Finite Group Databasebeta

← #368589

Subgroups — #368589

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic
2C2×#1988C211yesmaximal, characteristic
3C2×C4×C4⋊C8C211yesmaximal, characteristic
4#14714C211yesmaximal, characteristic
5C4×#389C211yesmaximal, characteristic
6#11315C212yesmaximal
7#11292C212yesmaximal
8#10880C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10849C211yesmaximal, characteristic
11#10820C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2212yes—
16C2×#391—21no—
17C2×#391—21no—
18C2×#389C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4—21no—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C43C2211yescharacteristic
25#1988—24no—
26#1988C418yescomplemented
27C4×C4⋊C8—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1436—21no—
38#1436—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C4×#95—21no—
46C23×Q8—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8C2×C414yes—
80C2×C4⋊C8D414yes—
81C2×C4×C8—24no—
82C2×C4×C8—24no—
83C2×C4×C8—42no—
84#391—44no—
85#391—44no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389Q814yescomplemented
90#384—22no—
91#384—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—44no—
96C4×C4⋊C4C2×C414yes—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C43—22no—
110C43—22no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—42no—
116#340—42no—
117#340C2×C418yes—
118#340C2×C418yes—
119#336—24no—
120C22×Q8—24no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8C22×C414yes—
153C22×C8C2×D414yes—
154C4⋊Q8—28no—
155C4⋊Q8—28no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C22×C414yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C2×Q812yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C4⋊C4#4112yes—
208C2×C4⋊C4#4112yes—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42D814yes—
236C2×C42SD1614yes—
237C2×C42C22×C412yes—
238C2×C42C2×D412yes—
239C2×C42C2×Q812yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C4⋊C8—28no—
246C4⋊C8—28no—
247C4×C8—48no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—42no—
260C24C22×D411yescharacteristic
261C2×Q8—48no—
262C2×Q8—48no—
263C2×Q8—48no—
264C2×Q8—84no—
265C2×Q8—84no—
266C2×Q8—84no—
267C2×Q8—84no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4C4×D412yes—
335C22×C4C4×D412yes—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4C22⋊Q812yes—
339C22×C4C22⋊Q812yes—
340C22×C4#12412yes—
341C22×C4#12412yes—
342C22×C4C2×D812yes—
343C22×C4C2×D812yes—
344C22×C4C2×D812yes—
345C22×C4C2×SD1612yes—
346C22×C4C2×SD1612yes—
347C22×C4C2×SD1612yes—
348C22×C4#13512yes—
349C22×C4#13512yes—
350C22×C4#13712yes—
351C22×C4#13712yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+411yescharacteristic
364C22×C42−1+411yescharacteristic
365C22×C42−1+411yescharacteristic
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—48no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C4⋊C4C4×Q814yes—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—28no—
414C42—28no—
415C42—44no—
416C42—44no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—48no—
422C42—84no—
423C42#11714yes—
424C42C4×D414yes—
425C42#12314yes—
426C42#12614yes—
427C23C2×C4×D411yescharacteristic
428C23C2×#12111yescharacteristic
429C23C2×C22⋊Q811yescharacteristic
430C23#52511yescharacteristic
431C23C2×#12411yescharacteristic
432C23#53611yescharacteristic
433C23#53711yescharacteristic
434C23C22×D811yescharacteristic
435C23C22×SD1611yescharacteristic
436C23C2×#13511yescharacteristic
437C23C2×#13511yescharacteristic
438C23C2×#13711yescharacteristic
439C23C2×#13711yescharacteristic
