Finite Group Databasebeta

← #6343233

Subgroups — #6343233

Order 512.

2832 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62813C211yesmaximal, characteristic
2C2×#2965C211yesmaximal, characteristic
3#60125C211yesmaximal, characteristic
4#17514C2116yesmaximal, complemented
5#17501C214yesmaximal, complemented
6#17486C212yesmaximal, complemented
7#17479C214yesmaximal, complemented
8C2×#1493C212yesmaximal, 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#2965—24no—
27#2965C2218yes—
28C2×#446—21no—
29C2×#436—21no—
30C2×#435—22no—
31#2925—24no—
32#2925C2218yes—
33#2921—22no—
34#2920C2214yes—
35C2×#431C2212yes—
36C2×#431C2211yescharacteristic
37C2×#423C2212yes—
38#2879—21no—
39#2873—21no—
40C2×C4×M16C2212yes—
41C22×C4×C8C2211yescharacteristic
42#1497C22116yescomplemented
43#1493C2214yes—
44#1493C2214yes—
45#1493C2218yes—
46#1493C2218yescomplemented
47#1491C22116yescomplemented
48#1487C2218yescomplemented
49#1485C22116yescomplemented
50#1481C2218yescomplemented
51#1479C22116yescomplemented
52C2×Q8⋊C8C2218yescomplemented
53C22×#41D411yescharacteristic
54C22×M16—22no—
55C22×M16—22no—
56C23×C8—21no—
57Q82—24no—
58Q82C2×C418yes—
59#557—24no—
60#557—28no—
61#557—28no—
62#557C2×C418yes—
63#554—28no—
64#554—24no—
65#554—28no—
66#554—216no—
67#554—216no—
68#554—28no—
69#554C2×C4116yes—
70#554C2×C418yes—
71D4×Q8—28no—
72D4×Q8—24no—
73D4×Q8C2×C418yes—
74D4×Q8C2×C4116yes—
75#544—28no—
76#544—24no—
77#544—28no—
78#544—24no—
79#544—24no—
80#544—22no—
81#544—24no—
82#544—22no—
83#543—24no—
84#543—24no—
85#543—24no—
86#533—28no—
87#533—216no—
88#533—24no—
89#533—22no—
90#533—22no—
91#533—24no—
92#533—22no—
93#533—24no—
94#533—22no—
95#533—24no—
96#533C2×C414yes—
97#533C2×C418yes—
98#533C2×C414yes—
99#533C2×C418yes—
100#533C2314yes—
101#533C2314yes—
102#533C2318yes—
103C2×C4⋊Q8C2×C412yes—
104C2×C4⋊Q8C2×C414yes—
105C2×C4⋊Q8C2×C412yes—
106C2×C4⋊Q8C2×C414yes—
107C2×#125—22no—
108C2×#125—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8C2×C414yes—
112C2×C22⋊Q8C2×C418yes—
113#520—24no—
114#520—24no—
115#519—22no—
116#518—22no—
117#518—24no—
118C4×#41—28no—
119C4×#41C2×C414yes—
120C4×#41D414yes—
121C4×#41D414yes—
122C4×#41C2314yes—
123C2×C4×Q8—22no—
124C2×C4×Q8—24no—
125C2×C4×Q8C2×C411yescharacteristic
126C2×C4×Q8D411yescharacteristic
127C2×C4×Q8C2314yes—
128C2×C4×Q8C2318yes—
129C2×C4×D4C2×C412yes—
130C2×C4×D4C2×C411yescharacteristic
131C2×C4×D4D412yes—
132C2×C4×D4D411yescharacteristic
133C2×#117—24no—
134C2×#117—22no—
135C2×#117D411yescharacteristic
136C2×#117D412yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#446—28no—
140#436—28no—
141#435—216no—
142#433—216no—
143#432—24no—
144#432C2318yes—
145#431—24no—
146#431—22no—
147#431—24no—
148#431C2318yes—
149#431C2318yes—
150#431C2314yes—
151#423—22no—
152#423—22no—
153#423C2314yes—
154#423C2314yes—
155C2×C4⋊C8—22no—
156C2×C4⋊C8—22no—
157C2×C4⋊C8—21no—
158C2×C4⋊C8C2314yes—
159#408—28no—
160#407—24no—
161C2×C22⋊C8—21no—
162C2×C22⋊C8—22no—
163#405—28no—
164C4×M16—24no—
165C4×M16—24no—
166C4×M16C2318yes—
167C2×#97—22no—
168C2×C4×C8—21no—
169C2×C4×C8—22no—
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2314yes—
173C2×C4×C8C2312yes—
174Q8⋊C8C23116yes—
175Q8⋊C8C23116yes—
176Q8⋊C8C23132yes—
177C2×#41—28no—
