Finite Group Databasebeta

← #6497524

Subgroups — #6497524

Order 512.

2148 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3239C212yesmaximal
2C2×#2868C211yesmaximal, characteristic, complemented
3C4×#529C211yesmaximal, characteristic
4#33196C212yesmaximal, complemented
5#33195C212yesmaximal, complemented
6#29866C212yesmaximal, complemented
7#29864C212yesmaximal, complemented
8#28726C214yesmaximal, complemented
9#28694C214yesmaximal, complemented
10#27648C214yesmaximal, complemented
11#27645C214yesmaximal
12#21841C212yesmaximal, complemented
13#20810C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8—21no—
15C22×C4⋊Q8—21no—
16C22×C4⋊Q8C2212yes—
17C2×#529C2211yescharacteristic
18#3239—216no—
19C2×#501C2214yes—
20C2×#489C2214yes—
21C2×#484C2214yes—
22C2×#481C2214yes—
23C2×#475C2214yes—
24C2×#474C2214yes—
25C2×#432C2212yes—
26#2868—22no—
27#2868—21no—
28#2868—21no—
29#2868C2214yes—
30#2868C2212yes—
31#2868C2212yes—
32C4×#125C2212yes—
33C4×#126—22no—
34C4×#124C2212yes—
35C4×C22⋊Q8C2212yes—
36C4×#123—22no—
37C4×#121C2212yes—
38#2298—21no—
39#2295C2211yescharacteristic
40#2293C2211yescharacteristic
41#2290—21no—
42C2×#384—22no—
43C2×#384C2214yes—
44#2283C2211yescharacteristic
45C42×Q8C2211yescharacteristic
46C42×D4—21no—
47C42×D4C2211yescharacteristic
48C4×#117—21no—
49C2×C43C2211yescharacteristic
50#2084C2218yes—
51#2083C2218yes—
52#2072C2218yes—
53#2071C2218yes—
54#1993C2218yes—
55#1988C2214yes—
56#1988C2214yes—
57#1948C2218yes—
58#1947C2218yes—
59#1842C2212yes—
60#1831C2212yes—
61#1824C2218yes—
62#1803C2218yes—
63#1802C2218yes—
64#1797C2214yes—
65#1767C2214yes—
66C23×Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—24no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—24no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—28no—
76C2×C4⋊Q8—28no—
77C2×C4⋊Q8—48no—
78C2×C4⋊Q8—48no—
79C2×C4⋊Q8C2314yes—
80C2×C4⋊Q8C2314yes—
81#529—24no—
82#529D418yes—
83C2×#126—21no—
84C2×#125C2311yescharacteristic
85C2×#124C2311yescharacteristic
86C2×#123—21no—
87C2×C22⋊Q8C2311yescharacteristic
88C2×#121C2311yescharacteristic
89C2×C4×Q8C2312yes—
90C2×C4×Q8C2311yescharacteristic
91C2×C4×D4—21no—
92C2×C4×D4—22no—
93C2×C4×D4C2312yes—
94C2×C4×D4C2311yescharacteristic
95C2×#117—21no—
96C2×#117—21no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4C2314yes—
101C22×C42—21no—
102C22×C42D411yescharacteristic
103C22×C42D411yescharacteristic
104C22×C42D412yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107#501—216no—
108#489—216no—
109#484—216no—
110#481—216no—
111#475—216no—
112#474—216no—
113#432—28no—
114C2×#107C2318yes—
115C2×#106C2318yes—
116C2×C4⋊C8C2314yes—
117C2×C4⋊C8C2318yes—
118C2×#103C2318yes—
119C2×#102C2318yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2314yes—
122#391C2312yes—
123#390C2312yes—
124#389C2312yes—
125#389C2312yes—
126#388—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#387—22no—
131#387—22no—
132#387—22no—
133#387—22no—
134#386—22no—
135#385—22no—
136#384—24no—
137#384—24no—
138#384—24no—
139#384—24no—
140#384—24no—
141#384—24no—
142#384C2318yes—
143#384C2318yes—
144#383—21no—
145#382C2312yes—
146#381—21no—
147#380C2312yes—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4C2312yes—
155C4×C4⋊C4C2312yes—
156C4×C4⋊C4C2311yescharacteristic
157C4×C4⋊C4C2311yescharacteristic
158C4×C22⋊C4—21no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4—22no—
161C4×C22⋊C4—22no—
162C4×C22⋊C4—22no—
163C4×C22⋊C4—22no—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—21no—
166#376—21no—
167#376—21no—
168C43—21no—
169C43D412yes—
170C43D411yescharacteristic
171C43D411yescharacteristic
172C43C2312yes—
173C43C2311yescharacteristic
174C43C2311yescharacteristic
175#340C23116yes—
176#340C23116yes—
177#336C2318yes—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—48no—
181C22×Q8—48no—
182C22×Q8—48no—
183C22×Q8C2411yescharacteristic
184C22×D4—21no—
185C22×D4C2411yescharacteristic
186C23×C4—21no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2411yescharacteristic
194C22×C8C2418yes—
195C4⋊Q8—28no—
196C4⋊Q8—28no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—48no—
200C4⋊Q8—48no—
201C4⋊Q8—416no—
202C4⋊Q8—416no—
203C4⋊Q8—48no—
204C4⋊Q8—48no—
205C4⋊Q8—48no—
206C4⋊Q8—48no—
207#126—28no—
208#125—22no—
209#125C2×D414yes—
210#124—22no—
211#124C2×D414yes—
212#123—28no—
213C22⋊Q8—22no—
214C22⋊Q8C2×D414yes—
