Finite Group Databasebeta

← #6386233

Subgroups — #6386233

Order 512.

2959 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2#60249C212yesmaximal, complemented
3#22515C214yesmaximal, complemented
4#22515C218yesmaximal, complemented
5Q8×#100C214yesmaximal, complemented
6#21832C214yesmaximal, complemented
7#21831C212yesmaximal, complemented
8#20729C214yesmaximal
9#20726C211yesmaximal, characteristic
10#20726C211yesmaximal, characteristic
11#3501—24no—
12#3501—24no—
13#3501—24no—
14#3501—24no—
15C2×#554C412yes—
16C2×#554C414yes—
17C2×#554C2212yes—
18C2×#554C2214yes—
19C2×D4×Q8C412yes—
20C2×D4×Q8C2212yes—
21C2×#544C2212yes—
22C2×#543C414yes—
23C2×#533C2211yescharacteristic
24C2×#533C2211yescharacteristic
25C22×C22⋊Q8C414yes—
26C2×#518C2212yes—
27C22×C4×D4C2211yescharacteristic
28#2987—24no—
29#2986—24no—
30#2985—24no—
31#2985—24no—
32Q8×M16C2218yescomplemented
33Q8×M16C2218yescomplemented
34#2921C2218yes—
35#2920C2212yes—
36#2920C2212yes—
37#1998C2218yescomplemented
38#1998C22116yescomplemented
39#1997C2214yescomplemented
40#1997C2214yescomplemented
41#1997C22116yescomplemented
42#1822C2218yes—
43#1821C2212yes—
44#1821C2212yes—
45#1800C2214yes—
46#1800C2218yes—
47#1800C2214yes—
48#1800C2218yes—
49#1783C2212yes—
50#1783C2212yes—
51C4×#100C2214yes—
52C4×#100C2218yes—
53C4×#100C2214yes—
54C4×#100C2218yes—
55C23×Q8C2×C414yes—
56C23×D4C2311yescharacteristic
57C24×C4C2×C412yes—
58#554—28no—
59#554—28no—
60#554—216no—
61#554—216no—
62#554—28no—
63#554—28no—
64#554—216no—
65#554—216no—
66D4×Q8—28no—
67D4×Q8—28no—
68D4×Q8—28no—
69D4×Q8—28no—
70#544—28no—
71#544—28no—
72#543—216no—
73#543—216no—
74#533—22no—
75#533—22no—
76#533—22no—
77#533—22no—
78#533D414yes—
79#533D414yes—
80#533D414yes—
81#533D414yes—
82C2×C4⋊Q8C2×C414yes—
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×C4⋊Q8C2×C411yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2314yes—
88C2×#125C2×C414yes—
89C2×#125C2×C412yes—
90C2×#125C2312yes—
91C2×#125C2314yes—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8C2×C412yes—
97C2×C22⋊Q8C2×C418yes—
98C2×C22⋊Q8C2×C414yes—
99C2×C22⋊Q8C2×C414yes—
100C2×C22⋊Q8C2×C418yes—
101C2×C22⋊Q8C2×C412yes—
102C2×C22⋊Q8C2×C418yes—
103C2×C22⋊Q8C2×C418yes—
104C2×C22⋊Q8C2×C418yes—
105C2×C22⋊Q8C2×C418yes—
106C2×C22⋊Q8C2×C412yes—
107C2×C22⋊Q8C2×C412yes—
108#518—28no—
109#518—28no—
110C2×C4×Q8C2×C412yes—
111C2×C4×Q8C2×C412yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4C2×C412yes—
119C2×C4×D4C2×C412yes—
120C2×C4×D4C2×C414yes—
121C2×C4×D4C2×C414yes—
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2314yes—
125C2×C4×D4C2314yes—
126C2×#117C2312yes—
127C2×#117C2312yes—
128C22×C4⋊C4C2×C418yes—
129C22×C4⋊C4C2×C414yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4C2×C412yes—
132C22×C22⋊C4C2×C418yes—
133C22×C42C2311yescharacteristic
134#446—28no—
135#446—28no—
136#446—216no—
137#446—216no—
138C8×Q8—28no—
139C8×Q8—28no—
140#433—216no—
141#433—216no—
142#432—22no—
143#432—22no—
144#432—22no—
145#432—22no—
146#431—22no—
147#431—22no—
148#431—22no—
149#431—22no—
150#423C2314yes—
151#423C2314yes—
152#423C23116yes—
153C2×#100C2318yes—
154C2×#100C2314yes—
155C2×#100C2314yes—
156#407C2314yes—
157#407C2314yes—
158C4×M16C2314yes—