440C23#57511yescharacteristic
441C23#57511yescharacteristic
442Q8—816no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
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—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4#38814yes—
518C2×C4#38814yes—
519C2×C4C4×D814yes—
520C2×C4C4×SD1614yes—
521C2×C4#46514yes—
522C2×C4#46614yes—
523C2×C4#47414yes—
524C2×C4#47514yes—
525C2×C4#48614yes—
526C2×C4#48814yes—
527C2×C4C2×#11711yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×C4×Q811yescharacteristic
530C2×C4C4×#4111yescharacteristic
531C2×C4#51911yescharacteristic
532C2×C4#52011yescharacteristic
533C2×C4C2×#12311yescharacteristic
534C2×C4C2×#12611yescharacteristic
535C2×C4#52911yescharacteristic
536C2×C4#53311yescharacteristic
537C2×C4#54111yescharacteristic
538C2×C4#54211yescharacteristic
539C2×C4#54611yescharacteristic
540C2×C4#54711yescharacteristic
541C2×C4#54811yescharacteristic
542C2×C4D4×Q811yescharacteristic
543C2×C4#55211yescharacteristic
544C2×C4#55211yescharacteristic
545C2×C4#55411yescharacteristic
546C2×C4#55611yescharacteristic
547C2×C4#55611yescharacteristic
548C2×C4#55611yescharacteristic
549C2×C4#55611yescharacteristic
550C2×C4#55611yescharacteristic
551C2×C4#55711yescharacteristic
552C2×C4#55711yescharacteristic
553C2×C4#55911yescharacteristic
554C2×C4#55911yescharacteristic
555C8—44no—
556C8—44no—
557C22C2×#38811yescharacteristic
558C22C4×C22⋊Q811yescharacteristic
559C22#232311yescharacteristic
560C22#241511yescharacteristic
561C22#241711yescharacteristic
562C22#256811yescharacteristic
563C22#259211yescharacteristic
564C22C2×C4×D811yescharacteristic
565C22C2×C4×SD1611yescharacteristic
566C22C4×#13711yescharacteristic
567C22C4×#13711yescharacteristic
568C22C2×#46511yescharacteristic
569C22C2×#46611yescharacteristic
570C22#305611yescharacteristic
571C22#305611yescharacteristic
572C22C2×#47411yescharacteristic
573C22C2×#47511yescharacteristic
574C22#307611yescharacteristic
575C22#307611yescharacteristic
576C22#310011yescharacteristic
577C22#310311yescharacteristic
578C22#310711yescharacteristic
579C22#310711yescharacteristic
580C22C2×#48611yescharacteristic
581C22C2×#48811yescharacteristic
582C22#313911yescharacteristic
583C22#314011yescharacteristic
584C22#322611yescharacteristic
585C22#322911yescharacteristic
586C22#323311yescharacteristic
587C22#323311yescharacteristic
588C22#324511yescharacteristic
589C22#324611yescharacteristic
590C22#324811yescharacteristic
591C22#324811yescharacteristic
592C4—24no—
593C4—24no—
594C4—24no—
595C4—44no—
596C4—42no—
597C4—42no—
598C4—82no—
599C4—84no—
600C4—82no—
601C4—82no—
602C4—162no—
603C4#193418yes—
604C4#235511yescharacteristic
605C4#235611yescharacteristic
606C4#236511yescharacteristic
607C4#236711yescharacteristic
608C4#244611yescharacteristic
609C4#244711yescharacteristic
610C4#251111yescharacteristic
611C4#251411yescharacteristic
612C2#1402111yescharacteristic
613C2C2×#193411yescharacteristic
614C2#2109411yescharacteristic
615C2C4×#47411yescharacteristic
616C2C4×#47511yescharacteristic
617C2#2211011yescharacteristic
618C2#2211211yescharacteristic
619C2#2470011yescharacteristic
620C2#2470311yescharacteristic
621C2#2483511yescharacteristic
622C2#2483811yescharacteristic
623C2#2921911yescharacteristic
624C2#2922211yescharacteristic
625C2#2934611yescharacteristic
626C2#2934911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1988C4semidirect
89. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19882
C2×C4×C4⋊C82
#147142
C4×#3892
#113152
#112922
#108802
#108492
#108202