C4×D411yescharacteristic
530C2×C4C2×C4×D411yescharacteristic
531C2×C4C2×C4×Q811yescharacteristic
532C2×C4C4×#4111yescharacteristic
533C2×C4#51911yescharacteristic
534C2×C4#52011yescharacteristic
535C2×C4C2×#12111yescharacteristic
536C2×C4C2×#12111yescharacteristic
537C2×C4C2×C22⋊Q811yescharacteristic
538C2×C4#52511yescharacteristic
539C2×C4#52911yescharacteristic
540C2×C4#53311yescharacteristic
541C2×C4#53611yescharacteristic
542C2×C4#53711yescharacteristic
543C2×C4#54111yescharacteristic
544C2×C4#54211yescharacteristic
545C2×C4#54611yescharacteristic
546C2×C4#54711yescharacteristic
547C2×C4#54811yescharacteristic
548C2×C4#54811yescharacteristic
549C2×C4#54811yescharacteristic
550C2×C4D4×Q811yescharacteristic
551C2×C4D4×Q811yescharacteristic
552C2×C4#55011yescharacteristic
553C2×C4#55211yescharacteristic
554C2×C4#55211yescharacteristic
555C2×C4#55411yescharacteristic
556C2×C4#55611yescharacteristic
557C8—42no—
558C8—42no—
559C8—82no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—48no—
569C22—44no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—82no—
576C22—82no—
577C22C2×#38611yescharacteristic
578C22Q8×C22⋊C411yescharacteristic
579C22#234111yescharacteristic
580C22#240711yescharacteristic
581C22#240911yescharacteristic
582C22#242111yescharacteristic
583C22#242311yescharacteristic
584C22C2×#45312yes—
585C22#302012yes—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—48no—
590C4—44no—
591C4—48no—
592C4—82no—
593C4—84no—
594C4—82no—
595C4#188714yes—
596C4C4×C22⋊Q811yescharacteristic
597C4#232311yescharacteristic
598C4#236011yescharacteristic
599C4#236211yescharacteristic
600C4#242611yescharacteristic
601C4#242911yescharacteristic
602C4#253611yescharacteristic
603C4#253811yescharacteristic
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—84no—
609C2#1396911yescharacteristic
610C2C2×#188711yescharacteristic
611C2#2095411yescharacteristic
612C2#2417012yes—
613C2#2432012yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. C2×#1997C2semidirect
6. #11210C2semidirect
9. #10891C2semidirect
10. #10852C2semidirect
11. #10851C2semidirect
21. #2542C4semidirect
96. #418Q8semidirect
338. C2×D4C4×Q8semidirect
426. C4⋊C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
#342162
C2×#19972
#206462
C4×#4182
#112102
#110542
#109582
#108912
#108522
#108512