Finite Group Databasebeta

← #118696

Subgroups — #118696

Order 512.

1172 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1797C211yesmaximal, characteristic
2C2×C4×C22⋊C8C211yesmaximal, characteristic
3C4×#380C211yesmaximal, characteristic
4C2×#1279C212yesmaximal, complemented
5C2×#1279C212yesmaximal, complemented
6C23×C42C2211yescharacteristic
7C22×C4⋊C8C2212yes—
8C22×C22⋊C8—21no—
9C22×C22⋊C8—21no—
10C22×C4×C8C2212yes—
11C2×#380C411yescharacteristic
12C2×#380C411yescharacteristic
13C2×C4×C4⋊C4C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15C2×C4×C22⋊C4—21no—
16#1797—24no—
17C4×C22⋊C8—24no—
18C2×#336—21no—
19C2×#336—21no—
20C4⋊#95C411yescharacteristic
21C4⋊#95C411yescharacteristic
22#1434—21no—
23C22⋊#95—21no—
24C4×#95—21no—
25#1279—24no—
26#1279—24no—
27C24×C4C2×C411yescharacteristic
28C24×C4D411yescharacteristic
29C24×C4D411yescharacteristic
30C23×C8—22no—
31C23×C8—22no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C42—22no—
39C22×C42—22no—
40C22×C42—22no—
41C22×C42—22no—
42C22×C42—22no—
43C22×C42—22no—
44C22×C42—21no—
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49C22×C42C2311yescharacteristic
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8D414yes—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—24no—
59C2×C22⋊C8—24no—
60C2×C22⋊C8—42no—
61C2×C22⋊C8—42no—
62C2×C4×C8—22no—
63C2×C4×C8—22no—
64C2×C4×C8—22no—
65C2×C4×C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8D414yes—
68#380—22no—
69#380—22no—
70#380—22no—
71#380C814yescomplemented
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C22⋊C4—42no—
77C4×C22⋊C4—42no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86#336—24no—
87#336—24no—
88C25C22⋊C411yescharacteristic
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C2×C812yes—
121C23×C4M1612yes—
122C23×C4D812yes—
123C23×C4SD1612yes—
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C22×C8—24no—
128C22×C8—24no—
129C22×C8—24no—
130C22×C8—24no—
131C22×C8—42no—
132C22×C8—42no—
133C22×C8—42no—
134C22×C8—42no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—42no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4C2×C812yes—
154C2×C4⋊C4C2×C812yes—
155C2×C4⋊C4M1612yes—
156C2×C4⋊C4M1612yes—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42—41no—
184C2×C42—42no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42—42no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42#4112yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198C2×C42#4112yes—
199C4⋊C8—24no—
200C4⋊C8—24no—
201C4⋊C8—42no—
202C4⋊C8—42no—
203C22⋊C8—216no—
204C22⋊C8—48no—
205C4×C8—24no—
206C4×C8—24no—
207C4×C8—42no—
208C4×C8—42no—
209#95—24no—
210#95—24no—
211#95—42no—
212#95—42no—
213#95—42no—
214#95—42no—
215#95—41no—
216#95—42no—
217#95—41no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—21no—
230C24C22⋊C812yes—
231C24#10212yes—
232C24#10412yes—
233C24C2×C22⋊C411yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
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—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4C22⋊C812yes—
327C22×C4#10212yes—
328C22×C4#10412yes—
329C22×C4#11711yescharacteristic
330C22×C4#11711yescharacteristic
331C22×C4C4×D412yes—
332C22×C4C4×D412yes—
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12112yes—
336C22×C4#12112yes—
337C22×C4#12312yes—
338C22×C4#12311yescharacteristic
339C22×C4#12312yes—
340C22×C4#12311yescharacteristic
341C22×C4C22×C811yescharacteristic
342C22×C4C2×M1611yescharacteristic
343C22×C4C2×M1611yescharacteristic
344C22×C4C2×M1611yescharacteristic
345C22×C4#13112yes—
346C22×C4#13112yes—
347C22×C4C2×D811yescharacteristic
348C22×C4C2×SD1611yescharacteristic
349C22×C4#13512yes—
350C22×C4AGL(1,ℤ/8)11yescharacteristic
351C22×C4#13711yescharacteristic
352C22×C4#13712yes—
353C2×C8—28no—
354C2×C8—44no—
355C2×C8—44no—
356C2×C8—44no—
357C2×C8—44no—
358C2×C8—44no—
359C2×C8—44no—
360C2×C8—44no—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—82no—
364C2×C8—82no—
