Finite Group Databasebeta

← #6344264

Subgroups — #6344264

Order 512.

2754 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3509C211yesmaximal, characteristic
2#60284C211yesmaximal, characteristic
3#60282C211yesmaximal, characteristic
4#17778C2116yesmaximal, complemented
5#17748C214yesmaximal, complemented
6#17746C212yesmaximal, complemented
7#17739C212yesmaximal, complemented
8#17734C214yesmaximal, complemented
9#3509—24no—
10#3509—24no—
11#3509C418yes—
12#3509C2218yes—
13C2×#557—21no—
14C2×#557—21no—
15C2×#556—21no—
16C2×#556—21no—
17C2×#552—22no—
18C2×#552—22no—
19C2×#541C414yes—
20C2×#536C412yes—
21C2×#533C2212yes—
22C2×#533C2212yes—
23C2×#533C2211yescharacteristic
24C22×C4⋊Q8C2211yescharacteristic
25C2×#519C412yes—
26C2×C4×#41C2211yescharacteristic
27#2991C2218yes—
28#2990C2218yes—
29#2983—21no—
30#2982—21no—
31#2975—22no—
32#2975—21no—
33#2973—22no—
34#2972—21no—
35#2971—24no—
36#2970—24no—
37#2920C2214yes—
38#2920C2212yes—
39#2920C2212yes—
40C2×#432C2214yes—
41#1561C2218yescomplemented
42#1559C22116yescomplemented
43#1558C2218yes—
44#1556C2214yes—
45#1556C2214yes—
46#1554C22116yescomplemented
47#1550C22116yescomplemented
48#1548C2218yescomplemented
49#1548C2218yescomplemented
50#1544C2218yescomplemented
51#1542C22116yescomplemented
52C22×#41C2×C411yescharacteristic
53C23×Q8C2×C412yes—
54#557—28no—
55#557—28no—
56#557—28no—
57#557—28no—
58#556—28no—
59#556—28no—
60#556—28no—
61#556—28no—
62#552—216no—
63#552—216no—
64#552—216no—
65#552—216no—
66#541—216no—
67#541C2×C4132yes—
68#536—28no—
69#536C2×C4116yes—
70#533—22no—
71#533—22no—
72#533—24no—
73#533—24no—
74#533—24no—
75#533—24no—
76#533C2×C418yes—
77#533C2×C418yes—
78#533C2×C414yes—
79#533C2318yes—
80#533C2318yes—
81#533C2314yes—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8C2×C412yes—
88C2×C4⋊Q8C2×C414yes—
89C2×C4⋊Q8C2314yes—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#126—24no—
94C2×#126—24no—
95C2×#125—24no—
96C2×#125—24no—
97C2×#125—22no—
98C2×#125—22no—
99C2×#125C2×C412yes—
100C2×#124—22no—
101C2×#123—24no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2×C414yes—
107C2×C22⋊Q8C2×C414yes—
108#519—28no—
109#519C2×C4116yes—
110C4×#41C2×C414yes—
111C4×#41D414yes—
112C4×#41D414yes—
113C4×#41C2314yes—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8C2×C414yes—
121C2×C4×Q8C2×C414yes—
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8D411yescharacteristic
124C2×C4×D4D412yes—
125C2×C4×D4D412yes—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×#117—22no—
129C2×#117—22no—
130C2×#117—24no—
131C2×#117C2×C412yes—
132C2×#117C2×C411yescharacteristic
133C2×#117C2×C412yes—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138#446—28no—
139C8×Q8—28no—
140#436—216no—
141#436—28no—
142#435—216no—
143C8×D4—28no—
144#432—24no—
145#432—24no—
146#432—22no—
147#432—22no—
148#432—24no—
149#432C2318yes—
150#432C2314yes—
151#432C2314yes—
152#431—22no—
153#431—24no—
154#431—22no—
155#431C2318yes—
156#431C2314yes—
157#431C2314yes—
158#423C2314yes—
159#423C2314yes—
160#423C2318yes—
161C2×C4⋊C8C2318yes—
162#407—22no—
163#407—22no—
164#407—24no—
165C2×C22⋊C8—24no—
166C4×M16—22no—
167C4×M16—22no—
