Finite Group Databasebeta

← #361213

Subgroups — #361213

Order 512.

1260 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#102C211yesmaximal, characteristic
2#13936C211yesmaximal, characteristic, complemented
3#13820C211yesmaximal, characteristic
4#11189C212yesmaximal
5#11121C212yesmaximal
6#10908C212yesmaximal, complemented
7#10861C211yesmaximal, characteristic, complemented
8#10807C211yesmaximal, characteristic, complemented
9#10806C212yesmaximal, complemented
10#10797C211yesmaximal, characteristic, complemented
11#10788C211yesmaximal, characteristic
12C22×C4×D4C2211yescharacteristic
13C22×#106C2212yes—
14C22×#102C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×C4×C8C2211yescharacteristic
18C2×#390C2212yes—
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4—21no—
22C4×#102C418yescomplemented
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1443—22no—
33#1437—22no—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38#1434—21no—
39#1434—21no—
40C22⋊#95—21no—
41C4×#95C2211yescharacteristic
42C23×D4C2311yescharacteristic
43C24×C4—21no—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4×D4—24no—
48C2×C4×D4—22no—
49C2×C4×D4—42no—
50C2×C4×D4C2×C414yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C42—21no—
60C22×C42—22no—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—42no—
66C2×#102—24no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102—24no—
70C2×#102—42no—
71C2×#102—42no—
72C2×#102C2×C414yes—
73C2×#102C2×C414yes—
74C2×C4×C8—42no—
75C2×C4×C8C2×C414yes—
76#390—24no—
77#390—24no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—42no—
81C4×C4⋊C4C2×C414yes—
82C4×C22⋊C4—44no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95D411yescharacteristic
94C2×#95D412yes—
95C2×#95D411yescharacteristic
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#340C2×C414yes—
101#340C2×C414yes—
102#336—24no—
103#336—22no—
104#336—42no—
105C25—21no—
106C22×D4—22no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4C22×C412yes—
115C22×D4#4114yes—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—44no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—82no—
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8C22×C412yes—
145C22×C8C22×C412yes—
146C22×C8#4112yes—
147C22×C8#4112yes—
148C4×D4—28no—
149C4×D4—44no—
150C4×D4—48no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4#4114yes—
176C2×C4⋊C4#4114yes—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—24no—
194C2×C42—44no—
195C2×C42—44no—
196C2×C42—82no—
197C2×C42—82no—
198C2×C42C22×C412yes—
199C2×C42#4112yes—
200#106—48no—
201#102—28no—
202#102—28no—
203#102—48no—
204C4×C8—44no—
205#95—24no—
206#95—22no—
207#95—42no—
208#95—44no—
209#95—44no—
210#95—42no—
211#95—42no—
212#95—82no—
213#95—82no—
214#95—81no—
215#95—81no—
216#95—82no—
217#95SD1614yes—
218C24—22no—
219C24—24no—
220C24—42no—
221C24—41no—
222C24—41no—
223C24—41no—
224C24—41no—
225C24—42no—
226C24—42no—
227C24—41no—
228C24—41no—
229C24C22×D411yescharacteristic
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4#11714yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—48no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—48no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—84no—
299C22×C4—88no—
300C22×C4—88no—
301C22×C4C4×D412yes—
302C22×C4C4×D412yes—
303C22×C4C2×SD1614yes—
304C22×C4C2×SD1612yes—
305C22×C4#13512yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C2×#4112yes—
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4112yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4H(5,𝔽2)12yes—
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C42−1+411yescharacteristic
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8#11714yes—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4#11714yes—
351C22⋊C4—48no—
352C22⋊C4—416no—
353C22⋊C4—88no—
354C42—22no—
355C42—22no—
356C42—24no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—84no—
362C42—88no—
363C23—24no—
364C23—44no—
365C23—44no—
366C23—42no—
367C23—44no—
368C23—44no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—44no—
375C23—44no—
376C23—81no—
377C23—81no—
378C23—82no—
379C23—81no—
380C23—81no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23C2×C4×D411yescharacteristic
387C23#52512yes—
388C23#54612yes—
389C23#54711yescharacteristic
390C23#55011yescharacteristic
391C23C22×SD1611yescharacteristic
392C23C2×#13511yescharacteristic
393C23#57512yes—
394C23#57612yes—
395C23#57711yescharacteristic
396C23#57711yescharacteristic
397D4—24no—
398D4—24no—
399D4—42no—
400D4—42no—
401D4—44no—
402D4—44no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—816no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4C4×SD1614yes—
457C2×C4C2×#11711yescharacteristic
458C2×C4C2×#11711yescharacteristic
459C2×C4C2×#11711yescharacteristic
460C2×C4C4×#4111yescharacteristic
461C2×C4#51811yescharacteristic
462C2×C4#52011yescharacteristic
463C2×C4#52912yes—
464C2×C4#52912yes—
465C2×C4#52912yes—
466C2×C4#53812yes—
467C2×C4#53912yes—
468C2×C4#55112yes—
469C2×C4#55112yes—
470C2×C4#55312yes—
471C2×C4#55312yes—
472C2×C4#55511yescharacteristic
473C2×C4#55612yes—
474C2×C4#55611yescharacteristic
475C8—24no—
476C8—42no—
477C8—82no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—84no—
486C22—84no—
487C22—84no—
488C22—84no—
489C22#229011yescharacteristic
490C22#234211yescharacteristic
491C22#235411yescharacteristic
492C22#246612yes—
493C22#246712yes—
494C22C2×C4×SD1611yescharacteristic
495C22#294311yescharacteristic
496C22#294811yescharacteristic
497C22#295211yescharacteristic
498C22#310112yes—
499C22#310212yes—
500C22#311812yes—
501C22#311912yes—
502C22#329512yes—
503C22#329712yes—
504C22#331712yes—
505C22#332112yes—
506C22#333511yescharacteristic
507C22#334011yescharacteristic
508C22#335512yes—
509C22#339211yescharacteristic
510C22#339411yescharacteristic
511C22#340312yes—
512C4—24no—
513C4—42no—
514C4—44no—
515C4—44no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4—84no—
520C4—162no—
521C4#228311yescharacteristic
522C4#231811yescharacteristic
523C4#234711yescharacteristic
524C4#235311yescharacteristic
525C4#251812yes—
526C4#252312yes—
527C2—84no—
528C2#1389211yescharacteristic
529C2#2111711yescharacteristic
530C2#2113411yescharacteristic
531C2#2256511yescharacteristic
532C2#2257411yescharacteristic
533C2#2280511yescharacteristic
534C2#2281611yescharacteristic
535C2#2703212yes—
536C2#2704012yes—
537C2#2706412yes—
538C2#2707212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13936C2semidirect
6. #10908C2semidirect
7. #10861C2semidirect
8. #10807C2semidirect
9. #10806C2semidirect
10. #10797C2semidirect
22. C4×#102C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1022
#139362
#138202
#111892
#111212
#109082
#108612
#108072
#108062
#107972
#107882