Finite Group Databasebeta

← #6344171

Subgroups — #6344171

Order 512.

2774 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62813C211yesmaximal, characteristic
2#60282C211yesmaximal, characteristic
3C2×#2990C211yesmaximal, characteristic
4#17774C218yesmaximal, complemented
5#17774C218yesmaximal, complemented
6#17755C214yesmaximal, complemented
7#17751C212yesmaximal, complemented
8#17715C212yesmaximal, complemented
9C2×#1552C214yesmaximal, complemented
10#3504C418yes—
11#3503C2214yes—
12#3503C2214yes—
13#3502—21no—
14#3502—21no—
15#3501—21no—
16#3501—22no—
17#3501—21no—
18#3501—22no—
19Q8×#41—24no—
20Q8×#41—24no—
21#3458C414yes—
22#3458C412yes—
23C2×#533C2214yes—
24C2×#533C2212yes—
25#3428C412yes—
26C2×C4×#41C2211yescharacteristic
27#2990—24no—
28#2990C2218yes—
29#2990C2218yes—
30#2983—21no—
31#2973—22no—
32#2972—21no—
33#2970—24no—
34C2×#446—21no—
35C2×#435—22no—
36C2×C8×D4—21no—
37#2920C2212yes—
38C2×#432C2214yes—
39C2×#432C2212yes—
40C2×#431C2211yescharacteristic
41C2×#423C2211yescharacteristic
42C22×C4⋊C8C2212yes—
43#1561C2218yescomplemented
44#1556C2218yescomplemented
45#1556C2218yescomplemented
46#1552C2214yes—
47#1552C2218yes—
48#1552C2214yes—
49#1552C22116yescomplemented
50#1552C22116yescomplemented
51#1549C2214yescomplemented
52#1549C2214yescomplemented
53#1546C2214yescomplemented
54#1546C2214yescomplemented
55#1544C2218yescomplemented
56C2×#332C2218yescomplemented
57C2×#332C2218yescomplemented
58C22×#41C2×C411yescharacteristic
59C22×M16—21no—
60C23×C8—22no—
61Q82—24no—
62Q82—24no—
63#557—28no—
64#557—28no—
65#557—24no—
66#557—24no—
67#554—216no—
68#554—216no—
69#554—28no—
70#554—28no—
71#554—24no—
72#554—24no—
73#554—28no—
74#554—28no—
75D4×Q8—24no—
76D4×Q8—24no—
77D4×Q8—28no—
78D4×Q8—28no—
79#544—24no—
80#544—24no—
81#544—22no—
82#544—22no—
83#544C2×C4116yes—
84#544C2×C4116yes—
85#544C2×C418yes—
86#544C2×C418yes—
87#543—24no—
88#543—24no—
89#543—24no—
90#533—216no—
91#533—28no—
92#533—24no—
93#533—22no—
94#533—24no—
95#533—22no—
96#533—22no—
97#533—24no—
98#533—22no—
99#533—24no—
100#533C2×C418yes—
101#533C2×C414yes—
102#533C2×C418yes—
103#533C2×C414yes—
104#533C2314yes—
105#533C2314yes—
106#533C2318yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×#125C2×C418yes—
112C2×#125C2×C414yes—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—22no—
117#520C2×C418yes—
118#520C2×C418yes—
119#519—22no—
120#518—24no—
121#518—22no—
122C4×#41—28no—
123C4×#41C2×C414yes—
124C4×#41D414yes—
125C4×#41D414yes—
126C4×#41C2314yes—
127C2×C4×Q8—24no—
128C2×C4×Q8—22no—
129C2×C4×Q8—24no—
130C2×C4×Q8—22no—
131C2×C4×Q8D411yescharacteristic
132C2×C4×Q8D411yescharacteristic
133C2×C4×D4D411yescharacteristic
134C2×C4×D4D412yes—
135C2×C4×D4D411yescharacteristic
136C2×C4×D4D412yes—
137C2×#117C2×C418yes—
138C2×#117C2×C414yes—
139C2×#117C2×C411yescharacteristic
140C2×#117C2×C412yes—
141C22×C42C2312yes—
142C22×C42C2311yescharacteristic
143#446—28no—
144#446—28no—
145#435—216no—
146#435—216no—
147C8×D4—28no—
148C8×D4—28no—
149#432—24no—
150#432—24no—
151#432—24no—
152#432—22no—
153#432C2314yes—
154#432C2318yes—
155#432C2318yes—
156#431—22no—
157#431—22no—
158#431C2314yes—
159#431C2314yes—
160#423—22no—
161#423C2314yes—
162#423C2314yes—
163C2×C4⋊C8—22no—
164C2×C4⋊C8—24no—
165C2×C4⋊C8—21no—
166C2×C4⋊C8C2318yes—
167C2×C4⋊C8C2318yes—
168C2×C4⋊C8C2311yescharacteristic
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2311yescharacteristic
171#407—22no—
172C2×C22⋊C8—21no—
173C2×C22⋊C8—22no—
174C2×C22⋊C8—24no—
175C4×M16—22no—
176C2×#97—22no—
177C2×#97—24no—
178C2×C4×C8—21no—
179#332C2318yes—
180#332C2318yes—