215#121—22no—
216#121C2×D414yes—
217C4×Q8—22no—
218C4×Q8—28no—
219C4×Q8C2×D414yes—
220C4×D4—28no—
221C4×D4—22no—
222C4×D4—28no—
223C4×D4—48no—
224C4×D4C2×D414yes—
225#117—28no—
226#117—44no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—416no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4C2418yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2411yescharacteristic
250C2×C4⋊C4C2411yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C42—28no—
261C2×C42—21no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—44no—
271C2×C42C2×D412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D414yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42#4114yes—
287C2×C42C2412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2412yes—
291#107—216no—
292#106—216no—
293C4⋊C8—28no—
294C4⋊C8—216no—
295#103—216no—
296#102—216no—
297C22⋊C8—28no—
298#97—28no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—21no—
307#95—22no—
308#95—21no—
309#95—22no—
310C24—21no—
311C24C22×D411yescharacteristic
312C2×Q8—21no—
313C2×Q8—24no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8—48no—
319C2×Q8—48no—
320C2×Q8—832no—
321C2×Q8C22×D412yes—
322C2×D4—24no—
323C2×D4—21no—
324C2×D4—24no—
325C2×D4—44no—
326C2×D4C22×D412yes—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4#12114yes—
352C22×C4#12114yes—
353C22×C4#13718yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D412yes—
364C22×C4C2×#4112yes—
365C22×C4C2511yescharacteristic
366C2×C8—28no—
367C2×C8—216no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—416no—
379C4⋊C4—416no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C4⋊C4—416no—
385C4⋊C4—416no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4C22×D412yes—
389C4⋊C4C22×D412yes—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C42—24no—
400C42—24no—
401C42—24no—
402C42—22no—
403C42—22no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—24no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—44no—
412C42—44no—
413C42—42no—
414C42—42no—
415C42—44no—
416C42#12114yes—
417C42#12114yes—
418C42AGL(1,ℤ/8)14yes—
419C42AGL(1,ℤ/8)14yes—
420C42C22×D412yes—
421C42C22×D412yes—
422C42C2×#4112yes—
423C42C2×#4112yes—
424C422−1+418yes—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—22no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23C2×#12112yes—
434C23C2×#13714yes—
435C23C23×D411yescharacteristic
436Q8—24no—
437Q8—42no—
438Q8—816no—
439Q8—816no—
440D4—24no—
441D4—48no—
442D4—42no—
443C2×C4—216no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—28no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—216no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—48no—
468C2×C4—48no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—48no—
474C2×C4—48no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—816no—
478C2×C4—88no—
479C2×C4C2×#12112yes—
480C2×C4C2×#12112yes—
481C2×C4C2×#12112yes—
482C2×C4C2×#12112yes—
483C2×C4C2×#12112yes—
484C2×C4#53614yes—
485C2×C4#53614yes—
486C2×C4C2×AGL(1,ℤ/8)14yes—
487C2×C4C2×AGL(1,ℤ/8)14yes—
488C2×C4C2×AGL(1,ℤ/8)12yes—
489C2×C4C2×AGL(1,ℤ/8)12yes—
490C2×C4C2×#13714yes—
491C2×C4C2×#13714yes—
492C2×C4C23×D411yescharacteristic
493C2×C4C22×#4111yescharacteristic
494C2×C4C2×2−1+414yes—
495C8—48no—
496C22—24no—
497C22—24no—
498C22—22no—
499C22—22no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—82no—
504C22—82no—
505C22#303914yes—
506C22C22×#12111yescharacteristic
507C22C2×#53612yes—
508C22C22×AGL(1,ℤ/8)11yescharacteristic
509C22C22×AGL(1,ℤ/8)11yescharacteristic
510C22C22×#13712yes—
511C4—24no—
512C4—24no—
513C4—24no—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—82no—
522C4—82no—
523C4—88no—
524C4#303814yes—
525C4#323918yescomplemented
526C4#349912yes—
527C4#350912yes—
528C2—24no—
529C2—82no—
530C2#3531611yescharacteristic
531C2C2×#303811yescharacteristic
532C2C2×#303911yescharacteristic
533C2C2×#323912yes—
534C2#6086012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2868C2semidirect
4. #33196C2semidirect
5. #33195C2semidirect
6. #29866C2semidirect
7. #29864C2semidirect
8. #28726C2semidirect
9. #28694C2semidirect
10. #27648C2semidirect
12. #21841C2semidirect
13. #20810C2semidirect
525. C4#3239semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
C2×#28682
C4×#5292
#331962
#331952
#298662
#298642
#287262
#286942
#276482
#276452
#218412
#208102