159C4×M16C2314yes—
160C4×M16C23116yes—
161#343C2318yes—
162#343C2318yes—
163#343C23116yes—
164#343C23116yes—
165C25C22×C412yes—
166C22×Q8—28no—
167C22×Q8—28no—
168C22×Q8—88no—
169C22×Q8C22×C412yes—
170C22×Q8C22×C412yes—
171C22×Q8C22×C418yes—
172C22×Q8C22×C412yes—
173C22×Q8C22×C412yes—
174C22×D4—28no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4C22×C412yes—
178C22×D4C2412yes—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—84no—
184C23×C4C22×C414yes—
185C23×C4C22×C418yes—
186C23×C4C22×C418yes—
187C23×C4C22×C412yes—
188C23×C4C22×C414yes—
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2411yescharacteristic
192C2×M16C2418yes—
193C2×M16C2418yes—
194C4⋊Q8—216no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—216no—
200C4⋊Q8C22⋊C414yes—
201C4⋊Q8C22⋊C414yes—
202C4⋊Q8C22⋊C414yes—
203C4⋊Q8C22⋊C414yes—
204#125—216no—
205#125—24no—
206#125—24no—
207#125—216no—
208#125C22⋊C418yes—
209#125C22⋊C418yes—
210C22⋊Q8—28no—
211C22⋊Q8—28no—
212C22⋊Q8—232no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—24no—
218C22⋊Q8—24no—
219C22⋊Q8—28no—
220C22⋊Q8—28no—
221C22⋊Q8—216no—
222C22⋊Q8—216no—
223C22⋊Q8—232no—
224C22⋊Q8—232no—
225C22⋊Q8—232no—
226C22⋊Q8—216no—
227C22⋊Q8—216no—
228C22⋊Q8C22⋊C414yes—
229C22⋊Q8C22⋊C414yes—
230C22⋊Q8C22⋊C414yes—
231C22⋊Q8C22⋊C414yes—
232C22⋊Q8C22⋊C418yes—
233C22⋊Q8C22⋊C418yes—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8C2×D418yes—
239C4×Q8C2×D418yes—
240C4×Q8C2×D418yes—
241C4×Q8C2×D418yes—
242C4×D4—216no—
243C4×D4—216no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—216no—
253C4×D4—216no—
254#117—24no—
255#117—24no—
256#117C2×D418yes—
257#117C2×D418yes—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—216no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C418yes—
269C2×C4⋊C4C22×C412yes—
270C2×C4⋊C4C22×C414yes—
271C2×C4⋊C4C22×C412yes—
272C2×C4⋊C4C22×C414yes—
273C2×C4⋊C4C22×C412yes—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C22×C412yes—
276C2×C4⋊C4C22×C414yes—
277C2×C4⋊C4C22×C412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4C22×C414yes—
286C2×C22⋊C4C22×C418yes—
287C2×C22⋊C4C22×C418yes—
288C2×C22⋊C4C22×C414yes—
289C2×C22⋊C4C22×C414yes—
290C2×C22⋊C4C22×C414yes—
291C2×C42C22×C414yes—
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2414yes—
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C4⋊C8—24no—
302C4⋊C8—24no—
303C4⋊C8—24no—
304C4⋊C8—24no—
305C4⋊C8—216no—
306C4⋊C8—216no—
307#100—216no—
308#100—28no—
309#100—28no—
310C22⋊C8—24no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314#97—24no—
315#97—24no—
316#97—216no—
317C4×C8—24no—
318C4×C8—24no—
319C4×C8—216no—
320C24—22no—
321C24—22no—
322C24—28no—
323C24—82no—
324C24—82no—
325C24C23×C411yescharacteristic
326C24C23×C414yes—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—24no—
330C2×Q8—216no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—82no—
334C2×Q8—88no—
335C2×Q8—82no—
336C2×Q8—816no—
337C2×Q8—816no—
338C2×Q8C2×C22⋊C414yes—
339C2×Q8C2×C22⋊C418yes—
340C2×Q8C2×C22⋊C414yes—
341C2×D4—216no—
342C2×D4—216no—
343C2×D4—28no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D42−1+418yes—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—42no—