181#332C23116yes—
182#332C23116yes—
183#332C2318yes—
184#332C2318yes—
185C2×#41—28no—
186C2×#41C22⋊C414yes—
187C2×#41C22×C414yes—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8C22⋊C411yescharacteristic
193C22×D4C22⋊C412yes—
194C22×D4C22⋊C411yescharacteristic
195C23×C4C22×C412yes—
196C23×C4C22×C411yescharacteristic
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D411yescharacteristic
201C2×M16—24no—
202C2×M16—24no—
203C2×M16—82no—
204C22×C8—21no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—28no—
208C22×C8—28no—
209C22×C8—84no—
210C4⋊Q8—216no—
211C4⋊Q8—216no—
212C4⋊Q8—28no—
213C4⋊Q8—28no—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C4⋊Q8—24no—
217C4⋊Q8—24no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C4⋊Q8—24no—
221C4⋊Q8—24no—
222C4⋊Q8C22×C418yes—
223C4⋊Q8C22×C418yes—
224C4⋊Q8C22×C414yes—
225C4⋊Q8C22×C414yes—
226C4⋊Q8C2414yes—
227C4⋊Q8C2414yes—
228#125—216no—
229#125—216no—
230#125—28no—
231#125—28no—
232#125—28no—
233#125—28no—
234#125—24no—
235#125—24no—
236#125C22×C418yes—
237#125C22×C418yes—
238#125C22×C4116yes—
239C22⋊Q8—24no—
240C22⋊Q8—28no—
241C22⋊Q8—24no—
242C22⋊Q8—28no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—28no—
246C22⋊Q8—28no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—28no—
250C22⋊Q8—28no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—216no—
254C22⋊Q8—216no—
255C22⋊Q8—216no—
256C22⋊Q8—216no—
257C22⋊Q8—216no—
258C22⋊Q8—216no—
259C4×Q8—28no—
260C4×Q8—28no—
261C4×Q8—28no—
262C4×Q8—28no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—28no—
266C4×Q8—28no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—28no—
274C4×Q8—28no—
275C4×Q8C22⋊C414yes—
276C4×Q8C22⋊C414yes—
277C4×Q8C2×D414yes—
278C4×Q8C2×D414yes—
279C4×D4—216no—
280C4×D4—216no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4C22⋊C418yes—
284C4×D4C22⋊C418yes—
285C4×D4C22⋊C414yes—
286C4×D4C22⋊C414yes—
287C4×D4C2×D414yes—
288C4×D4C2×D414yes—
289C4×D4C2×D418yes—
290C4×D4C2×D418yes—
291#117—24no—
292#117—28no—
293#117—28no—
294#117—24no—
295#117C22⋊C414yes—
296#117C22⋊C418yes—
297#117C22×C418yes—
298#117C22×C418yes—
299#117C22×C4116yes—
300#117C22×C414yes—
301#117C22×C418yes—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C22⋊C412yes—
311C2×C4⋊C4C22⋊C412yes—
312C2×C4⋊C4C22⋊C411yescharacteristic
313C2×C4⋊C4C22⋊C411yescharacteristic
314C2×C4⋊C4C22×C418yes—
315C2×C4⋊C4C22×C418yes—
316C2×C4⋊C4C22×C414yes—
317C2×C4⋊C4C22×C414yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4C22⋊C412yes—
321C2×C22⋊C4C22⋊C412yes—
322C2×C22⋊C4C22⋊C412yes—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42C22×C412yes—
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C414yes—
328C2×C42C22×C412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D414yes—
334C2×C42C2×D412yes—
335C2×C42C2×D414yes—
336C2×C42C2×D412yes—
337C2×C42C2414yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C4⋊C8—21no—
341C4⋊C8—22no—
342C4⋊C8—28no—
343C4⋊C8—24no—
344C4⋊C8—21no—
345C4⋊C8—28no—
346C4⋊C8—24no—
347C4⋊C8C2412yes—
348C4⋊C8C2414yes—
349C4⋊C8C2418yes—
350C4⋊C8C2412yes—
351C22⋊C8—24no—
352C22⋊C8—24no—
353C22⋊C8—28no—
354C22⋊C8—28no—
355C22⋊C8—24no—
356#97—24no—
357#97—28no—
358#97—28no—
359C4×C8—24no—
360C4×C8—24no—
361C24C2×C22⋊C411yescharacteristic
362C24C2×C22⋊C412yes—
363#41—216no—
364#41—216no—
365C2×Q8—24no—
366C2×Q8—28no—
367C2×Q8—24no—
368C2×Q8—28no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—88no—
373C2×Q8—84no—
374C2×Q8—84no—
375C2×Q8—82no—
376C2×Q8—84no—
377C2×Q8—82no—
378C2×Q8C2×C22⋊C414yes—
379C2×D4—28no—
380C2×D4#10118yes—
381C2×D4C2×C22⋊C418yes—
