Finite Group Databasebeta

← #364049

Subgroups — #364049

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1818C211yesmaximal, characteristic
3#13795C211yesmaximal, characteristic
4#11373C211yesmaximal, characteristic, complemented
5#11371C211yesmaximal, characteristic, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11358C211yesmaximal, characteristic, complemented
8#10807C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10#10731C212yesmaximal
11#10730C212yesmaximal
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16#2314—24no—
17#2314C418yescomplemented
18C2×#388—22no—
19C2×#387—22no—
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380C2212yes—
23C2×C4×C4⋊C4C2212yes—
24#1818—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C4⋊#95—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C22⋊#95—22no—
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#117—24no—
47C2×#117—24no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—24no—
51C2×#117—42no—
52C2×#117C2×C414yes—
53C2×#117D414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—42no—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D412yes—
70C22×C42—21no—
71C22×C42—22no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77#388—216no—
78#387—216no—
79#384—22no—
80#384C2×C414yescomplemented
81#382—22no—
82#382C2×C414yescomplemented
83#380—24no—
84#380C2×C418yes—
85C4×C4⋊C4—24no—
86C4×C4⋊C4C2×C418yescomplemented
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—24no—
103#336—22no—
104#336—22no—
105C25C2×D411yescharacteristic
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—24no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—82no—
122C23×C4—82no—
123C23×C4C22×C412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4#4114yes—
129C23×C4#4114yes—
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145#117—216no—
146#117—216no—
147#117—48no—
148#117—48no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—48no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4C22⋊C418yes—
182C2×C4⋊C4C22×C414yes—
183C2×C4⋊C4C22×C414yes—
184C2×C4⋊C4#4112yes—
185C2×C4⋊C4#4112yes—
186C2×C4⋊C4#4112yes—
187C2×C4⋊C4#4112yes—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4C22⋊C418yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—28no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—82no—
206C2×C42—82no—
207C2×C42C22×C412yes—
208C2×C42C22×C412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C22⋊C8—48no—
212#95—28no—
213#95—28no—
214#95—28no—
215#95—82no—
216#95—82no—
217#95—82no—
218#95—82no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24C2×C22⋊C412yes—
228C24#12312yes—
229C24#12312yes—
230C24C22×D411yescharacteristic
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—48no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—88no—
297C22×C4—88no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4C2×C22⋊C414yes—
301C22×C4C2×C22⋊C414yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4#11718yes—
304C22×C4#12318yes—
305C22×C4#12312yes—
306C22×C4#12312yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C2×#4112yes—
310C22×C4C2×#4112yes—
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4H(5,𝔽2)12yes—
316C22×C42−1+412yes—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—216no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—88no—
350C22⋊C4—216no—
351C22⋊C4—48no—
352C22⋊C4—88no—
353C42—24no—
354C42—24no—
355C42—24no—
356C42—24no—
357C42—48no—
358C42—88no—
359C23—22no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23—42no—
381C23#38114yes—
382C23C22×C22⋊C411yescharacteristic
383C23C2×#12311yescharacteristic
384C23C2×#12311yescharacteristic
385C23#52511yescharacteristic
386C23#52511yescharacteristic
387C23#53511yescharacteristic
388C23#53611yescharacteristic
389C23#57612yes—
390C23#57712yes—
391C23#57712yes—
392C23#57812yes—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—816no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4#38114yes—
453C2×C4C2×#11712yes—
454C2×C4C4×#4111yescharacteristic
455C2×C4C4×#4111yescharacteristic
456C2×C4#52012yes—
457C2×C4C2×#12312yes—
458C2×C4#53211yescharacteristic
459C2×C4#53211yescharacteristic
460C2×C4#53712yes—
461C2×C4#55112yes—
462C2×C4#55112yes—
463C2×C4#55112yes—
464C2×C4#55112yes—
465C2×C4#55212yes—
466C2×C4#55212yes—
467C2×C4#55212yes—
468C2×C4#55212yes—
469C8—82no—
470C8—82no—
471C22—24no—
472C22—28no—
473C22—42no—
474C22—42no—
475C22—44no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—44no—
481C22—44no—
482C22C2×#38111yescharacteristic
483C22#229711yescharacteristic
484C22#231111yescharacteristic
485C22#248211yescharacteristic
486C22#248211yescharacteristic
487C22#248411yescharacteristic
488C22#248411yescharacteristic
489C22#289712yes—
490C22#289812yes—
491C22#309612yes—
492C22#309712yes—
493C22#309812yes—
494C22#309912yes—
495C22#311812yes—
496C22#312112yes—
497C22#312412yes—
498C22#312512yes—
499C22#322011yescharacteristic
500C22#322111yescharacteristic
501C22#322211yescharacteristic
502C22#322311yescharacteristic
503C22#324112yes—
504C22#324212yes—
505C4—24no—
506C4—44no—
507C4—44no—
508C4—84no—
509C4—84no—
510C4—82no—
511C4—82no—
512C4—82no—
513C4—82no—
514C4#234312yes—
515C4#234312yes—
516C4#249812yes—
517C4#249812yes—
518C2—44no—
519C2#1395811yescharacteristic
520C2#2070312yes—
521C2#2139712yes—
522C2#2187211yescharacteristic
523C2#2187311yescharacteristic
524C2#2759411yescharacteristic
525C2#2759511yescharacteristic
526C2#2759611yescharacteristic
527C2#2759711yescharacteristic
528C2#2762212yes—
529C2#2762312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11373C2semidirect
5. #11371C2semidirect
6. #11358C2semidirect
7. #11358C2semidirect
8. #10807C2semidirect
9. #10806C2semidirect
17. #2314C4semidirect
80. #384C2×C4semidirect
82. #382C2×C4semidirect
86. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#18182
#137952
#113732
#113712
#113582
#108072
#108062
#107312
#107302