Finite Group Databasebeta

← #6343179

Subgroups — #6343179

Order 512.

2785 conjugacy classes of subgroups.

The subgroup lattice has 564 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#17514C2116yesmaximal, complemented
5#17496C214yesmaximal, complemented
6#17486C212yesmaximal, complemented
7#17464C214yesmaximal, complemented
8C2×#1481C212yesmaximal, complemented
9#3504—24no—
10#3503C414yes—
11#3503C414yes—
12#3502—21no—
13#3502C412yes—
14#3501—22no—
15#3501—21no—
16#3501C414yes—
17#3501C412yes—
18Q8×#41—24no—
19Q8×#41C2218yes—
20#3458—21no—
21#3458—22no—
22C2×#533C2212yes—
23C2×#533C2214yes—
24#3428—21no—
25C2×C4×#41C2211yescharacteristic
26#2989—24no—
27#2983—21no—
28#2975—21no—
29#2973—22no—
30#2965C2218yes—
31#2925—24no—
32#2925C2218yes—
33#2920C2214yes—
34C2×#433—22no—
35C2×#432C2212yes—
36C2×#431C2212yes—
37C2×#431C2212yes—
38C2×#423C2211yescharacteristic
39C22×C4⋊C8C2211yescharacteristic
40C2×#408—21no—
41C2×#405—21no—
42#1497C22116yescomplemented
43#1493C2218yescomplemented
44#1491C22116yescomplemented
45#1487C2218yescomplemented
46#1485C22116yescomplemented
47#1481C2214yes—
48#1481C2214yes—
49#1481C2218yes—
50#1481C2218yescomplemented
51#1479C22116yescomplemented
52C2×Q8⋊C8C2218yescomplemented
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56Q82—24no—
57Q82C2×C418yes—
58#557—24no—
59#557—28no—
60#557—28no—
61#557C2×C418yes—
62#554—28no—
63#554—24no—
64#554—28no—
65#554—216no—
66#554—216no—
67#554—28no—
68#554C2×C4116yes—
69#554C2×C418yes—
70D4×Q8—28no—
71D4×Q8—24no—
72D4×Q8C2×C418yes—
73D4×Q8C2×C4116yes—
74#544—28no—
75#544—24no—
76#544—28no—
77#544—24no—
78#544—24no—
79#544—22no—
80#544—24no—
81#544—22no—
82#543—24no—
83#543—24no—
84#543—24no—
85#533—28no—
86#533—216no—
87#533—24no—
88#533—22no—
89#533—22no—
90#533—24no—
91#533—22no—
92#533—24no—
93#533—22no—
94#533—24no—
95#533C2×C414yes—
96#533C2×C418yes—
97#533C2×C414yes—
98#533C2×C418yes—
99#533C2314yes—
100#533C2314yes—
101#533C2318yes—
102C2×C4⋊Q8C2×C412yes—
103C2×C4⋊Q8C2×C414yes—
104C2×C4⋊Q8C2×C412yes—
105C2×C4⋊Q8C2×C414yes—
106C2×#125—22no—
107C2×#125—24no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8C2×C414yes—
111C2×C22⋊Q8C2×C418yes—
112#520—24no—
113#520—24no—
114#519—22no—
115#518—22no—
116#518—24no—
117C4×#41—28no—
118C4×#41C2×C414yes—
119C4×#41D414yes—
120C4×#41D414yes—
121C4×#41C2314yes—
122C2×C4×Q8—22no—
123C2×C4×Q8—24no—
124C2×C4×Q8C2×C411yescharacteristic
125C2×C4×Q8D411yescharacteristic
126C2×C4×Q8C2314yes—
127C2×C4×Q8C2318yes—
128C2×C4×D4C2×C412yes—
129C2×C4×D4C2×C411yescharacteristic
130C2×C4×D4D412yes—
131C2×C4×D4D411yescharacteristic
132C2×#117—24no—
133C2×#117—22no—
134C2×#117D411yescharacteristic
135C2×#117D412yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#446—28no—
139#436—28no—
140#435—216no—
141#433—216no—
142#432—24no—
143#432—24no—
144#432—24no—
145#432C2318yes—
146#431—24no—
147#431—22no—
148#431—22no—
149#431C2318yes—
150#431C2314yes—
151#431C2314yes—
152#431C2318yes—
153#423—22no—
154#423—24no—
155#423C2314yes—
156C2×C4⋊C8—21no—
157C2×C4⋊C8—22no—
158C2×C4⋊C8—21no—
159C2×C4⋊C8—22no—
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2314yes—
163C2×C4⋊C8C2312yes—
164#408—28no—
165#407—24no—
166C2×C22⋊C8—22no—
167C2×C22⋊C8—22no—
168C2×C22⋊C8—22no—
169#405—28no—
170C4×M16C2318yes—
171C2×#97—22no—
172C2×C4×C8—22no—
173C2×C4×C8C2314yes—
174Q8⋊C8C23116yes—
175Q8⋊C8C23116yes—
176Q8⋊C8C23132yes—
177C2×#41—24no—
178C2×#41—24no—
179C2×#41C22⋊C414yes—
180C2×#41C2×D414yes—
181C22×Q8—24no—
182C22×Q8—22no—