365C2×C8—82no—
366C2×C8—82no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C22⋊C4—48no—
376C22⋊C4—48no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—81no—
384C42—81no—
385C42—82no—
386C42—82no—
387C42—81no—
388C42—81no—
389C42#12112yes—
390C42#12112yes—
391C42#12312yes—
392C42#12312yes—
393C42#12412yes—
394C42#12412yes—
395C42#12612yes—
396C42#12612yes—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23D4⋊C814yes—
432C23#38111yescharacteristic
433C23#38611yescharacteristic
434C23#38611yescharacteristic
435C23C2×C22⋊C811yescharacteristic
436C23#40711yescharacteristic
437C23#40811yescharacteristic
438C23#40811yescharacteristic
439C23C2×#10211yescharacteristic
440C23#41611yescharacteristic
441C23#41711yescharacteristic
442C23#41911yescharacteristic
443C23C2×#10411yescharacteristic
444C23C2×#10411yescharacteristic
445C23#42111yescharacteristic
446C23#42111yescharacteristic
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—48no—
465C2×C4—48no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4#38812yes—
515C2×C4#38812yes—
516C2×C4#39412yes—
517C2×C4#39412yes—
518C2×C4#39712yes—
519C2×C4#39712yes—
520C2×C4#43111yescharacteristic
521C2×C4#43111yescharacteristic
522C2×C4#43211yescharacteristic
523C2×C4#43211yescharacteristic
524C2×C4#43312yes—
525C2×C4#43312yes—
526C2×C4C8×D412yes—
527C2×C4C8×D412yes—
528C2×C4#43512yes—
529C2×C4#43512yes—
530C2×C4#43512yes—
531C2×C4#43512yes—
532C2×C4#43612yes—
533C2×C4#43612yes—
534C2×C4#44711yescharacteristic
535C2×C4#44811yescharacteristic
536C2×C4#45212yes—
537C2×C4#46512yes—
538C2×C4#46612yes—
539C2×C4#46812yes—
540C2×C4#47012yes—
541C2×C4#48011yescharacteristic
542C2×C4#48012yes—
543C2×C4#48111yescharacteristic
544C2×C4#48212yes—
545C2×C4#48312yes—
546C2×C4#48512yes—
547C2×C4#48512yes—
548C8—48no—
549C8—84no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22#144011yescharacteristic
566C22C2×D4⋊C811yescharacteristic
567C22#147611yescharacteristic
568C22#147711yescharacteristic
569C22#147911yescharacteristic
570C22#148011yescharacteristic
571C22#148211yescharacteristic
572C22#148311yescharacteristic
573C22#148511yescharacteristic
574C22#148611yescharacteristic
575C22#148811yescharacteristic
576C22#148911yescharacteristic
577C22#149111yescharacteristic
578C22#149211yescharacteristic
579C22#149411yescharacteristic
580C22#149511yescharacteristic
581C22#149711yescharacteristic
582C22#181711yescharacteristic
583C22#181811yescharacteristic
584C22#182311yescharacteristic
585C22#182511yescharacteristic
586C22#182611yescharacteristic
587C22#182611yescharacteristic
588C22#186911yescharacteristic
589C22#186911yescharacteristic
590C22#187011yescharacteristic
591C22#187011yescharacteristic
592C22#187111yescharacteristic
593C22#187111yescharacteristic
594C22#187311yescharacteristic
595C22#187411yescharacteristic
596C22#189311yescharacteristic
597C22#189411yescharacteristic
598C22#189611yescharacteristic
599C22#189711yescharacteristic
600C4—22no—
601C4—22no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4#192312yes—
610C4#192312yes—
611C4#192412yes—
612C4#192412yes—
613C4#203312yes—
614C4#203512yes—
615C4#207512yes—
616C4#207712yes—
617C2—44no—
618C2#870611yescharacteristic
619C2#870811yescharacteristic
620C2#870911yescharacteristic
621C2#871111yescharacteristic
622C2#893811yescharacteristic
623C2#893911yescharacteristic
624C2#894111yescharacteristic
625C2#894411yescharacteristic
626C2#907211yescharacteristic
627C2#907311yescharacteristic
628C2#907611yescharacteristic
629C2#907711yescharacteristic
630C2#1114711yescharacteristic
631C2#1119911yescharacteristic
632C2#1120111yescharacteristic
subgroupG/H
centerC23×C4C2×D4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC24C2×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1279C2semidirect
5. C2×#1279C2semidirect
71. #380C8semidirect

Maximal subgroups

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

subgroupindex
C2×#17972
C2×C4×C22⋊C82
C4×#3802
C2×#12792