Finite Group Databasebeta

← #397966

Subgroups — #397966

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×#1862C211yesmaximal, characteristic
3C2×#1802C212yesmaximal
4#15143C211yesmaximal, characteristic, complemented
5D4×#95C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11324C212yesmaximal, complemented
8#11322C211yesmaximal, characteristic, complemented
9#11232C212yesmaximal, complemented
10#11227C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2212yes—
15C2×#474—28no—
16C2×#474—24no—
17C22×#107C2212yes—
18C22×#107C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C2×#395—21no—
25C2×#395—21no—
26C2×#389C2211yescharacteristic
27C2×#384C2212yes—
28C22×#95—21no—
29#1862—24no—
30#1802—28no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1444—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C22⋊#95—21no—
43C22⋊#95—22no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—22no—
60C2×C4×D4—24no—
61C2×C4×D4—42no—
62C2×C4×D4—42no—
63C2×C4×D4—42no—
64C2×C4×D4D414yes—
65C2×C4×D4Q818yescomplemented
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C42C2311yescharacteristic
78C22×C42C2312yes—
79#474—232no—
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—22no—
84C2×#107—42no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—42no—
90C2×#102—22no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—22no—
94C2×#102—24no—
95C2×#102—22no—
96C2×#102D418yes—
97C2×#102Q814yescomplemented
98#395—44no—
99#395—44no—
100#389—22no—
101#389—22no—
102#384—24no—
103#384—42no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—44no—
112C2×#95D411yescharacteristic
113C2×#95D412yes—
114C2×#95D411yescharacteristic
115#340—24no—
116#340—24no—
117#340—22no—
118#340—42no—
119#340—42no—
120#336—22no—
121#336—24no—
122C25—21no—
123C22×Q8—44no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—82no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4C2×D414yes—
134C22×D4C2×D412yes—
135C22×D4C2×Q814yes—
136C22×D4C2×Q812yes—
137C22×D4#4112yes—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—28no—
144C23×C4—21no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—42no—
149C23×C4—44no—
150C23×C4—42no—
151C23×C4—44no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2411yescharacteristic
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8C2×D414yes—
176C22×C8C2×Q812yes—
177C4⋊Q8—28no—
178C4⋊Q8—48no—
179C4⋊Q8—44no—
180C4×D4—28no—
181C4×D4—216no—
182C4×D4—44no—
183C4×D4—48no—
184C4×D4—48no—
185C4×D4—48no—
186C4×D4—48no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×Q814yes—
223C2×C4⋊C4C2×Q812yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42C2×D412yes—
236C2×C42C2×Q814yes—
237C2×C42#4112yes—
238C2×C42#4114yes—
239#107—216no—
240#107—44no—
241C4⋊C8—28no—
242C4⋊C8—48no—
243#102—216no—
244#102—28no—
245#102—44no—
246#95—22no—
247#95—24no—
248#95—44no—
249#95—42no—
250#95—42no—
251#95—44no—
252#95—44no—
253#95—82no—
254#95—81no—
255#95D814yes—
256C24—22no—
257C24—24no—
258C24—24no—
259C24—22no—
260C24—22no—
261C24—41no—
262C24—41no—
263C24—42no—
264C24—42no—
265C24—41no—
266C24—41no—
267C24C22×D411yescharacteristic
268C2×Q8—48no—
269C2×Q8—88no—
270C2×Q8—88no—
271C2×Q8—84no—
272C2×Q8—84no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—24no—
285C2×D4—22no—
286C2×D4#12014yes—
287C2×D4C22⋊Q814yes—
288C2×D4C22⋊Q818yes—
289C2×D4C4⋊Q814yes—
290C2×D4C4⋊Q818yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—48no—
312C22×C4—416no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—48no—
336C22×C4—416no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—88no—
343C22×C4—84no—
344C22×C4—88no—
345C22×C4—88no—
346C22×C4—88no—
347C22×C4—88no—
348C22×C4—88no—
