Finite Group Databasebeta

← #196735

Subgroups — #196735

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2988C211yesmaximal, characteristic
2#35297C211yesmaximal, characteristic
3#20721C211yesmaximal, characteristic
4C2×#1552C211yesmaximal, characteristic, complemented
5C2×#1544C211yesmaximal, characteristic, complemented
6#9132C212yesmaximal, complemented
7#9131C212yesmaximal, complemented
8#9130C212yesmaximal, complemented
9#9129C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic, complemented
11#8039C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#2988—24no—
15#2988—24no—
16C2×C8×Q8—21no—
17C2×C8×D4—22no—
18C2×C8×D4—21no—
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2212yes—
22C2×#431C2211yescharacteristic
23C22×C4⋊C8C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25#2857C2212yes—
26#2857C2212yes—
27#2625—21no—
28#2616—21no—
29#2584C412yes—
30#2581—21no—
31#2562C412yes—
32#2536—21no—
33#2485C414yes—
34#2415—22no—
35#2323—22no—
36#2285C2211yescharacteristic
37#1818—21no—
38#1797C2212yes—
39#1797C2212yes—
40#1780—21no—
41#1750—22no—
42#1552—24no—
43#1544—24no—
44C2×#332C2211yescharacteristic
45C2×#332C2211yescharacteristic
46C2×#332C2211yescharacteristic
47C2×#332C2211yescharacteristic
48C2×#332C2212yes—
49C2×#332C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54C22×#41D411yescharacteristic
55C23×C8—22no—
56C23×C8—21no—
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125C2×C412yes—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—22no—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41—22no—
71C4×#41—22no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C412yes—
78C2×C4×D4D412yes—
79C2×C4×D4D411yescharacteristic
80C2×#117—21no—
81C2×#117C2×C412yes—
82C2×#117D412yes—
83C2×#117D411yescharacteristic
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86C8×Q8—28no—
87C8×D4—216no—
88C8×D4—28no—
89#432—22no—
90#432—22no—
91#431—22no—
92#431—24no—
93#431—24no—
94#431—22no—
95#431—22no—
96#431D414yes—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—24no—
102C2×C4⋊C8D414yes—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2312yes—
108C2×C4⋊C8C2311yescharacteristic
109C2×C22⋊C8—22no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×#97—21no—
116C2×C4×C8—22no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119#400—22no—
120#398—22no—
121#397—24no—
122#395—22no—
123#394—22no—
124#394—24no—
125#391—22no—
126#391—22no—
127#389—24no—
128#389—22no—
129#389—22no—
130#389C2×C414yes—
131#389C2×C414yes—
132#388—24no—
133#388—24no—
134#387—24no—
135#384C2312yes—
136#382—21no—
137#380—22no—
138#380—22no—
139#380—21no—
140#380C2×C414yes—
141#380C2×C414yes—
142#380C2×C412yes—
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2×C412yes—
145#336—22no—
146#336—22no—
147#332—22no—
148#332—22no—
149#332—22no—
150#332—22no—
151#332—24no—
152#332—24no—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—24no—
156C2×#41—24no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8C22⋊C411yescharacteristic
160C22×D4C22⋊C411yescharacteristic
161C22×D4C22⋊C412yes—
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—28no—
172C22×C8—24no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—81no—
177C22×C8—82no—
178C22×C8—81no—
179C22×C8—82no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188#125—24no—
189#125—24no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C22⋊Q8—44no—
195C22⋊Q8—44no—
196C22⋊Q8—82no—
197C22⋊Q8—82no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214#117—22no—
215#117—22no—
216#117—24no—
217#117—24no—
218#117—24no—
219#117—24no—
220#117—81no—
221#117—81no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4C22⋊C412yes—
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C414yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22⋊C412yes—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42C22×C412yes—
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4112yes—
259C2×C42#4112yes—
260C2×C42C2411yescharacteristic
261C4⋊C8—28no—
262C4⋊C8—24no—
263C4⋊C8—22no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—22no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—22no—
272C4⋊C8C2×D412yes—
273C4⋊C8C2×D414yes—
274C4⋊C8C2×D412yes—
275C22⋊C8—28no—
276C22⋊C8—24no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279C22⋊C8—44no—
280C22⋊C8—44no—
281#97—42no—
282C4×C8—28no—
283C4×C8—24no—
284C4×C8—42no—
285#95—22no—
286#95—22no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291C24C2×C22⋊C411yescharacteristic
292C24C2×C22⋊C412yes—
293#41—28no—
294#41—28no—
295#41—216no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C412yes—
343C22×C4C4×D412yes—
344C22×C4C4×D412yes—
345C22×C4#12012yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12312yes—
349C22×C4#13514yes—
350C22×C4#13514yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—22no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—84no—
371C2×C8—88no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—44no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—84no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—44no—
397C22⋊C4—48no—
398C22⋊C4—82no—
399C22⋊C4—84no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—44no—
407C42—82no—
408C42C4×D412yes—
409C42C4×D412yes—
410C42#12012yes—
411C42#12112yes—
412C42#12112yes—
413C42#12312yes—
414C42C22×D411yescharacteristic
415C42C22×D411yescharacteristic
416C42C2×#4111yescharacteristic
417C42C2×#4111yescharacteristic
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—21no—
425C23—22no—
426C23—21no—
427C23—22no—
428C23#38612yes—
429C23#41614yes—
430C23C22×C22⋊C411yescharacteristic
431Q8—28no—
432Q8—81no—
433Q8—81no—
434Q8—81no—
435Q8—81no—
436Q8—81no—
437Q8—81no—
438Q8—82no—
439D4—28no—
440D4—216no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4#38612yes—
482C2×C4#38612yes—
483C2×C4#38612yes—
484C2×C4#41614yes—
485C2×C4C2×C4×D411yescharacteristic
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C2×#12011yescharacteristic
488C2×C4C2×#12111yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12311yescharacteristic
491C2×C4C2×#13512yes—
492C2×C4C2×#13512yes—
493C2×C4#57611yescharacteristic
494C2×C4#57711yescharacteristic
495C2×C4#57711yescharacteristic
496C2×C4#57811yescharacteristic
497C8—44no—
498C8—42no—
499C8—42no—
500C8—81no—
501C8—81no—
502C8—82no—
503C22—24no—
504C22—24no—
505C22—22no—
506C22—22no—
507C22—24no—
508C22—22no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22C2×#38611yescharacteristic
516C22#288912yes—
517C22C2×#41612yes—
518C22#289711yescharacteristic
519C22#289811yescharacteristic
520C4—22no—
521C4—22no—
522C4—24no—
523C4—21no—
524C4—21no—
525C4—22no—
526C4—42no—
527C4—42no—
528C4—84no—
529C4—82no—
530C4—82no—
531C4#301211yescharacteristic
532C4#301311yescharacteristic
533C4#305011yescharacteristic
534C4#305011yescharacteristic
535C4#305012yes—
536C4#309811yescharacteristic
537C4#309911yescharacteristic
538C2—44no—
539C2—42no—
540C2#1772512yes—
541C2#1772712yes—
542C2#2091511yescharacteristic
543C2#2096711yescharacteristic
544C2#2099111yescharacteristic
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×#1552C2semidirect
5. C2×#1544C2semidirect
6. #9132C2semidirect
7. #9131C2semidirect
8. #9130C2semidirect
9. #9129C2semidirect
10. #8046C2semidirect
11. #8039C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29882
#352972
#207212
C2×#15522
C2×#15442
#91322
#91312
#91302
#91292
#80462
#80392