Finite Group Databasebeta

← #196656

Subgroups — #196656

Order 512.

1220 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×#2925C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#21847C211yesmaximal, characteristic
4C2×#1550C211yesmaximal, characteristic, complemented
5C2×#1542C211yesmaximal, characteristic, complemented
6#9138C212yesmaximal, complemented
7#9136C212yesmaximal, complemented
8#9130C212yesmaximal, complemented
9#9128C212yesmaximal, complemented
10#8043C211yesmaximal, characteristic, complemented
11C2×#1296C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#2925—24no—
15#2925—24no—
16C2×#433—22no—
17C2×#432C2212yes—
18C2×#431C2211yescharacteristic
19C2×#431C2212yes—
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#408—21no—
24C2×#405—21no—
25#2857C2212yes—
26#2857C2212yes—
27#2629—22no—
28#2582—21no—
29#2582C412yes—
30#2538—21no—
31#2538C412yes—
32#2417—22no—
33#2417C414yes—
34#2316—21no—
35#2311—21no—
36C2×#384C2211yescharacteristic
37#1984—21no—
38#1954—21no—
39#1869—22no—
40#1797C2212yes—
41#1797C2212yes—
42#1550—24no—
43#1542—24no—
44C2×#333C2211yescharacteristic
45C2×#333C2211yescharacteristic
46C2×#333C2212yes—
47C2×#332C2211yescharacteristic
48C2×#332C2211yescharacteristic
49C2×#332C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54C22×#41D411yescharacteristic
55C22×M16—21no—
56C23×C8—21no—
57C23×C8—21no—
58#533—22no—
59#533—22no—
60#533—22no—
61#533—22no—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8C2×C412yes—
68C2×C22⋊Q8C2×C414yes—
69C4×#41—22no—
70C4×#41—22no—
71C4×#41—22no—
72C4×#41—22no—
73C2×C4×Q8—21no—
74C2×C4×Q8C2×C412yes—
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8D411yescharacteristic
77C2×C4×D4C2×C412yes—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4D412yes—
80C2×C4×D4D411yescharacteristic
81C2×#117—21no—
82C2×#117D411yescharacteristic
83C2×#117D412yes—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2×C412yes—
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89#433—216no—
90#432—24no—
91#432—24no—
92#431—24no—
93#431—24no—
94#431—22no—
95#431—22no—
96#423—22no—
97#423D414yes—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8D414yes—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2312yes—
111#408—28no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117#405—28no—
118C2×#97—22no—
119C2×C4×C8—22no—
120C2×C4×C8—21no—
121#398—24no—
122#398—24no—
123#396—24no—
124#396—24no—
125#391—22no—
126#391—22no—
127#391C2×C414yes—
128#390—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389C2×C414yes—
135#386—24no—
136#384—21no—
137#384—21no—
138#384C2×C412yes—
139#384C2312yes—
140#380—22no—
141#380—22no—
142#380—22no—
143#380—24no—
144#380C2×C414yes—
145#336—22no—
146#336—22no—
147#333—22no—
148#333—22no—
149#333—24no—
150#332—22no—
151#332—22no—
152#332—24no—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C22×Q8—21no—
160C22×Q8C22⋊C411yescharacteristic
161C22×Q8C22×C412yes—
162C22×D4C22⋊C412yes—
163C22×D4C22⋊C411yescharacteristic
164C23×C4—22no—
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C2×M16—24no—
172C2×M16—42no—
173C2×M16—42no—
174C22×C8—24no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—42no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8—81no—
185C22×C8—81no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194#125—42no—
195#125—42no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200C22⋊Q8—82no—
201C22⋊Q8—82no—
202C22⋊Q8—84no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219#117—22no—
220#117—22no—
221#117—24no—
222#117—24no—
223#117—42no—
224#117—42no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—81no—
240C2×C4⋊C4—81no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C412yes—
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C412yes—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C414yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4C22⋊C412yes—
252C2×C22⋊C4C22⋊C412yes—
253C2×C22⋊C4C22⋊C412yes—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42C22×C412yes—
