Finite Group Databasebeta

← #6433200

Subgroups — #6433200

Order 512.

2220 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3093C212yesmaximal, complemented
2#34633C211yesmaximal, characteristic
3C2×#2317C211yesmaximal, characteristic, complemented
4#29769C218yesmaximal, complemented
5#29766C214yesmaximal, complemented
6#29765C214yesmaximal, complemented
7#23657C212yesmaximal, complemented
8#23617C214yesmaximal, complemented
9#23508C212yesmaximal, complemented
10#20909C212yesmaximal, complemented
11#20627C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#525C2212yes—
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#3093—216no—
18C2×#485C2218yes—
19C2×#482C2218yes—
20C2×#428C2214yes—
21C2×#423C2212yes—
22C2×#417C2214yes—
23C2×#417C2214yes—
24C2×#407C2212yes—
25#2621C2212yes—
26#2620C2214yes—
27#2619C2212yes—
28#2618—21no—
29#2618—21no—
30#2617—21no—
31#2617—21no—
32#2392—22no—
33#2392—22no—
34#2317—22no—
35#2317—22no—
36#2317C2214yes—
37#2317C2214yes—
38#2297C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#385—21no—
42C2×#385—21no—
43#2285C2211yescharacteristic
44C2×#384—21no—
45C2×#382—21no—
46#2085C2218yescomplemented
47#2084C2214yescomplemented
48#2083C2214yescomplemented
49#2068C2218yescomplemented
50#2067C2218yescomplemented
51#2066C2218yescomplemented
52#2065C2218yescomplemented
53#2043C2214yescomplemented
54#2042C2218yescomplemented
55#2041C2214yescomplemented
56#1874C2218yes—
57#1872C2218yes—
58#1798C2214yes—
59#1798C2214yes—
60#1736C2214yes—
61#1728C2212yes—
62C22×#41C2312yes—
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C22×M16C2314yes—
66#532—44no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#124C2312yes—
70#525—28no—
71#525—28no—
72C2×#123—22no—
73C2×C22⋊Q8C2314yes—
74C2×#121—21no—
75C2×#121—21no—
76C2×#121C2314yes—
77C2×#120—22no—
78C2×C4×D4—24no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×#117—22no—
82C2×#117—24no—
83C2×#117—44no—
84C2×#117D412yes—
85C2×#117D412yes—
86C2×#117D418yescomplemented
87C2×#117C2312yes—
88C2×#117C2314yes—
89C2×#117C2312yes—
90C2×#117C2312yes—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4D411yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4—21no—
102C22×C22⋊C4D411yescharacteristic
103C22×C22⋊C4D411yescharacteristic
104C22×C42—21no—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#485—232no—
108#482—232no—
109#428—216no—
110C2×#107C2318yes—
111C2×#106C2318yes—
112#423—44no—
113C2×C4⋊C8C2314yes—
114C2×C4⋊C8C2314yes—
115#417—216no—
116#417—48no—
117C2×#103C2318yes—
118C2×#103C2318yes—
119C2×#102C2318yes—
120C2×#102C2318yes—
121#407—28no—
122C2×C22⋊C8C2318yes—
123#398—22no—
124#398—22no—
125#397—22no—
126#397—22no—
127#396—24no—
128#396—24no—
129#394—22no—
130#394—22no—
131#394—24no—
132#394—24no—
133#392—22no—
134#392—22no—
135#391C2314yes—
136#390C2314yes—
137#389—22no—
138#389—22no—
139#389—24no—
140#389C2318yes—
141#389C2314yes—
142#389C2314yes—
143#386—22no—
144#386—22no—
145#385—24no—
146#385—24no—
147#385—24no—
148#385—24no—
149#384—24no—
150#384—24no—
151#384C2312yes—
152#382—24no—
153#382—24no—
154#382C2312yes—
155#380—21no—
156#380—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C22⋊C4—24no—
160C2×#95—22no—
161C2×#95—22no—
162#340C2318yes—
163#340C2318yes—
164#336C2314yes—
165#336C2314yes—
166C25C2×D411yescharacteristic
167C2×#41—24no—
168C2×#41—24no—
169C2×#41—28no—
170C2×#41—28no—
171C22×Q8C2412yes—
172C22×D4—22no—
173C22×D4—24no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C2412yes—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—44no—
184C23×C4—84no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4C2×D414yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4#4114yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C2×M16—28no—
196C2×M16—28no—
197C2×M16—28no—
198C22×C8C2418yes—
199C4⋊Q8—42no—
200#127—42no—
201#124—44no—
202#123—48no—
203C22⋊Q8—28no—
204C22⋊Q8—28no—
205#121—28no—
206#121—28no—
207#121—44no—
208#121—44no—
209#120—48no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—416no—
213#117—216no—
214#117—28no—
215#117—24no—
216#117—24no—
217#117—44no—
218#117—44no—
219#117—42no—
220#117—42no—
221#117—42no—
222#117—42no—
223#117C2×D4116yes—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—84no—