183C22×Q8C22⋊C411yescharacteristic
184C22×Q8C22×C414yes—
185C22×Q8C22×C418yes—
186C22×D4C22⋊C412yes—
187C22×D4C22⋊C411yescharacteristic
188C23×C4C22×C412yes—
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×D411yescharacteristic
194C2×M16—24no—
195C2×M16—28no—
196C2×M16—44no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—24no—
200C22×C8—22no—
201C22×C8—24no—
202C22×C8—44no—
203C4⋊Q8—216no—
204C4⋊Q8—28no—
205C4⋊Q8—216no—
206C4⋊Q8—28no—
207C4⋊Q8—24no—
208C4⋊Q8—22no—
209C4⋊Q8—24no—
210C4⋊Q8—22no—
211C4⋊Q8—24no—
212C4⋊Q8—22no—
213C4⋊Q8—24no—
214C4⋊Q8—22no—
215C4⋊Q8C22×C414yes—
216C4⋊Q8C22×C414yes—
217C4⋊Q8C22×C418yes—
218C4⋊Q8C22×C414yes—
219C4⋊Q8C22×C414yes—
220C4⋊Q8C22×C418yes—
221#125—216no—
222#125—28no—
223#125—216no—
224#125—28no—
225#125—28no—
226#125—24no—
227#125—28no—
228#125—24no—
229#125—24no—
230#125—24no—
231#125—28no—
232C22⋊Q8—28no—
233C22⋊Q8—24no—
234C22⋊Q8—28no—
235C22⋊Q8—24no—
236C22⋊Q8—28no—
237C22⋊Q8—24no—
238C22⋊Q8—28no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242C22⋊Q8—28no—
243C22⋊Q8—216no—
244C22⋊Q8—216no—
245C22⋊Q8—216no—
246C22⋊Q8—216no—
247C22⋊Q8—216no—
248C22⋊Q8—216no—
249C22⋊Q8C22×C418yes—
250C22⋊Q8C22×C418yes—
251C22⋊Q8C22×C4116yes—
252C4×Q8—28no—
253C4×Q8—28no—
254C4×Q8—28no—
255C4×Q8—24no—
256C4×Q8—28no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—28no—
260C4×Q8—24no—
261C4×Q8—28no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—28no—
265C4×Q8C22⋊C414yes—
266C4×Q8C22⋊C414yes—
267C4×Q8C22×C418yes—
268C4×Q8C22×C4116yes—
269C4×Q8C22×C414yes—
270C4×Q8C2×D414yes—
271C4×Q8C2418yes—
272C4×D4—216no—
273C4×D4—28no—
274C4×D4—216no—
275C4×D4—28no—
276C4×D4C22⋊C418yes—
277C4×D4C22⋊C414yes—
278C4×D4C22⋊C418yes—
279C4×D4C22⋊C414yes—
280C4×D4C22×C418yes—
281C4×D4C22×C414yes—
282C4×D4C2×D418yes—
283C4×D4C2×D414yes—
284#117—28no—
285#117—24no—
286#117—28no—
287#117—24no—
288#117—24no—
289#117—28no—
290#117—24no—
291#117C22⋊C418yes—
292#117C22⋊C414yes—
293#117C2×D414yes—
294#117C2×D418yes—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4C22⋊C412yes—
306C2×C4⋊C4C22⋊C411yescharacteristic
307C2×C4⋊C4C22⋊C412yes—
308C2×C4⋊C4C22⋊C411yescharacteristic
309C2×C4⋊C4C22×C414yes—
310C2×C4⋊C4C22×C418yes—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4C22⋊C412yes—
314C2×C22⋊C4C22⋊C412yes—
315C2×C22⋊C4C22⋊C412yes—
316C2×C42—28no—
317C2×C42—24no—
318C2×C42C22×C412yes—
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C414yes—
321C2×C42C22×C412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D414yes—
328C2×C42C2×D412yes—
329C2×C42C2×D414yes—
330C2×C42C2412yes—
331C2×C42C2414yes—
332C2×C42C2411yescharacteristic
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—24no—
336C4⋊C8—24no—
337C4⋊C8—24no—
338C4⋊C8—24no—
339C4⋊C8—22no—
340C4⋊C8—28no—
341C4⋊C8C2412yes—
342C4⋊C8C2412yes—
343C4⋊C8C2414yes—
344C22⋊C8—24no—
345C22⋊C8—28no—
346C22⋊C8—28no—
347C22⋊C8—24no—
348C22⋊C8—28no—
349#97—28no—
350#97—28no—
351C4×C8—28no—
352C4×C8—24no—
353C4×C8C2418yes—
354C24C2×C22⋊C412yes—
355C24C2×C22⋊C411yescharacteristic
356#41—28no—
357#41—28no—
358#41—216no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—48no—
366C2×Q8—416no—
367C2×Q8—84no—
368C2×Q8—88no—
369C2×Q8C2×C22⋊C414yes—
370C2×Q8C23×C418yes—
371C2×D4—28no—
372C2×D4—24no—
373C2×D4C2×C22⋊C418yes—
374C2×D4C2×C22⋊C414yes—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—24no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—81no—
384C22×C4—82no—
