Finite Group Databasebeta

← #198319

Subgroups — #198319

Order 512.

1062 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2#60248C211yesmaximal, characteristic
3#21879C211yesmaximal, characteristic
4#17715C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#9252C214yesmaximal, complemented
7#9250C214yesmaximal, complemented
8#8004C211yesmaximal, characteristic
9#7934C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3078—21no—
13#3078—21no—
14#3078—22no—
15#3078—21no—
16#3078—21no—
17#3078—22no—
18C2×#479C414yes—
19C2×#477C2212yes—
20C2×#475C2212yes—
21#2984—22no—
22#2984—22no—
23#2983—22no—
24#2983—21no—
25#2982—21no—
26Q8×M16—22no—
27Q8×M16C2214yes—
28#2920—22no—
29#2920C2211yescharacteristic
30#2920C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#428C412yes—
33C2×#423C2211yescharacteristic
34C2×#423C2211yescharacteristic
35C2×#419C412yes—
36#1997C2212yes—
37#1966C2212yes—
38#1901—22no—
39#1801—22no—
40#1552—22no—
41#1549C2212yes—
42#1546C2212yes—
43#1544—22no—
44#1487—22no—
45#1487C2214yes—
46C2×Q8⋊C8—21no—
47C2×Q8⋊C8—21no—
48#1292C2214yes—
49#1283C2214yes—
50C23×Q8C2×C411yescharacteristic
51C22×M16—21no—
52C22×M16C2×C412yes—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125C2×C412yes—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8C2×C412yes—
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D412yes—
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8D411yescharacteristic
80C2×C4×Q8D412yes—
81C2×C4×Q8C2312yes—
82C2×#117C2×C412yes—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#479—24no—
88#479—24no—
89#479—24no—
90#479—24no—
91#477—22no—
92#477—22no—
93#477—22no—
94#477—22no—
95#475—22no—
96#475—22no—
97#475—22no—
98#475—22no—
99#446—28no—
100#446—24no—
101#446—24no—
102#446—28no—
103C8×Q8—24no—
104C8×Q8—24no—
105#432—24no—
106#432—24no—
107#432—21no—
108#432—21no—
109#432—21no—
110#432D412yes—
111#431—24no—
112#431—24no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431—21no—
119#431D412yes—
120#431C2312yes—
121#428—24no—
122#428—22no—
123#428—22no—
124C2×#107C2×C414yes—
125C2×#106C2×C414yes—
126#423—24no—
127#423—21no—
128#423—21no—
129#423—21no—
130#423—21no—
131#423—21no—
132#423—21no—
133#423—21no—
134#423D412yes—
135#423C2312yes—
136#423C2312yes—
137C2×C4⋊C8—21no—
138C2×C4⋊C8—21no—
139C2×C4⋊C8—21no—
140C2×C4⋊C8C2×C412yes—
141C2×C4⋊C8C2312yes—
142#419—22no—
143#419—24no—
144#419—22no—
145C2×#103C2×C414yes—
146C2×#103C2×C414yes—
147C2×#101—24no—
148#407—21no—
149#407D412yes—
150C2×C22⋊C8—21no—
151C4×M16—24no—
152C4×M16—21no—
153C4×M16C2312yes—
154C2×C4×C8—21no—
155#343—24no—
156#332—24no—
157#332—24no—
158Q8⋊C8—24no—
159Q8⋊C8—24no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—24no—
166C22×Q8—42no—
167C22×Q8C22⋊C411yescharacteristic
168C22×Q8C22⋊C412yes—
169C22×Q8C22⋊C411yescharacteristic
170C22×Q8C22×C412yes—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4111yescharacteristic
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—44no—
181C2×M16—41no—
182C2×M16—82no—
183C2×M16C2×D414yes—
184C22×C8—41no—
185C22×C8—41no—
186C22×C8C22×C414yes—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195#125—22no—
196#125—22no—
197#125—22no—
198#125—22no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—21no—
210C4×Q8—22no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—22no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—22no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8C22⋊C412yes—
232C4×Q8C22⋊C412yes—
233C4×Q8C22⋊C412yes—
234C4×Q8C22⋊C412yes—
235C4×Q8C2×D412yes—
236C4×Q8C2×D412yes—
237C4×Q8C2×D412yes—
238C4×Q8C2×D412yes—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C22⋊C412yes—
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C22⋊C4—22no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C22×C411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42#4111yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42C2411yescharacteristic