244C2×C4⋊C4—84no—
245C2×C4⋊C4C2×D418yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2414yes—
252C2×C4⋊C4C2414yes—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4C2×D412yes—
265C2×C22⋊C4C2×D412yes—
266C2×C22⋊C4C2×D412yes—
267C2×C22⋊C4C2×D412yes—
268C2×C42—28no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D412yes—
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2412yes—
279#107—216no—
280#106—216no—
281C4⋊C8—44no—
282C4⋊C8—44no—
283#103—216no—
284#103—48no—
285#102—216no—
286#102—48no—
287C22⋊C8—216no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—22no—
294#95—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—42no—
299C24—42no—
300C24#12114yes—
301C24C22×D412yes—
302C24C22×D411yescharacteristic
303#41—416no—
304#41—416no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—48no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—84no—
322C2×D4—84no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—216no—
328C22×C4—216no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—28no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—48no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—88no—
356C22×C4—816no—
357C22×C4#12114yes—
358C22×C4#12118yes—
359C22×C4C22×D412yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D412yes—
366C22×C4C22×D414yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D412yes—
369C22×C4C2×#4114yes—
370C22×C4C2×#4112yes—
371C22×C4H(5,𝔽2)14yes—
372C22×C42−1+414yes—
373C22×C4C2511yescharacteristic
374M16—416no—
375C2×C8—28no—
376C2×C8—28no—
377C2×C8—28no—
378C4⋊C4—28no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—28no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—88no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4C22×D418yes—
398C22⋊C4—216no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C42—216no—
411C42—24no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—44no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—24no—
423C23—28no—
424C23—22no—
425C23—22no—
426C23—48no—
427C23—44no—
428C23—44no—
429C23—48no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23C2×#12114yes—
436C23C2×#12112yes—
437C23#53412yes—
438C23#53612yes—
439C23C23×D411yescharacteristic
440Q8—44no—
441Q8—44no—
442D4—44no—
443D4—44no—
444D4—84no—
445D4—84no—
446D4—816no—
447D4—88no—
448C2×C4—28no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—216no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—28no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—416no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—88no—
486C2×C4—816no—
487C2×C4—88no—
488C2×C4C2×#12114yes—
489C2×C4C2×#12112yes—
490C2×C4#53412yes—
491C2×C4#53612yes—
492C2×C4#53718yes—
493C2×C4#53918yes—
494C2×C4C23×D411yescharacteristic
495C2×C4C22×#4111yescharacteristic
496C2×C4C2×H(5,𝔽2)12yes—
497C2×C4C2×2−1+412yes—
498C8—48no—
499C22—24no—
500C22—22no—
501C22—22no—
502C22—24no—
503C22—28no—
504C22—24no—
505C22—416no—
506C22—42no—
507C22—42no—
508C22—44no—
509C22—84no—
510C22—82no—
511C22—82no—
512C22—84no—
513C22—82no—
514C22#261814yes—
515C22C22×#12111yescharacteristic
516C22C2×#53411yescharacteristic
517C22C2×#53611yescharacteristic
518C22#358512yes—
519C22#358612yes—
520C4—24no—
521C4—48no—
522C4—42no—
523C4—41no—
524C4—41no—
525C4—42no—
526C4—41no—
527C4—41no—
528C4—84no—
529C4—84no—
530C4—82no—
531C4#261814yes—
532C4C2×#53712yes—
533C4C2×#53912yes—
534C2—24no—
535C2—44no—
536C2—162no—
537C2C2×#261811yescharacteristic
538C2#6042512yes—
539C2#6075311yescharacteristic
540C2#6075411yescharacteristic
541C2#6086712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34633C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3093C2semidirect
3. C2×#2317C2semidirect
4. #29769C2semidirect
5. #29766C2semidirect
6. #29765C2semidirect
7. #23657C2semidirect
8. #23617C2semidirect
9. #23508C2semidirect
10. #20909C2semidirect
11. #20627C2semidirect
46. #2085C22semidirect
47. #2084C22semidirect
48. #2083C22semidirect
49. #2068C22semidirect
50. #2067C22semidirect
51. #2066C22semidirect
52. #2065C22semidirect
53. #2043C22semidirect
54. #2042C22semidirect
55. #2041C22semidirect
86. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30932
#346332
C2×#23172
#297692
#297662
#297652
#236572
#236172
#235082
#209092
#206272