Finite Group Databasebeta

← #214570

Subgroups — #214570

Order 512.

1005 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#21847C211yesmaximal, characteristic
3#20721C211yesmaximal, characteristic
4#9138C212yesmaximal, complemented
5#9132C212yesmaximal, complemented
6#9131C212yesmaximal, complemented
7#9130C212yesmaximal, complemented
8#8726C212yesmaximal, complemented
9#8340C212yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2857C2212yes—
17#2857C2212yes—
18#2628—21no—
19#2628—21no—
20#2568—21no—
21#2567C412yes—
22#2563C412yes—
23#2542—21no—
24#2487C412yes—
25#2487C412yes—
26#2409—21no—
27#2409—21no—
28#2317—21no—
29#2311—21no—
30#2285C2211yescharacteristic
31#1984—21no—
32#1954—21no—
33#1869—21no—
34#1869—21no—
35#1818—21no—
36#1797C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39#1797C2212yes—
40#1780—21no—
41#1750—21no—
42#1750—21no—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1296C2212yes—
51C22×#41D411yescharacteristic
52C23×Q8—21no—
53C23×C8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—41no—
57C2×C4⋊Q8—41no—
58C2×C4⋊Q8—41no—
59C2×C4⋊Q8—41no—
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8C2312yes—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C4×#41—41no—
65C4×#41—41no—
66C4×#41—41no—
67C4×#41—41no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×#117—21no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117D411yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2×C412yes—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#432—41no—
86#432—41no—
87#431—41no—
88#431—41no—
89#431—41no—
90#431—41no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—41no—
93C2×C4⋊C8—41no—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108#400—22no—
109#400—22no—
110#399—22no—
111#399—22no—
112#395—22no—
113#395—22no—
114#393—22no—
115#393—22no—
116#391—22no—
117#391—22no—
118#391—22no—
119#390—22no—
120#389—22no—
121#389—22no—
122#389C2×C414yes—
123#389C2×C414yes—
124#386—22no—
125#386—22no—
126#385—22no—
127#385—22no—
128#384—22no—
129#384C2312yes—
130#382—22no—
131#382—21no—
132#380—22no—
133#380—22no—
134#380—21no—
135#380C2×C414yes—
136#380C2×C414yes—
137#380C2×C412yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2×C412yes—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#332—42no—
145#332—42no—
146#332—42no—
147#332—42no—
148C2×#41—24no—
149C2×#41—24no—
150C2×#41—41no—
151C2×#41—41no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—82no—
157C22×Q8—82no—
158C22×Q8C22⋊C411yescharacteristic
159C22×D4C22⋊C411yescharacteristic
160C22×D4C22⋊C411yescharacteristic
161C22×D4C22⋊C411yescharacteristic
162C23×C4—22no—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—42no—
177C22×C8—82no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187C4⋊Q8—42no—
188C22⋊Q8—82no—
189C22⋊Q8—82no—
190C22⋊Q8—82no—
191C22⋊Q8—82no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—41no—
212#117—41no—
213#117—41no—
214#117—44no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C22×C414yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22⋊C411yescharacteristic
239C2×C22⋊C4C22⋊C411yescharacteristic
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C22⋊C411yescharacteristic
242C2×C22⋊C4C22⋊C411yescharacteristic
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2411yescharacteristic
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—44no—
279C22⋊C8—44no—
280C22⋊C8—82no—
281C22⋊C8—82no—
282#97—42no—
283C4×C8—42no—
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⋊C411yescharacteristic
293C24C2×C22⋊C411yescharacteristic
294#41—48no—
295#41—48no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—44no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—84no—
307C2×Q8—88no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—84no—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4#13514yes—
364C22×C4#13714yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—84no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—44no—
406C22⋊C4—84no—
407C22⋊C4—84no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—44no—
427C42H(5,𝔽2)11yescharacteristic
428C42H(5,𝔽2)11yescharacteristic
429C42H(5,𝔽2)11yescharacteristic
430C42H(5,𝔽2)11yescharacteristic
431C23—21no—
432C23—21no—
433C23—22no—
434C23—22no—
435C23—21no—
436C23—21no—
437C23—22no—
438C23—22no—
439C23—21no—
440C23—22no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23#41612yes—
445C23#41912yes—
446C23C22×C22⋊C411yescharacteristic
447Q8—44no—
448Q8—88no—
449Q8—82no—
450Q8—82no—
451Q8—82no—
452Q8—82no—
453D4—44no—
454D4—44no—
455D4—44no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
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—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—88no—
504C2×C4#41612yes—
505C2×C4#41912yes—
506C2×C4#51811yescharacteristic
507C2×C4#51811yescharacteristic
508C2×C4#53411yescharacteristic
509C2×C4#53411yescharacteristic
510C2×C4#53411yescharacteristic
511C2×C4#53411yescharacteristic
512C2×C4C2×#13512yes—
513C2×C4C2×#13712yes—
514C2×C4#57612yes—
515C2×C4#57712yes—
516C8—82no—
517C8—82no—
518C8—82no—
519C8—82no—
520C22—22no—
521C22—22no—
522C22—24no—
523C22—24no—
524C22—41no—
525C22—41no—
526C22—42no—
527C22—42no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22#230811yescharacteristic
536C22#288912yes—
537C22C2×#41611yescharacteristic
538C22C2×#41911yescharacteristic
539C22#289712yes—
540C4—22no—
541C4—22no—
542C4—21no—
543C4—21no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—84no—
552C4—84no—
553C4#319012yes—
554C4#319012yes—
555C4#319812yes—
556C4#319911yescharacteristic
557C4#320011yescharacteristic
558C2—42no—
559C2—42no—
560C2—81no—
561C2#1772712yes—
562C2#1776112yes—
563C2#2175611yescharacteristic
564C2#2176211yescharacteristic
565C2#2177011yescharacteristic
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. #9138C2semidirect
5. #9132C2semidirect
6. #9131C2semidirect
7. #9130C2semidirect
8. #8726C2semidirect
9. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#218472
#207212
#91382
#91322
#91312
#91302
#87262
#83402