Finite Group Databasebeta

← #6344062

Subgroups — #6344062

Order 512.

2765 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62813C211yesmaximal, characteristic
2#60277C211yesmaximal, characteristic
3C2×#2925C211yesmaximal, characteristic
4#17761C218yesmaximal, complemented
5#17760C218yesmaximal, complemented
6#17734C214yesmaximal, complemented
7#17718C214yesmaximal, complemented
8#17708C212yesmaximal, complemented
9C2×#1542C212yesmaximal, complemented
10#3504—24no—
11#3503C2214yes—
12#3503C2214yes—
13#3502—21no—
14#3502C412yes—
15#3501—22no—
16#3501—21no—
17#3501C414yes—
18#3501C412yes—
19Q8×#41—24no—
20Q8×#41C418yes—
21#3458—21no—
22#3458—22no—
23C2×#533C2212yes—
24C2×#533C2214yes—
25#3428—21no—
26C2×C4×#41C2211yescharacteristic
27#2989C2218yes—
28#2983—21no—
29#2975—21no—
30#2973—22no—
31#2965—24no—
32#2925—24no—
33#2925C2218yes—
34#2920C2214yes—
35C2×#433—22no—
36C2×#432C2212yes—
37C2×#431C2212yes—
38C2×#431C2212yes—
39C2×#423C2211yescharacteristic
40C22×C4⋊C8C2211yescharacteristic
41C2×#408—21no—
42C2×#405—21no—
43#1559C22116yescomplemented
44#1558C2218yescomplemented
45#1556C2218yescomplemented
46#1554C22116yescomplemented
47#1550C2218yescomplemented
48#1549C2214yescomplemented
49#1548C2214yescomplemented
50#1548C2218yescomplemented
51#1546C2218yescomplemented
52#1542C2214yes—
53#1542C2214yes—
54#1542C2218yes—
55#1542C2218yescomplemented
56C2×#333C2214yescomplemented
57C2×#332C2214yescomplemented
58C22×#41D411yescharacteristic
59C22×M16—21no—
60C23×C8—21no—
61Q82—24no—
62Q82C2×C418yes—
63#557—28no—
64#557—28no—
65#557—24no—
66#557C2×C418yes—
67#554—28no—
68#554—216no—
69#554—216no—
70#554—28no—
71#554—28no—
72#554—24no—
73#554C2×C4116yes—
74#554C2×C418yes—
75D4×Q8—28no—
76D4×Q8—24no—
77D4×Q8C2×C418yes—
78D4×Q8C2×C4116yes—
79#544—24no—
80#544—24no—
81#544—28no—
82#544—28no—
83#544—24no—
84#544—22no—
85#544—24no—
86#544—22no—
87#543—24no—
88#543—24no—
89#543—24no—
90#533—216no—
91#533—28no—
92#533—24no—
93#533—22no—
94#533—22no—
95#533—24no—
96#533—22no—
97#533—24no—
98#533—22no—
99#533—24no—
100#533C2×C414yes—
101#533C2×C418yes—
102#533C2×C414yes—
103#533C2×C418yes—
104#533C2314yes—
105#533C2314yes—
106#533C2318yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×#125—22no—
112C2×#125—24no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8C2×C414yes—
116C2×C22⋊Q8C2×C418yes—
117#520—24no—
118#520—24no—
119#519—22no—
120#518—22no—
121#518—24no—
122C4×#41—28no—
123C4×#41C2×C414yes—
124C4×#41D414yes—
125C4×#41D414yes—
126C4×#41C2314yes—
127C2×C4×Q8—22no—
128C2×C4×Q8—24no—
129C2×C4×Q8C2×C414yes—
130C2×C4×Q8C2×C418yes—
131C2×C4×Q8C2×C411yescharacteristic
132C2×C4×Q8D411yescharacteristic
133C2×C4×D4C2×C412yes—
134C2×C4×D4C2×C411yescharacteristic
135C2×C4×D4D412yes—
136C2×C4×D4D411yescharacteristic
137C2×#117—24no—
138C2×#117—22no—
139C2×#117D411yescharacteristic
140C2×#117D412yes—
141C22×C42C2311yescharacteristic
142C22×C42C2312yes—
143#446—28no—
144#436—28no—
145#435—216no—
146#433—216no—
147#432—24no—
148#432—24no—
149#432C2318yes—
150#432C2318yes—
151#431—24no—
152#431—22no—
153#431—22no—
154#431—24no—
155#431C2314yes—
156#431C2314yes—
157#431C2318yes—
158#423—22no—
159#423C2318yes—
160#423C2314yes—
161C2×C4⋊C8—21no—
162C2×C4⋊C8—22no—
163C2×C4⋊C8—21no—
164C2×C4⋊C8C2314yes—
165C2×C4⋊C8C2314yes—
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2311yescharacteristic
168C2×C4⋊C8C2312yes—
169#408—28no—
170#407—24no—
171C2×C22⋊C8—22no—
172C2×C22⋊C8—22no—
173C2×C22⋊C8—22no—
174#405—28no—
175C4×M16—24no—
176C2×#97—22no—
177C2×C4×C8—22no—
178C2×C4×C8—22no—
179#333C2318yes—
180#333C2318yes—
181#333C23116yes—
182#332C2318yes—
183#332C2318yes—
