Finite Group Databasebeta

← #463370

Subgroups — #463370

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C212yesmaximal
2#15195C211yesmaximal, characteristic
3#14130C211yesmaximal, characteristic
4#11267C212yesmaximal, complemented
5#11256C212yesmaximal, complemented
6#11228C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11042C211yesmaximal, characteristic
9#10730C212yesmaximal
10C23×C4⋊C4C2211yescharacteristic
11C22×#107C2212yes—
12C22×C4⋊C8C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C2×#389C2212yes—
15C2×#385—21no—
16C2×#384C2211yescharacteristic
17C2×#380C2212yes—
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20#2076—28no—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30#1444—21no—
31#1442—21no—
32#1442—21no—
33#1442—21no—
34#1439—21no—
35#1439—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C22⋊#95—21no—
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C22×C4⋊C4—24no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—41no—
50C22×C4⋊C4—41no—
51C22×C4⋊C4—42no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#107—24no—
63C2×#107—24no—
64C2×#107—42no—
65C2×C4⋊C8—41no—
66C2×C4⋊C8—42no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—42no—
70#389—24no—
71#389—42no—
72#385—44no—
73#384—41no—
74#384—41no—
75#380—22no—
76#380—24no—
77#380—24no—
78#380D414yes—
79C4×C22⋊C4—41no—
80C4×C22⋊C4—41no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D411yescharacteristic
91C2×#95D411yescharacteristic
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#340—42no—
99#336—24no—
100#336—41no—
101C25C2×D411yescharacteristic
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—44no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—82no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4#4112yes—
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
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#4114yes—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
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—44no—
164C2×C4⋊C4—48no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4#4114yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4Q1614yes—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—22no—
197#107—48no—
198C4⋊C8—44no—
199C22⋊C8—48no—
200#95—24no—
201#95—41no—
202#95—41no—
203#95—44no—
204#95—42no—
205#95—44no—
206#95—42no—
207#95—44no—
208#95—82no—
209#95—82no—
210#95—82no—
211C24—21no—
212C24—21no—
213C24—22no—
214C24—22no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—22no—
219C24#12112yes—
220C24C2×Q1612yes—
221C24#13712yes—
222C24C22×D411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—48no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
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—88no—
292C22×C4—88no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4#12112yes—
296C22×C4C2×Q1612yes—
297C22×C4C2×Q1612yes—
298C22×C4AGL(1,ℤ/8)12yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4112yes—
302C22×C4C2×#4112yes—
303C22×C4H(5,𝔽2)11yescharacteristic
304C22×C4H(5,𝔽2)11yescharacteristic
305C22×C4H(5,𝔽2)12yes—
306C22×C42−1+411yescharacteristic
307C22×C42−1+411yescharacteristic
308C22×C42−1+412yes—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—48no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—88no—
341C4⋊C4—84no—
342C4⋊C4—84no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—48no—
346C22⋊C4—88no—
347C42—42no—
348C42—42no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C23—24no—
354C23—24no—
355C23—22no—
356C23—22no—
357C23—24no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—42no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23#46214yes—
376C23C2×#12111yescharacteristic
377C23#53411yescharacteristic
378C23#53611yescharacteristic
379C23#54612yes—
380C23#54712yes—
381C23C22×Q1611yescharacteristic
382C23C2×AGL(1,ℤ/8)11yescharacteristic
383C23C2×#13711yescharacteristic
384C23#57511yescharacteristic
385C23#57612yes—
386C23#57712yes—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—816no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
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—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4#52912yes—
448C2×C4#53711yescharacteristic
449C2×C4#53711yescharacteristic
450C2×C4#53812yes—
451C2×C4#53912yes—
452C2×C4#53911yescharacteristic
453C2×C4#53911yescharacteristic
454C2×C4#53912yes—
455C2×C4#54112yes—
456C2×C4#54212yes—
457C2×C4#54612yes—
458C2×C4#54712yes—
459C2×C4#55111yescharacteristic
460C2×C4#55212yes—
461C2×C4#55911yescharacteristic
462C2×C4H(3,𝔽4)11yescharacteristic
463C2×C4#56312yes—
464C2×C4#56311yescharacteristic
465C8—44no—
466C8—82no—
467C22—24no—
468C22—28no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—42no—
476C22—42no—
477C22—44no—
478C22#242512yes—
479C22#261811yescharacteristic
480C22#268411yescharacteristic
481C22#269812yes—
482C22#272411yescharacteristic
483C22C2×#46211yescharacteristic
484C22#303611yescharacteristic
485C22#304512yes—
486C22#318911yescharacteristic
487C22#319311yescharacteristic
488C22#319911yescharacteristic
489C22#320011yescharacteristic
490C22#323011yescharacteristic
491C22#323611yescharacteristic
492C22#324112yes—
493C22#329912yes—
494C22#331412yes—
495C22#332112yes—
496C22#332512yes—
497C22#333812yes—
498C22#334412yes—
499C22#334512yes—
500C22#335512yes—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—81no—
506C4—84no—
507C4—84no—
508C4—162no—
509C4—162no—
510C4#251711yescharacteristic
511C4#265212yes—
512C4#269512yes—
513C4#271211yescharacteristic
514C4#272211yescharacteristic
515C4#283511yescharacteristic
516C2—44no—
517C2#1647011yescharacteristic
518C2#2520612yes—
519C2#2522812yes—
520C2#2975711yescharacteristic
521C2#2976511yescharacteristic
522C2#3126311yescharacteristic
523C2#3127011yescharacteristic
524C2#3169212yes—
525C2#3171012yes—
526C2#3246011yescharacteristic
527C2#3246811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11267C2semidirect
5. #11256C2semidirect
6. #11228C2semidirect
7. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
#151952
#141302
#112672
#112562
#112282
#112192
#110422
#107302