358C22×C4—48no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4—81no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—82no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C23×C411yescharacteristic
375C22×C4C23×C411yescharacteristic
376C22×C4C23×C414yes—
377C22×C4C23×C414yes—
378C22×C4C23×C414yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C23×C411yescharacteristic
381C22×C4C23×C411yescharacteristic
382C22×C4C23×C414yes—
383C22×C4C23×C414yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C2511yescharacteristic
387M16—28no—
388M16—28no—
389M16—216no—
390C2×C8—28no—
391C2×C8—28no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—216no—
398C4⋊C4—216no—
399C4⋊C4—216no—
400C4⋊C4—28no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4C2×C22⋊C418yes—
406C4⋊C4C2×C22⋊C418yes—
407C4⋊C4C2×C22⋊C414yes—
408C4⋊C4C2×C22⋊C414yes—
409C4⋊C4C2×C22⋊C418yes—
410C4⋊C4C2×C22⋊C418yes—
411C4⋊C4C2×C22⋊C4116yes—
412C4⋊C4C2×C22⋊C418yes—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—216no—
416C22⋊C4—216no—
417C22⋊C4—28no—
418C22⋊C4—28no—
419C22⋊C4C2×C22⋊C418yes—
420C22⋊C4C2×C22⋊C418yes—
421C42—28no—
422C42C2×C22⋊C412yes—
423C42C2×C22⋊C412yes—
424C42C2×C22⋊C412yes—
425C42C2×C22⋊C412yes—
426C42C22×D412yes—
427C42C22×D412yes—
428C42C22×D4116yes—
429C42C22×D412yes—
430C42C22×D412yes—
431C23—28no—
432C23—28no—
433C23—24no—
434C23—22no—
435C23—24no—
436C23—41no—
437C23—41no—
438C23—44no—
439C23—44no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—816no—
445C23C22×C22⋊C411yescharacteristic
446C23C22×C22⋊C411yescharacteristic
447C23#51918yes—
448C23C24×C411yescharacteristic
449Q8—28no—
450Q8—44no—
451Q8—48no—
452Q8—44no—
453Q8—44no—
454Q8—84no—
455Q8—816no—
456D4—216no—
457D4—216no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—816no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—816no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4C22×C22⋊C412yes—
488C2×C4C22×C22⋊C412yes—
489C2×C4C22×C22⋊C412yes—
490C2×C4C22×C22⋊C411yescharacteristic
491C2×C4C22×C22⋊C411yescharacteristic
492C2×C4C22×C22⋊C411yescharacteristic
493C2×C4C22×C22⋊C418yes—
494C2×C4C22×C22⋊C418yes—
495C2×C4C22×C22⋊C411yescharacteristic
496C2×C4C23×D412yes—
497C2×C4C2×2−1+414yes—
498C8—48no—
499C8—48no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—28no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—48no—
510C22—48no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—88no—
515C22—88no—
516C22C23×C22⋊C411yescharacteristic
517C22C2×#51912yes—
518C4—22no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—28no—
523C4—22no—
524C4—22no—
525C4—28no—
526C4—44no—
527C4—44no—
528C4—44no—
529C4#288812yes—
530C4#348614yes—
531C2—22no—
532C2—41no—
533C2—41no—
534C2—48no—
535C2#3417311yescharacteristic
536C2C2×#288811yescharacteristic
537C2#6007611yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60249C2semidirect
3. #22515C2semidirect
4. #22515C2semidirect
5. Q8×#100C2semidirect
6. #21832C2semidirect
7. #21831C2semidirect
32. Q8×M16C22semidirect
33. Q8×M16C22semidirect
37. #1998C22semidirect
38. #1998C22semidirect
39. #1997C22semidirect
40. #1997C22semidirect
41. #1997C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
#602492
#225152
Q8×#1002
#218322
#218312
#207292
#207262