184#332C23116yes—
185C2×#41—24no—
186C2×#41—24no—
187C2×#41C22⋊C414yes—
188C2×#41C2×D414yes—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8C22⋊C411yescharacteristic
192C22×Q8C22×C414yes—
193C22×Q8C22×C418yes—
194C22×D4C22⋊C412yes—
195C22×D4C22⋊C411yescharacteristic
196C23×C4C22×C412yes—
197C23×C4C22×C411yescharacteristic
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D412yes—
201C23×C4C2×D411yescharacteristic
202C2×M16—24no—
203C2×M16—28no—
204C2×M16—44no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—24no—
208C22×C8—22no—
209C22×C8—24no—
210C22×C8—44no—
211C4⋊Q8—216no—
212C4⋊Q8—28no—
213C4⋊Q8—216no—
214C4⋊Q8—28no—
215C4⋊Q8—24no—
216C4⋊Q8—24no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—24no—
221C4⋊Q8—22no—
222C4⋊Q8—24no—
223C4⋊Q8C22×C414yes—
224C4⋊Q8C22×C418yes—
225C4⋊Q8C22×C414yes—
226C4⋊Q8C22×C418yes—
227C4⋊Q8C2414yes—
228C4⋊Q8C2414yes—
229#125—216no—
230#125—28no—
231#125—216no—
232#125—28no—
233#125—28no—
234#125—24no—
235#125—28no—
236#125—24no—
237#125—24no—
238#125—28no—
239#125—24no—
240C22⋊Q8—28no—
241C22⋊Q8—24no—
242C22⋊Q8—28no—
243C22⋊Q8—24no—
244C22⋊Q8—28no—
245C22⋊Q8—24no—
246C22⋊Q8—28no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—28no—
251C22⋊Q8—216no—
252C22⋊Q8—216no—
253C22⋊Q8—216no—
254C22⋊Q8—216no—
255C22⋊Q8—216no—
256C22⋊Q8—216no—
257C22⋊Q8C22×C418yes—
258C22⋊Q8C22×C418yes—
259C22⋊Q8C22×C4116yes—
260C4×Q8—28no—
261C4×Q8—28no—
262C4×Q8—28no—
263C4×Q8—24no—
264C4×Q8—28no—
265C4×Q8—24no—
266C4×Q8—28no—
267C4×Q8—24no—
268C4×Q8—28no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—28no—
273C4×Q8C22⋊C414yes—
274C4×Q8C22⋊C414yes—
275C4×Q8C22×C418yes—
276C4×Q8C22×C418yes—
277C4×Q8C22×C4116yes—
278C4×Q8C22×C414yes—
279C4×Q8C2×D414yes—
280C4×D4—216no—
281C4×D4—28no—
282C4×D4—216no—
283C4×D4—28no—
284C4×D4C22⋊C418yes—
285C4×D4C22⋊C414yes—
286C4×D4C22⋊C418yes—
287C4×D4C22⋊C414yes—
288C4×D4C22×C418yes—
289C4×D4C22×C414yes—
290C4×D4C2×D418yes—
291C4×D4C2×D414yes—
292#117—28no—
293#117—24no—
294#117—28no—
295#117—24no—
296#117—24no—
297#117—28no—
298#117—24no—
299#117C22⋊C418yes—
300#117C22⋊C414yes—
301#117C2×D414yes—
302#117C2×D418yes—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4C22⋊C412yes—
314C2×C4⋊C4C22⋊C411yescharacteristic
315C2×C4⋊C4C22⋊C412yes—
316C2×C4⋊C4C22⋊C411yescharacteristic
317C2×C4⋊C4C22×C414yes—
318C2×C4⋊C4C22×C418yes—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4C22⋊C412yes—
322C2×C22⋊C4C22⋊C412yes—
323C2×C22⋊C4C22⋊C412yes—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42C22×C412yes—
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C414yes—
329C2×C42C22×C412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D414yes—
336C2×C42C2×D412yes—
337C2×C42C2×D414yes—
338C2×C42C2412yes—
339C2×C42C2414yes—
340C2×C42C2411yescharacteristic
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8—24no—
344C4⋊C8—24no—
345C4⋊C8—22no—
346C4⋊C8—24no—
347C4⋊C8—28no—
348C4⋊C8C2412yes—
349C4⋊C8C2412yes—
350C4⋊C8C2418yes—
351C4⋊C8C2414yes—
352C22⋊C8—24no—
353C22⋊C8—28no—
354C22⋊C8—28no—
355C22⋊C8—24no—
356C22⋊C8—28no—
357#97—28no—
358#97—28no—
359C4×C8—24no—
360C4×C8—28no—
361C4×C8—24no—
362C24C2×C22⋊C412yes—
363C24C2×C22⋊C411yescharacteristic
364#41—28no—
365#41—28no—
366#41—216no—
367C2×Q8—28no—
368C2×Q8—24no—
369C2×Q8—28no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—84no—
374C2×Q8—88no—
375C2×Q8—84no—
376C2×Q8—88no—
377C2×Q8C2×C22⋊C414yes—
378C2×Q8C23×C418yes—
379C2×D4—28no—
380C2×D4—24no—
381C2×D4C2×C22⋊C418yes—
382C2×D4C2×C22⋊C414yes—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—81no—
