Finite Group Databasebeta

← #204991

Subgroups — #204991

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35180C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic, complemented
3C2×#1518C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic, complemented
6C2×#1502C211yesmaximal, characteristic
7#13804C211yesmaximal, characteristic, complemented
8#9716C214yesmaximal, complemented
9#9713C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2850C2214yes—
15#2639—22no—
16#2592—22no—
17#2568C412yescomplemented
18#2563C412yescomplemented
19#2562—22no—
20#2548C412yescomplemented
21#2547C412yescomplemented
22#2311—21no—
23#2311—21no—
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1939—22no—
29#1939—22no—
30#1797C2214yes—
31#1518—24no—
32#1517—24no—
33#1503—24no—
34#1502—24no—
35C2×#328C2212yes—
36C2×#328C2212yes—
37C2×#328C2214yes—
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2214yes—
41#1436—22no—
42C4⋊#95—21no—
43#1434C2212yes—
44C4×#95C2212yes—
45C22×#41—21no—
46#533—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50#532—22no—
51#532—22no—
52#532—22no—
53#532—22no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#127C2×C411yescharacteristic
58C2×#125—21no—
59C2×#124—21no—
60C2×#124C2×C414yes—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8C2312yes—
63C2×#121—21no—
64C2×#121C2312yes—
65C2×C4×Q8—21no—
66C2×C4×Q8C2×C412yes—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42—22no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#431—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—24no—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89#400—24no—
90#397—24no—
91#394—24no—
92#393—24no—
93#391—22no—
94#391—22no—
95#391C2×C414yes—
96#390—22no—
97#390—22no—
98#390C2×C414yes—
99#388—24no—
100#388—24no—
101#386—24no—
102#386—24no—
103#382—22no—
104#382C2×C414yes—
105#382D414yes—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—44no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4D414yes—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95D412yes—
122C2×#95C2311yescharacteristic
123C2×#95C2312yes—
124C2×#95C2311yescharacteristic
125#336—24no—
126#328—24no—
127#328—24no—
128#328—28no—
129#327—24no—
130#327—24no—
131#327—28no—
132C2×#41—42no—
133C2×#41—42no—
134C2×#41—42no—
135C2×#41—42no—
136C2×#41—42no—
137C2×#41—42no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8C22×C412yes—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C22×C412yes—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162#127—22no—
163#127—22no—
164#125—42no—
165#125—42no—
166#124—42no—
167#124—42no—
168#124—84no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—22no—
179#121—42no—
180#121—42no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—42no—
188C4×D4—42no—
189#117—42no—
190#117—42no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—44no—
225C2×C42—44no—
226C2×C42—82no—
227C2×C42—82no—
228C2×C42C22×C414yes—
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D414yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4112yes—
237C2×C42#4112yes—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—84no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C4×C8—42no—
246C4×C8—42no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—21no—
253#95—21no—
254#95—22no—
255#95—44no—
256#95—44no—
257#95—82no—
258#95C22⋊C414yes—
259#95C22⋊C414yes—
260#95C2×D412yes—
261#95C2×D412yes—
262#95C2×D414yes—
263C24—22no—
264C24C22×D411yescharacteristic
265#41—84no—
266#41—84no—
267#41—84no—
268#41—84no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—81no—
294C2×D4—81no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4C2×C22⋊C414yes—
335C22×C4C2×C22⋊C414yes—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4#12112yes—
343C22×C4#12112yes—
344C22×C4#13512yes—
345C22×C4#13512yes—
346C22×C4AGL(1,ℤ/8)12yes—
347C22×C4#13712yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—48no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—48no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—44no—
382C22⋊C4—84no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C22⋊C4—84no—
386C42—44no—
387C42—44no—
388C42—44no—
389C42—48no—
390C42—88no—
391C42C4×D412yes—
392C42C4×D412yes—
393C42C4×D412yes—
394C42C4×D412yes—
395C42#12412yes—
396C42#12412yes—
397C42#12712yes—
398C42#12712yes—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23C22×C22⋊C411yescharacteristic
411C23C2×#12111yescharacteristic
412C23C2×#12111yescharacteristic
413C23C2×#13511yescharacteristic
414C23C2×#13511yescharacteristic
415C23C2×AGL(1,ℤ/8)11yescharacteristic
416C23C2×#13711yescharacteristic
417Q8—82no—
418Q8—82no—
419Q8—84no—
420Q8—84no—
421D4—82no—
422D4—82no—
423D4—84no—
424D4—82no—
425D4—82no—
426D4—82no—
427D4—82no—
428D4—84no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—88no—
484C2×C4#39012yes—
485C2×C4#39012yes—
486C2×C4#39012yes—
487C2×C4#39012yes—
488C2×C4#41612yes—
489C2×C4#41612yes—
490C2×C4#41712yes—
491C2×C4#41712yes—
492C2×C4#46314yes—
493C2×C4#46314yes—
494C2×C4#46414yes—
495C2×C4#46414yes—
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×#12411yescharacteristic
499C2×C4C2×#12711yescharacteristic
500C8—82no—
501C8—82no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22C2×#39011yescharacteristic
522C22#288311yescharacteristic
523C22#288411yescharacteristic
524C22C2×#41611yescharacteristic
525C22C2×#41711yescharacteristic
526C22#290111yescharacteristic
527C22#290111yescharacteristic
528C22C2×#46311yescharacteristic
529C22C2×#46311yescharacteristic
530C22C2×#46411yescharacteristic
531C22C2×#46411yescharacteristic
532C4—22no—
533C4—22no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—44no—
539C4—44no—
540C4—44no—
541C4—82no—
542C4—84no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4#198314yes—
547C4#198314yes—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—162no—
553C2#1761611yescharacteristic
554C2#1761711yescharacteristic
555C2#1762811yescharacteristic
556C2#1763011yescharacteristic
557C2#2121011yescharacteristic
558C2#2121411yescharacteristic
559C2C2×#198311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21951C2semidirect
3. C2×#1518C2semidirect
5. C2×#1503C2semidirect
7. #13804C2semidirect
8. #9716C2semidirect
9. #9713C2semidirect
17. #2568C4semidirect
18. #2563C4semidirect
20. #2548C4semidirect
21. #2547C4semidirect

Maximal subgroups

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

subgroupindex
#351802
#219512
C2×#15182
C2×#15172
C2×#15032
C2×#15022
#138042
#97162
#97132