168C4×M16—24no—
169C2×#97—24no—
170#333C23132yes—
171#332C23116yes—
172#332C23116yes—
173C2×#41—44no—
174C2×#41C22⋊C414yes—
175C2×#41C22×C414yes—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—48no—
181C22×Q8C22⋊C411yescharacteristic
182C22×Q8C22×C414yes—
183C22×Q8C22×C418yes—
184C22×D4C22⋊C411yescharacteristic
185C22×D4C22⋊C412yes—
186C23×C4—22no—
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C2×M16—24no—
194C2×M16—24no—
195C2×M16—28no—
196C22×C8—28no—
197C4⋊Q8—216no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—24no—
204C4⋊Q8—24no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8C22×C414yes—
210C4⋊Q8C22×C418yes—
211C4⋊Q8C22×C412yes—
212C4⋊Q8C22×C412yes—
213C4⋊Q8C2414yes—
214C4⋊Q8C2412yes—
215C4⋊Q8C2412yes—
216#126—232no—
217#126—232no—
218#125—232no—
219#125—232no—
220#125—28no—
221#125—28no—
222#125—24no—
223#125—28no—
224#125—28no—
225#125C22×C418yes—
226#124—216no—
227#123—232no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—28no—
231C22⋊Q8—28no—
232C22⋊Q8—28no—
233C22⋊Q8—28no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8—28no—
237C22⋊Q8—28no—
238C22⋊Q8C22×C4116yes—
239C22⋊Q8C22×C4116yes—
240C4×Q8—216no—
241C4×Q8—216no—
242C4×Q8—28no—
243C4×Q8—28no—
244C4×Q8—24no—
245C4×Q8—24no—
246C4×Q8—28no—
247C4×Q8—28no—
248C4×Q8—28no—
249C4×Q8—28no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8C22⋊C414yes—
253C4×Q8C22⋊C414yes—
254C4×Q8C22×C4116yes—
255C4×Q8C22×C4116yes—
256C4×Q8C2×D414yes—
257C4×Q8C2×D414yes—
258C4×D4C22⋊C418yes—
259C4×D4C22⋊C418yes—
260C4×D4C22⋊C414yes—
261C4×D4C22⋊C414yes—
262C4×D4C2×D418yes—
263C4×D4C2×D418yes—
264C4×D4C2×D414yes—
265C4×D4C2×D414yes—
266#117—28no—
267#117—28no—
268#117—24no—
269#117—28no—
270#117—28no—
271#117—232no—
272#117C22⋊C414yes—
273#117C22⋊C418yes—
274#117C22×C418yes—
275#117C22×C414yes—
276#117C22×C418yes—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C22⋊C411yescharacteristic
294C2×C4⋊C4C22⋊C411yescharacteristic
295C2×C4⋊C4C22⋊C412yes—
296C2×C4⋊C4C22⋊C412yes—
297C2×C4⋊C4C22×C412yes—
298C2×C4⋊C4C22×C412yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4C22⋊C412yes—
303C2×C22⋊C4C22⋊C412yes—
304C2×C22⋊C4C22⋊C412yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42C22×C412yes—
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C412yes—
311C2×C42C22×C414yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2412yes—
321C2×C42C2414yes—
322C2×C42C2411yescharacteristic
323C4⋊C8—28no—
324C4⋊C8—24no—
325C4⋊C8—28no—
326C4⋊C8—24no—
327C4⋊C8C2418yes—
328C4⋊C8C2418yes—
329C22⋊C8—24no—
330C22⋊C8—28no—
331C22⋊C8—24no—
332C22⋊C8—24no—
333C22⋊C8—24no—
334C22⋊C8—28no—
335#97—24no—
336#97—28no—
337#97—24no—
338C4×C8—24no—
339C4×C8—28no—
340C4×C8—24no—
341C24C2×C22⋊C412yes—
342C24C2×C22⋊C411yescharacteristic
343#41—48no—
344#41—48no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—28no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—24no—
352C2×Q8—416no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—416no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8C2×C22⋊C414yes—
359C2×Q8C23×C418yes—
360C2×D4—24no—
361C2×D4—44no—
362C2×D4C2×C22⋊C414yes—