286#107—24no—
287#107—24no—
288#106—24no—
289#106—24no—
290C4⋊C8—28no—
291C4⋊C8—28no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—22no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300C4⋊C8—22no—
301C4⋊C8—22no—
302C4⋊C8—41no—
303C4⋊C8—41no—
304C4⋊C8—41no—
305C4⋊C8C2×D414yes—
306#103—24no—
307#103—24no—
308#103—24no—
309#103—24no—
310#101—84no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—41no—
317#97—28no—
318#97—22no—
319#97—22no—
320C4×C8—28no—
321C4×C8—22no—
322C4×C8—22no—
323C4×C8—22no—
324C4×C8—41no—
325C24C4×D411yescharacteristic
326C2×Q8—22no—
327C2×Q8—21no—
328C2×Q8—22no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—44no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—82no—
347C2×Q8#10114yescomplemented
348C2×Q8C2×C22⋊C412yes—
349C2×Q8C2×C22⋊C412yes—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—44no—
365C22×C4—81no—
366C22×C4—82no—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4#12011yescharacteristic
375C22×C4#12011yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12111yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12311yescharacteristic
381C22×C4#12311yescharacteristic
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385M16—24no—
386M16—24no—
387M16—44no—
388M16—42no—
389M16—42no—
390M16—44no—
391M16—44no—
392M16—82no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—44no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—82no—
400C2×C8—82no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4C2×C22⋊C412yes—
426C4⋊C4C2×C22⋊C412yes—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—42no—
438C42—42no—
439C42C2×C22⋊C411yescharacteristic
440C42C2×C22⋊C411yescharacteristic
441C42C4×D411yescharacteristic
442C42C4×D411yescharacteristic
443C42#12011yescharacteristic
444C42#12011yescharacteristic
445C42#12111yescharacteristic
446C42#12111yescharacteristic
447C42#12111yescharacteristic
448C42#12111yescharacteristic
449C42#12311yescharacteristic
450C42#12311yescharacteristic
451C42C22×D411yescharacteristic
452C42C22×D411yescharacteristic
453C42C22×D411yescharacteristic
454C42C2×#4111yescharacteristic
455C23—22no—
456C23—22no—
457C23—41no—
458C23#38611yescharacteristic
459C23#38611yescharacteristic
460C23C2×C4×D411yescharacteristic
461Q8—22no—
462Q8—24no—
463Q8—22no—
464Q8—22no—
465Q8—44no—
466Q8—44no—
467Q8—42no—
468Q8—82no—
469Q8—82no—
470Q8—82no—
471Q8—81no—
472Q8—81no—
473Q8C2×#10114yes—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4#38611yescharacteristic
511C2×C4#38611yescharacteristic
512C2×C4#38611yescharacteristic
513C2×C4#38611yescharacteristic
514C2×C4#38611yescharacteristic
515C2×C4#38611yescharacteristic
516C2×C4C2×#10112yes—
517C2×C4#41312yes—
518C2×C4C22×C22⋊C411yescharacteristic
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×#12011yescharacteristic
521C2×C4C2×#12111yescharacteristic
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×#12311yescharacteristic
524C2×C4#57712yes—
525C2×C4#57812yes—
526C8—44no—
527C8—44no—
528C8—44no—
529C8—82no—
530C8—82no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—44no—
539C22#190112yes—
540C22C2×#38611yescharacteristic
541C22#294712yes—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—24no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—82no—
556C4#190112yes—
557C4C22×#10111yescharacteristic
558C4C2×#41311yescharacteristic
559C4#301312yes—
560C4#304311yescharacteristic
561C4#304411yescharacteristic
562C2—22no—
563C2—22no—
564C2—41no—
565C2#2093011yescharacteristic
566C2C2×#190111yescharacteristic
567C2#2099511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9252C2semidirect
7. #9250C2semidirect
347. C2×Q8#101semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#602482
#218792
#177152
C2×#14872
#92522
#92502
#80042
#79342