178C2×#41C22⋊C414yes—
179C2×#41C2×D414yes—
180C22×Q8—24no—
181C22×Q8—22no—
182C22×Q8C22⋊C411yescharacteristic
183C22×Q8C22×C414yes—
184C22×Q8C22×C418yes—
185C22×D4C22⋊C412yes—
186C22×D4C22⋊C411yescharacteristic
187C23×C4C22×C412yes—
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D411yescharacteristic
193C2×M16—24no—
194C2×M16—28no—
195C2×M16—24no—
196C2×M16—48no—
197C2×M16—48no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—24no—
201C22×C8—22no—
202C22×C8—24no—
203C22×C8—44no—
204C4⋊Q8—216no—
205C4⋊Q8—28no—
206C4⋊Q8—216no—
207C4⋊Q8—28no—
208C4⋊Q8—24no—
209C4⋊Q8—22no—
210C4⋊Q8—24no—
211C4⋊Q8—22no—
212C4⋊Q8—24no—
213C4⋊Q8—22no—
214C4⋊Q8—24no—
215C4⋊Q8—22no—
216C4⋊Q8C22×C414yes—
217C4⋊Q8C22×C414yes—
218C4⋊Q8C22×C418yes—
219C4⋊Q8C22×C414yes—
220C4⋊Q8C22×C414yes—
221C4⋊Q8C22×C418yes—
222#125—216no—
223#125—28no—
224#125—216no—
225#125—28no—
226#125—28no—
227#125—24no—
228#125—28no—
229#125—24no—
230#125—24no—
231#125—24no—
232#125—28no—
233C22⋊Q8—28no—
234C22⋊Q8—24no—
235C22⋊Q8—28no—
236C22⋊Q8—24no—
237C22⋊Q8—28no—
238C22⋊Q8—24no—
239C22⋊Q8—28no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242C22⋊Q8—24no—
243C22⋊Q8—28no—
244C22⋊Q8—216no—
245C22⋊Q8—216no—
246C22⋊Q8—216no—
247C22⋊Q8—216no—
248C22⋊Q8—216no—
249C22⋊Q8—216no—
250C22⋊Q8C22×C418yes—
251C22⋊Q8C22×C418yes—
252C22⋊Q8C22×C4116yes—
253C4×Q8—28no—
254C4×Q8—28no—
255C4×Q8—28no—
256C4×Q8—24no—
257C4×Q8—28no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—28no—
261C4×Q8—24no—
262C4×Q8—28no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—28no—
266C4×Q8C22⋊C414yes—
267C4×Q8C22⋊C414yes—
268C4×Q8C22×C418yes—
269C4×Q8C22×C4116yes—
270C4×Q8C22×C414yes—
271C4×Q8C2×D414yes—
272C4×Q8C2418yes—
273C4×D4—216no—
274C4×D4—28no—
275C4×D4—216no—
276C4×D4—28no—
277C4×D4C22⋊C418yes—
278C4×D4C22⋊C414yes—
279C4×D4C22⋊C418yes—
280C4×D4C22⋊C414yes—
281C4×D4C22×C418yes—
282C4×D4C22×C414yes—
283C4×D4C2×D418yes—
284C4×D4C2×D414yes—
285#117—28no—
286#117—24no—
287#117—28no—
288#117—24no—
289#117—24no—
290#117—28no—
291#117—24no—
292#117C22⋊C418yes—
293#117C22⋊C414yes—
294#117C2×D414yes—
295#117C2×D418yes—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4C22⋊C412yes—
307C2×C4⋊C4C22⋊C411yescharacteristic
308C2×C4⋊C4C22⋊C412yes—
309C2×C4⋊C4C22⋊C411yescharacteristic
310C2×C4⋊C4C22×C414yes—
311C2×C4⋊C4C22×C418yes—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4C22⋊C412yes—
315C2×C22⋊C4C22⋊C412yes—
316C2×C22⋊C4C22⋊C412yes—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42C22×C412yes—
320C2×C42C22×C411yescharacteristic
321C2×C42C22×C414yes—
322C2×C42C22×C412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D414yes—
329C2×C42C2×D412yes—
330C2×C42C2×D414yes—
331C2×C42C2412yes—
332C2×C42C2414yes—
333C2×C42C2411yescharacteristic
334C4⋊C8—24no—
335C4⋊C8—24no—
336C4⋊C8—28no—
337C4⋊C8—24no—
338C4⋊C8—24no—
339C4⋊C8C2418yes—
340C22⋊C8—24no—
341C22⋊C8—28no—
342C22⋊C8—28no—
343C22⋊C8—28no—
344#97—28no—
345#97—28no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8—24no—
349C4×C8—22no—
350C4×C8—28no—
351C4×C8C2412yes—
352C4×C8C2412yes—
353C4×C8C2414yes—
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—28no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—24no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4C2×C22⋊C414yes—