363C2×D4C2×C22⋊C418yes—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—48no—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C414yes—
384C22×C4C2×C22⋊C414yes—
385C22×C4C2×C22⋊C414yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C23×C411yescharacteristic
388C22×C4C23×C412yes—
389C22×C4C23×C414yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D414yes—
395C22×C4C22×D414yes—
396M16—84no—
397M16—84no—
398M16—88no—
399C2×C8—28no—
400C2×C8—28no—
401C2×C8—88no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—28no—
405C4⋊C4—28no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—216no—
409C4⋊C4—216no—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C4⋊C4—216no—
413C4⋊C4—216no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4C2×C22⋊C414yes—
421C4⋊C4C2×C22⋊C414yes—
422C4⋊C4C2×C22⋊C418yes—
423C4⋊C4C2×C22⋊C418yes—
424C4⋊C4C23×C414yes—
425C4⋊C4C23×C414yes—
426C22⋊C4—28no—
427C22⋊C4—216no—
428C22⋊C4—216no—
429C22⋊C4C2×C22⋊C418yes—
430C22⋊C4C2×C22⋊C418yes—
431C22⋊C4C2×C22⋊C418yes—
432C42—216no—
433C42—216no—
434C42—28no—
435C42C2×C22⋊C414yes—
436C42C2×C22⋊C412yes—
437C42C2×C22⋊C414yes—
438C42C2×C22⋊C414yes—
439C42C2×C22⋊C412yes—
440C42C2×C22⋊C412yes—
441C42C23×C411yescharacteristic
442C42C23×C414yes—
443C42C23×C412yes—
444C42C22×D414yes—
445C42C22×D412yes—
446C42C22×D412yes—
447C42C22×D411yescharacteristic
448C42C22×D411yescharacteristic
449C42C22×D414yes—
450C42C2511yescharacteristic
451C23—22no—
452C23—24no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23C22×C22⋊C414yes—
458C23C22×C22⋊C412yes—
459C23C22×C22⋊C411yescharacteristic
460Q8—28no—
461Q8—48no—
462Q8—44no—
463Q8—44no—
464Q8—48no—
465Q8—44no—
466Q8—44no—
467Q8—416no—
468D4—28no—
469D4—48no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—416no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4#41314yes—
498C2×C4C22×C22⋊C414yes—
499C2×C4C22×C22⋊C414yes—
500C2×C4C22×C22⋊C411yescharacteristic
501C2×C4C22×C22⋊C411yescharacteristic
502C2×C4C22×C22⋊C411yescharacteristic
503C2×C4C22×C22⋊C412yes—
504C2×C4C22×C22⋊C412yes—
505C2×C4C22×C22⋊C414yes—
506C2×C4C22×C22⋊C412yes—
507C2×C4C24×C411yescharacteristic
508C2×C4C23×D411yescharacteristic
509C2×C4C23×D411yescharacteristic
510C8—48no—
511C8—48no—
512C22—21no—
513C22—21no—
514C22—22no—
515C22—24no—
516C22—22no—
517C22—22no—
518C22—24no—
519C22—24no—
520C22—44no—
521C22—84no—
522C22C2×#41312yes—
523C22C23×C22⋊C411yescharacteristic
524C4—21no—
525C4—21no—
526C4—24no—
527C4—21no—
528C4—21no—
529C4—24no—
530C4—24no—
531C4—22no—
532C4—22no—
533C4—48no—
534C4—44no—
535C4—44no—
536C4C2×#41314yes—
537C4#358612yes—
538C4#358612yes—
539C2—22no—
540C2—42no—
541C2—44no—
542C2C22×#41311yescharacteristic
543C2#6008811yescharacteristic
544C2#6008811yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17778C2semidirect
5. #17748C2semidirect
6. #17746C2semidirect
7. #17739C2semidirect
8. #17734C2semidirect
41. #1561C22semidirect
42. #1559C22semidirect
46. #1554C22semidirect
47. #1550C22semidirect
48. #1548C22semidirect
49. #1548C22semidirect
50. #1544C22semidirect
51. #1542C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35092
#602842
#602822
#177782
#177482
#177462
#177392
#177342