385C22×C4C2×C22⋊C414yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C414yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C414yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4#13714yes—
398C22×C4#13714yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C23×C414yes—
401C22×C4C23×C412yes—
402C22×C4C22×D414yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C22×D414yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408M16—48no—
409M16—88no—
410C2×C8—28no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—48no—
417C2×C8—44no—
418C2×C8—84no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—28no—
423C4⋊C4—28no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—24no—
427C4⋊C4—28no—
428C4⋊C4—24no—
429C4⋊C4—28no—
430C4⋊C4—216no—
431C4⋊C4—216no—
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—28no—
435C4⋊C4—28no—
436C4⋊C4C2×C22⋊C418yes—
437C4⋊C4C2×C22⋊C414yes—
438C4⋊C4C2×C22⋊C418yes—
439C4⋊C4C2×C22⋊C414yes—
440C4⋊C4C23×C418yes—
441C22⋊C4—28no—
442C22⋊C4—216no—
443C22⋊C4—28no—
444C22⋊C4C2×C22⋊C418yes—
445C22⋊C4C2×C22⋊C418yes—
446C22⋊C4C2×C22⋊C418yes—
447C42—216no—
448C42—28no—
449C42—28no—
450C42C2×C22⋊C414yes—
451C42C2×C22⋊C412yes—
452C42C2×C22⋊C414yes—
453C42C2×C22⋊C412yes—
454C42C2×C22⋊C414yes—
455C42C2×C22⋊C412yes—
456C42C23×C411yescharacteristic
457C42C23×C414yes—
458C42C23×C412yes—
459C42C22×D412yes—
460C42C22×D414yes—
461C42C22×D411yescharacteristic
462C42C22×D414yes—
463C42C22×D411yescharacteristic
464C42C22×D412yes—
465C42C2511yescharacteristic
466C23—24no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23#41914yes—
472C23C22×C22⋊C414yes—
473C23C22×C22⋊C411yescharacteristic
474C23C22×C22⋊C412yes—
475Q8—28no—
476Q8—216no—
477Q8—48no—
478Q8—84no—
479Q8—88no—
480Q8—84no—
481Q8—88no—
482D4—216no—
483D4—28no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—84no—
505C2×C4—88no—
506C2×C4—84no—
507C2×C4—88no—
508C2×C4#41914yes—
509C2×C4#41918yes—
510C2×C4C22×C22⋊C414yes—
511C2×C4C22×C22⋊C411yescharacteristic
512C2×C4C22×C22⋊C411yescharacteristic
513C2×C4C22×C22⋊C414yes—
514C2×C4C22×C22⋊C412yes—
515C2×C4C22×C22⋊C412yes—
516C2×C4C22×C22⋊C414yes—
517C2×C4C22×C22⋊C412yes—
518C2×C4C22×C22⋊C411yescharacteristic
519C2×C4C2×#13712yes—
520C2×C4C2×#13712yes—
521C2×C4C2×#13714yes—
522C2×C4C2×#13714yes—
523C2×C4C24×C411yescharacteristic
524C2×C4C23×D411yescharacteristic
525C2×C4C23×D411yescharacteristic
526C8—48no—
527C8—42no—
528C8—42no—
529C8—44no—
530C22—21no—
531C22—22no—
532C22—21no—
533C22—22no—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—44no—
538C22—84no—
539C22C2×#41912yes—
540C22C2×#41914yes—
541C22C23×C22⋊C411yescharacteristic
542C4—21no—
543C4—22no—
544C4—24no—
545C4—24no—
546C4—21no—
547C4—21no—
548C4—22no—
549C4—24no—
550C4—48no—
551C4—48no—
552C4C2×#41912yes—
553C4C2×#41914yes—
554C4#357312yes—
555C4C22×#13711yescharacteristic
556C4C22×#13711yescharacteristic
557C2—22no—
558C2—24no—
559C2—44no—
560C2#6006511yescharacteristic
561C2C22×#41911yescharacteristic
562C2#6009311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C23×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C2×C4 ▷ C2 ▷ 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. #17514C2semidirect
5. #17496C2semidirect
6. #17486C2semidirect
7. #17464C2semidirect
8. C2×#1481C2semidirect
42. #1497C22semidirect
43. #1493C22semidirect
44. #1491C22semidirect
45. #1487C22semidirect
46. #1485C22semidirect
50. #1481C22semidirect
51. #1479C22semidirect
52. C2×Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#628132
#602772
C2×#29252
#175142
#174962
#174862
#174642
C2×#14812