392C22×C4—82no—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C414yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C414yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C414yes—
402C22×C4C2×C22⋊C412yes—
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C412yes—
405C22×C4#13714yes—
406C22×C4#13714yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C23×C414yes—
409C22×C4C23×C412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D414yes—
412C22×C4C22×D412yes—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416M16—48no—
417M16—88no—
418C2×C8—22no—
419C2×C8—28no—
420C2×C8—24no—
421C2×C8—22no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—48no—
425C2×C8—44no—
426C2×C8—84no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—216no—
436C4⋊C4—28no—
437C4⋊C4—216no—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4C2×C22⋊C418yes—
445C4⋊C4C2×C22⋊C414yes—
446C4⋊C4C2×C22⋊C418yes—
447C4⋊C4C2×C22⋊C414yes—
448C4⋊C4C23×C418yes—
449C22⋊C4—28no—
450C22⋊C4—216no—
451C22⋊C4—28no—
452C22⋊C4C2×C22⋊C418yes—
453C22⋊C4C2×C22⋊C418yes—
454C22⋊C4C2×C22⋊C418yes—
455C42—216no—
456C42—28no—
457C42—28no—
458C42C2×C22⋊C414yes—
459C42C2×C22⋊C414yes—
460C42C2×C22⋊C412yes—
461C42C2×C22⋊C412yes—
462C42C2×C22⋊C414yes—
463C42C2×C22⋊C412yes—
464C42C23×C414yes—
465C42C23×C412yes—
466C42C23×C411yescharacteristic
467C42C22×D414yes—
468C42C22×D412yes—
469C42C22×D414yes—
470C42C22×D411yescharacteristic
471C42C22×D412yes—
472C42C22×D411yescharacteristic
473C42C2511yescharacteristic
474C23—24no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23#41914yes—
480C23C22×C22⋊C414yes—
481C23C22×C22⋊C411yescharacteristic
482C23C22×C22⋊C412yes—
483Q8—28no—
484Q8—48no—
485Q8—48no—
486Q8—84no—
487Q8—88no—
488Q8—84no—
489Q8—88no—
490D4—216no—
491D4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—22no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—84no—
513C2×C4—88no—
514C2×C4—84no—
515C2×C4—88no—
516C2×C4#41914yes—
517C2×C4#41918yes—
518C2×C4C22×C22⋊C414yes—
519C2×C4C22×C22⋊C411yescharacteristic
520C2×C4C22×C22⋊C411yescharacteristic
521C2×C4C22×C22⋊C414yes—
522C2×C4C22×C22⋊C412yes—
523C2×C4C22×C22⋊C414yes—
524C2×C4C22×C22⋊C412yes—
525C2×C4C22×C22⋊C412yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×#13712yes—
528C2×C4C2×#13712yes—
529C2×C4C2×#13714yes—
530C2×C4C2×#13714yes—
531C2×C4C24×C411yescharacteristic
532C2×C4C23×D411yescharacteristic
533C2×C4C23×D411yescharacteristic
534C8—48no—
535C8—42no—
536C8—42no—
537C8—44no—
538C22—21no—
539C22—22no—
540C22—21no—
541C22—22no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—44no—
546C22—84no—
547C22C2×#41912yes—
548C22C2×#41914yes—
549C22C23×C22⋊C411yescharacteristic
550C4—21no—
551C4—21no—
552C4—22no—
553C4—24no—
554C4—24no—
555C4—21no—
556C4—22no—
557C4—24no—
558C4—48no—
559C4—48no—
560C4C2×#41912yes—
561C4C2×#41914yes—
562C4C22×#13711yescharacteristic
563C4C22×#13711yescharacteristic
564C4#358412yes—
565C2—22no—
566C2—24no—
567C2—44no—
568C2#6007611yescharacteristic
569C2C22×#41911yescharacteristic
570C2#6008611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C23×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17761C2semidirect
5. #17760C2semidirect
6. #17734C2semidirect
7. #17718C2semidirect
8. #17708C2semidirect
9. C2×#1542C2semidirect
43. #1559C22semidirect
44. #1558C22semidirect
45. #1556C22semidirect
46. #1554C22semidirect
47. #1550C22semidirect
48. #1549C22semidirect
49. #1548C22semidirect
50. #1548C22semidirect
51. #1546C22semidirect
55. #1542C22semidirect
56. C2×#333C22semidirect
57. C2×#332C22semidirect

Maximal subgroups

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

subgroupindex
#628132
#602772
C2×#29252
#177612
#177602
#177342
#177182
#177082
C2×#15422