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42#4112yes—
268C2×C42#4112yes—
269C2×C42C2411yescharacteristic
270C4⋊C8—24no—
271C4⋊C8—24no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—22no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—42no—
282C4⋊C8C2×D412yes—
283C4⋊C8C2×D412yes—
284C4⋊C8C2×D414yes—
285C22⋊C8—28no—
286C22⋊C8—28no—
287C22⋊C8—42no—
288C22⋊C8—82no—
289C22⋊C8—82no—
290#97—28no—
291C4×C8—28no—
292C4×C8—42no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297C24C2×C22⋊C412yes—
298C24C2×C22⋊C411yescharacteristic
299#41—44no—
300#41—44no—
301#41—44no—
302#41—44no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C4×D412yes—
357C22×C4C4×D412yes—
358C22×C4#12012yes—
359C22×C4#12112yes—
360C22×C4#12112yes—
361C22×C4#12312yes—
362C22×C4#13512yes—
363C22×C4#13512yes—
364C22×C4#13712yes—
365C22×C4#13712yes—
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369M16—48no—
370C2×C8—22no—
371C2×C8—24no—
372C2×C8—22no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—48no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—84no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—44no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—44no—
418C22⋊C4—84no—
419C22⋊C4—84no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—44no—
427C42C4×D412yes—
428C42C4×D412yes—
429C42#12012yes—
430C42#12112yes—
431C42#12112yes—
432C42#12312yes—
433C42C22×D411yescharacteristic
434C42C22×D411yescharacteristic
435C42C2×#4111yescharacteristic
436C42C2×#4111yescharacteristic
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23#38612yes—
449C23#41612yes—
450C23#41912yes—
451C23C22×C22⋊C411yescharacteristic
452Q8—44no—
453Q8—82no—
454Q8—82no—
455Q8—81no—
456Q8—81no—
457Q8—81no—
458Q8—81no—
459D4—48no—
460D4—44no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—42no—
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—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—164no—
502C2×C4#38612yes—
503C2×C4#38612yes—
504C2×C4#38612yes—
505C2×C4#41612yes—
506C2×C4#41912yes—
507C2×C4#45214yes—
508C2×C4#45214yes—
509C2×C4#48514yes—
510C2×C4#48514yes—
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C2×C4×D411yescharacteristic
513C2×C4C2×#12011yescharacteristic
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12111yescharacteristic
516C2×C4C2×#12311yescharacteristic
517C2×C4C2×#13511yescharacteristic
518C2×C4C2×#13511yescharacteristic
519C2×C4C2×#13711yescharacteristic
520C2×C4C2×#13711yescharacteristic
521C2×C4#57611yescharacteristic
522C2×C4#57711yescharacteristic
523C2×C4#57711yescharacteristic
524C2×C4#57811yescharacteristic
525C8—42no—
526C8—42no—
527C8—44no—
528C8—81no—
529C8—81no—
530C8—82no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22#187314yes—
545C22C2×#38611yescharacteristic
546C22#288811yescharacteristic
547C22#288911yescharacteristic
548C22C2×#41611yescharacteristic
549C22C2×#41911yescharacteristic
550C22#289711yescharacteristic
551C22#289811yescharacteristic
552C4—21no—
553C4—21no—
554C4—22no—
555C4—22no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—84no—
561C4—82no—
562C4#187314yes—
563C4C2×#45211yescharacteristic
564C4C2×#45211yescharacteristic
565C4#306011yescharacteristic
566C4#306111yescharacteristic
567C4#306211yescharacteristic
568C4#306311yescharacteristic
569C4C2×#48511yescharacteristic
570C4C2×#48511yescharacteristic
571C2—24no—
572C2—82no—
573C2#1772411yescharacteristic
574C2#1772511yescharacteristic
575C2#1776011yescharacteristic
576C2#1776111yescharacteristic
577C2C2×#187311yescharacteristic
578C2#2097611yescharacteristic
579C2#2099211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1550C2semidirect
5. C2×#1542C2semidirect
6. #9138C2semidirect
7. #9136C2semidirect
8. #9130C2semidirect
9. #9128C2semidirect
10. #8043C2semidirect
11. C2×#1296C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29252
#352802
#218472
C2×#15502
C2×#15422
#91382
#91362
#91302
#91282
#80432
C2×#12962