382C2×D4C2×C22⋊C414yes—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4C2×C22⋊C414yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C412yes—
403C22×C4C2×C22⋊C412yes—
404C22×C4C2×C22⋊C414yes—
405C22×C4C2×C22⋊C414yes—
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C23×C411yescharacteristic
408C22×C4C23×C414yes—
409C22×C4C23×C412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D414yes—
412C22×C4C22×D414yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416M16—84no—
417M16—84no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—28no—
421C2×C8—24no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—44no—
425C2×C8—88no—
426C2×C8—88no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—24no—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—216no—
435C4⋊C4—28no—
436C4⋊C4—216no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4C2×C22⋊C418yes—
444C4⋊C4C2×C22⋊C418yes—
445C4⋊C4C2×C22⋊C414yes—
446C4⋊C4C2×C22⋊C414yes—
447C4⋊C4C23×C418yes—
448C4⋊C4C23×C418yes—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—216no—
452C22⋊C4C2×C22⋊C418yes—
453C22⋊C4C2×C22⋊C418yes—
454C22⋊C4C2×C22⋊C418yes—
455C42—28no—
456C42—216no—
457C42—28no—
458C42C2×C22⋊C414yes—
459C42C2×C22⋊C412yes—
460C42C2×C22⋊C414yes—
461C42C2×C22⋊C414yes—
462C42C2×C22⋊C412yes—
463C42C2×C22⋊C412yes—
464C42C23×C411yescharacteristic
465C42C23×C414yes—
466C42C23×C412yes—
467C42C22×D412yes—
468C42C22×D412yes—
469C42C22×D411yescharacteristic
470C42C22×D414yes—
471C42C22×D414yes—
472C42C22×D411yescharacteristic
473C42C2511yescharacteristic
474C23—22no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23C2×#10118yes—
480C23C22×C22⋊C412yes—
481C23C22×C22⋊C411yescharacteristic
482C23C22×C22⋊C414yes—
483Q8—28no—
484Q8—48no—
485Q8—44no—
486Q8—44no—
487Q8—88no—
488Q8—84no—
489Q8—88no—
490Q8—84no—
491D4—216no—
492D4C2×#101116yes—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—88no—
519C2×C4—84no—
520C2×C4—88no—
521C2×C4—84no—
522C2×C4C2×#10114yes—
523C2×C4C22×C22⋊C411yescharacteristic
524C2×C4C22×C22⋊C414yes—
525C2×C4C22×C22⋊C411yescharacteristic
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C22×C22⋊C412yes—
528C2×C4C22×C22⋊C414yes—
529C2×C4C22×C22⋊C412yes—
530C2×C4C22×C22⋊C414yes—
531C2×C4C22×C22⋊C412yes—
532C2×C4C24×C411yescharacteristic
533C2×C4C23×D411yescharacteristic
534C2×C4C23×D411yescharacteristic
535C8—48no—
536C8—42no—
537C8—44no—
538C8—42no—
539C22—21no—
540C22—24no—
541C22—21no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—48no—
547C22—82no—
548C22C22×#10112yes—
549C22C22×#10118yes—
550C22C23×C22⋊C411yescharacteristic
551C4—21no—
552C4—21no—
553C4—24no—
554C4—21no—
555C4—24no—
556C4—22no—
557C4—21no—
558C4—24no—
559C4—22no—
560C4—48no—
561C4—48no—
562C4C22×#10114yes—
563C4#358412yes—
564C4#358511yescharacteristic
565C4#358611yescharacteristic
566C2—22no—
567C2—28no—
568C2—42no—
569C2C23×#10111yescharacteristic
570C2#6008611yescharacteristic
571C2#6008911yescharacteristic
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. #17774C2semidirect
5. #17774C2semidirect
6. #17755C2semidirect
7. #17751C2semidirect
8. #17715C2semidirect
9. C2×#1552C2semidirect
43. #1561C22semidirect
44. #1556C22semidirect
45. #1556C22semidirect
49. #1552C22semidirect
50. #1552C22semidirect
51. #1549C22semidirect
52. #1549C22semidirect
53. #1546C22semidirect
54. #1546C22semidirect
55. #1544C22semidirect
56. C2×#332C22semidirect
57. C2×#332C22semidirect

Maximal subgroups

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

subgroupindex
#628132
#602822
C2×#29902
#177742
#177552
#177512
#177152
C2×#15522