349C22×C4—88no—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4C22⋊Q814yes—
353C22×C4C22⋊Q814yes—
354C22×C4C2×D812yes—
355C22×C4C2×D814yes—
356C22×C4AGL(1,ℤ/8)12yes—
357C22×C4#13714yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D412yes—
360C22×C4C22×Q812yes—
361C22×C4C22×Q811yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4112yes—
364C22×C4C2×#4111yescharacteristic
365C22×C4H(5,𝔽2)12yes—
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C42−1+411yescharacteristic
368C2×C8—24no—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—48no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—88no—
405C4⋊C4—88no—
406C4⋊C4—84no—
407C22⋊C4—48no—
408C22⋊C4—416no—
409C42—22no—
410C42—24no—
411C42—22no—
412C42—24no—
413C42—24no—
414C42—24no—
415C42—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—28no—
420C23—28no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—44no—
438C23—44no—
439C23C2×#12111yescharacteristic
440C23C2×C22⋊Q812yes—
441C23D4212yes—
442C23#54611yescharacteristic
443C23D4×Q811yescharacteristic
444C23C22×D811yescharacteristic
445C23C2×AGL(1,ℤ/8)11yescharacteristic
446C23C2×#13712yes—
447C23#57612yes—
448C23#57711yescharacteristic
449C23#57811yescharacteristic
450Q8—816no—
451D4—24no—
452D4—24no—
453D4—24no—
454D4—28no—
455D4—28no—
456D4#39518yescomplemented
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—816no—
492C2×C4—832no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4#45914yes—
513C2×C4#47418yescomplemented
514C2×C4C2×#12011yescharacteristic
515C2×C4C2×C22⋊Q811yescharacteristic
516C2×C4C2×C22⋊Q812yes—
517C2×C4C2×C4⋊Q811yescharacteristic
518C2×C4C2×C4⋊Q812yes—
519C2×C4#53211yescharacteristic
520C2×C4#53312yes—
521C2×C4#53411yescharacteristic
522C2×C4#53711yescharacteristic
523C2×C4#54312yes—
524C2×C4#54412yes—
525C2×C4#54612yes—
526C2×C4#54612yes—
527C2×C4#54712yes—
528C2×C4#55112yes—
529C2×C4#55211yescharacteristic
530C2×C4#55412yes—
531C2×C4#55411yescharacteristic
532C8—42no—
533C8—44no—
534C8—81no—
535C22—28no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—48no—
540C22—48no—
541C22—44no—
542C22—44no—
543C22—48no—
544C22—48no—
545C22—44no—
546C22#241011yescharacteristic
547C22#241711yescharacteristic
548C22#243712yes—
549C22#244012yes—
550C22#285711yescharacteristic
551C22C2×#45911yescharacteristic
552C22#303911yescharacteristic
553C22#304111yescharacteristic
554C22#304611yescharacteristic
555C22C2×#47412yes—
556C22#307912yes—
557C22#308112yes—
558C22#308312yes—
559C22#326412yes—
560C22#326612yes—
561C22#327612yes—
562C22D4×D812yes—
563C22#329411yescharacteristic
564C22#331412yes—
565C22#333011yescharacteristic
566C22Q8×D811yescharacteristic
567C22#338411yescharacteristic
568C22#338512yes—
569C4—24no—
570C4—28no—
571C4—42no—
572C4—44no—
573C4—82no—
574C4—84no—
575C4—84no—
576C4—82no—
577C4—161no—
578C4C2×#39511yescharacteristic
579C4#243011yescharacteristic
580C4#243812yes—
581C4#254912yes—
582C4#256411yescharacteristic
583C4#265911yescharacteristic
584C2—48no—
585C2#1475811yescharacteristic
586C2#2435411yescharacteristic
587C2#2437611yescharacteristic
588C2#2481111yescharacteristic
589C2#2483311yescharacteristic
590C2#2595012yes—
591C2#2597012yes—
592C2#2604012yes—
593C2#2605812yes—
594C2#3319511yescharacteristic
595C2#3320311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #15143C2semidirect
6. #11384C2semidirect
7. #11324C2semidirect
8. #11322C2semidirect
9. #11232C2semidirect
10. #11227C2semidirect
65. C2×C4×D4Q8semidirect
97. C2×#102Q8semidirect
456. D4#395semidirect
513. C2×C4#474semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#18622
C2×#18022
#151432
D4×#952
#113842
#113242
#113222
#112322
#112272
#107882