385C22×C4C2×C22⋊C414yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C414yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4#13718yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C23×C414yes—
399C22×C4C23×C412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D414yes—
402C22×C4C22×D412yes—
403C22×C4C22×D414yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406M16—416no—
407M16—48no—
408M16—48no—
409C2×C8—24no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—28no—
414C2×C8—48no—
415C2×C8—42no—
416C2×C8—44no—
417C2×C8—48no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—28no—
422C4⋊C4—28no—
423C4⋊C4—24no—
424C4⋊C4—28no—
425C4⋊C4—24no—
426C4⋊C4—28no—
427C4⋊C4—24no—
428C4⋊C4—28no—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4C2×C22⋊C418yes—
436C4⋊C4C2×C22⋊C414yes—
437C4⋊C4C2×C22⋊C418yes—
438C4⋊C4C2×C22⋊C414yes—
439C4⋊C4C23×C418yes—
440C22⋊C4—28no—
441C22⋊C4—216no—
442C22⋊C4—28no—
443C22⋊C4C2×C22⋊C418yes—
444C22⋊C4C2×C22⋊C418yes—
445C22⋊C4C2×C22⋊C418yes—
446C42—216no—
447C42—28no—
448C42—28no—
449C42C2×C22⋊C414yes—
450C42C2×C22⋊C412yes—
451C42C2×C22⋊C414yes—
452C42C2×C22⋊C412yes—
453C42C2×C22⋊C414yes—
454C42C2×C22⋊C412yes—
455C42C23×C411yescharacteristic
456C42C23×C414yes—
457C42C23×C412yes—
458C42C22×D411yescharacteristic
459C42C22×D412yes—
460C42C22×D414yes—
461C42C22×D414yes—
462C42C22×D411yescharacteristic
463C42C22×D412yes—
464C42C2511yescharacteristic
465C23—24no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23#41914yes—
471C23C22×C22⋊C414yes—
472C23C22×C22⋊C411yescharacteristic
473C23C22×C22⋊C412yes—
474Q8—28no—
475Q8—216no—
476Q8—48no—
477Q8—84no—
478Q8—88no—
479Q8—84no—
480Q8—88no—
481D4—216no—
482D4—28no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—84no—
503C2×C4—88no—
504C2×C4—84no—
505C2×C4—88no—
506C2×C4#41914yes—
507C2×C4#41918yes—
508C2×C4C22×C22⋊C414yes—
509C2×C4C22×C22⋊C411yescharacteristic
510C2×C4C22×C22⋊C414yes—
511C2×C4C22×C22⋊C412yes—
512C2×C4C22×C22⋊C412yes—
513C2×C4C22×C22⋊C414yes—
514C2×C4C22×C22⋊C412yes—
515C2×C4C22×C22⋊C411yescharacteristic
516C2×C4C22×C22⋊C411yescharacteristic
517C2×C4C2×#13714yes—
518C2×C4C2×#13714yes—
519C2×C4C2×#13714yes—
520C2×C4C24×C411yescharacteristic
521C2×C4C23×D411yescharacteristic
522C2×C4C23×D411yescharacteristic
523C8—24no—
524C8—42no—
525C8—44no—
526C8—48no—
527C22—21no—
528C22—22no—
529C22—21no—
530C22—22no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—44no—
535C22—84no—
536C22C2×#41912yes—
537C22C2×#41914yes—
538C22C23×C22⋊C411yescharacteristic
539C4—21no—
540C4—22no—
541C4—24no—
542C4—24no—
543C4—21no—
544C4—22no—
545C4—24no—
546C4—48no—
547C4—48no—
548C4C2×#41912yes—
549C4C2×#41914yes—
550C4#290112yes—
551C4#357312yes—
552C4C22×#13712yes—
553C2—22no—
554C2—24no—
555C2—44no—
556C2#6006511yescharacteristic
557C2C22×#41911yescharacteristic
558C2C2×#290111yescharacteristic
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. #17501C2semidirect
6. #17486C2semidirect
7. #17479C2semidirect
8. C2×#1493C2semidirect
42. #1497C22semidirect
46. #1493C22semidirect
47. #1491C22semidirect
48. #1487C22semidirect
49. #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
C2×#29652
#601252
#175142
#175012
#174862
#174792
C2×#14932