Finite Group Databasebeta

← #361056

Subgroups — #361056

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#102C211yesmaximal, characteristic
2#13918C211yesmaximal, characteristic, complemented
3#13820C211yesmaximal, characteristic
4#11189C212yesmaximal
5#11066C212yesmaximal
6#10884C212yesmaximal, complemented
7#10820C211yesmaximal, characteristic, complemented
8#10806C211yesmaximal, characteristic, complemented
9#10806C212yesmaximal, complemented
10#10797C211yesmaximal, characteristic, complemented
11#10788C211yesmaximal, characteristic
12C22×C4×D4C2211yescharacteristic
13C22×#102C2211yescharacteristic
14C22×#102C2211yescharacteristic
15C22×#102C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#390C2212yes—
18C2×#389C2212yes—
19C2×#382—21no—
20C2×C4×C4⋊C4C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4—21no—
23C4×#102C418yescomplemented
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1442—22no—
35#1437—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
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⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C42—21no—
61C22×C42—22no—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×#102—24no—
65C2×#102—22no—
66C2×#102—22no—
67C2×#102—24no—
68C2×#102—42no—
69C2×#102—42no—
70C2×#102C2×C414yes—
71C2×#102C2×C414yes—
72C2×C4×C8—42no—
73C2×C4×C8C2×C414yes—
74#390—24no—
75#390—24no—
76#389—24no—
77#389—42no—
78#382—44no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—42no—
82C4×C4⋊C4C2×C414yes—
83C4×C22⋊C4—44no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
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—24no—
99#340—24no—
100#340—42no—
101#340C2×C414yes—
102#340C2×C414yes—
103#336—22no—
104#336—42no—
105#336—42no—
106C25—21no—
107C22×D4—22no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4C22×C412yes—
116C22×D4#4114yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—24no—
122C23×C4—22no—
123C23×C4—44no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—82no—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8C22×C412yes—
148C22×C8C22×C412yes—
149C4×D4—28no—
150C4×D4—44no—
151C4×D4—48no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
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—21no—
164C2×C4⋊C4—22no—
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—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4#4114yes—
178C2×C4⋊C4#4114yes—
179C2×C4⋊C4#4112yes—
180C2×C4⋊C4#4112yes—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—44no—
201C2×C42—44no—
202C2×C42—82no—
203C2×C42C22×C412yes—
204C2×C42#4112yes—
205#102—28no—
206#102—28no—
207#102—48no—
208C4×C8—44no—
209#95—24no—
210#95—22no—
211#95—42no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—44no—
217#95—42no—
218#95—81no—
219#95—81no—
220#95D814yes—
221C24—22no—
222C24—24no—
223C24—42no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—41no—
228C24—42no—
229C24—41no—
230C24—42no—
231C24—41no—
232C24C22×D411yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4#11714yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—48no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—48no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—84no—
304C22×C4—88no—
305C22×C4—88no—
306C22×C4C4×D412yes—
307C22×C4C4×D412yes—
308C22×C4C2×D814yes—
309C22×C4C2×D812yes—
310C22×C4#13512yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C2×#4112yes—
313C22×C4C2×#4112yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4111yescharacteristic
318C22×C4H(5,𝔽2)12yes—
319C22×C4H(5,𝔽2)11yescharacteristic
320C22×C42−1+411yescharacteristic
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—88no—
355C4⋊C4#11714yes—
356C4⋊C4#11714yes—
357C22⋊C4—48no—
358C22⋊C4—416no—
359C22⋊C4—88no—
360C42—22no—
361C42—22no—
362C42—24no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—48no—
368C42—88no—
369C23—24no—
370C23—44no—
371C23—44no—
372C23—42no—
373C23—42no—
374C23—44no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—44no—
381C23—44no—
382C23—81no—
383C23—81no—
384C23—82no—
385C23—82no—
386C23—81no—
387C23—82no—
388C23—81no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23C2×C4×D411yescharacteristic
393C23#52512yes—
394C23#54612yes—
395C23#54611yescharacteristic
396C23#54811yescharacteristic
397C23C22×D811yescharacteristic
398C23C2×#13511yescharacteristic
399C23#57512yes—
400C23#57711yescharacteristic
401C23#57711yescharacteristic
402C23#57712yes—
403D4—24no—
404D4—24no—
405D4—42no—
406D4—42no—
407D4—44no—
408D4—44no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—816no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4C4×D814yes—
465C2×C4C2×#11711yescharacteristic
466C2×C4C2×#11711yescharacteristic
467C2×C4C2×#11711yescharacteristic
468C2×C4C4×#4111yescharacteristic
469C2×C4#51811yescharacteristic
470C2×C4#52011yescharacteristic
471C2×C4#52912yes—
472C2×C4#52912yes—
473C2×C4#52912yes—
474C2×C4#53812yes—
475C2×C4#53912yes—
476C2×C4#55112yes—
477C2×C4#55212yes—
478C2×C4#55211yescharacteristic
479C2×C4#55312yes—
480C2×C4#55311yescharacteristic
481C2×C4#55512yes—
482C2×C4#55912yes—
483C8—42no—
484C8—42no—
485C8—82no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—82no—
490C22—82no—
491C22—84no—
492C22—84no—
493C22—84no—
494C22—84no—
495C22—82no—
496C22—82no—
497C22#229011yescharacteristic
498C22#234211yescharacteristic
499C22#235211yescharacteristic
500C22#245912yes—
501C22#246012yes—
502C22C2×C4×D811yescharacteristic
503C22#294311yescharacteristic
504C22#294911yescharacteristic
505C22#295211yescharacteristic
506C22#310012yes—
507C22#310212yes—
508C22#312312yes—
509C22#312512yes—
510C22#329412yes—
511C22#329411yescharacteristic
512C22#329612yes—
513C22#330311yescharacteristic
514C22#331712yes—
515C22#332112yes—
516C22#332612yes—
517C22#335811yescharacteristic
518C22#336011yescharacteristic
519C22#337112yes—
520C4—24no—
521C4—42no—
522C4—44no—
523C4—44no—
524C4—44no—
525C4—82no—
526C4—82no—
527C4—84no—
528C4—162no—
529C4#228311yescharacteristic
530C4#231811yescharacteristic
531C4#234611yescharacteristic
532C4#235711yescharacteristic
533C4#251712yes—
534C4#251812yes—
535C2—84no—
536C2#1388811yescharacteristic
537C2#2111311yescharacteristic
538C2#2113311yescharacteristic
539C2#2256511yescharacteristic
540C2#2257011yescharacteristic
541C2#2276511yescharacteristic
542C2#2276611yescharacteristic
543C2#2672012yes—
544C2#2672812yes—
545C2#2675212yes—
546C2#2676012yes—
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. #13918C2semidirect
6. #10884C2semidirect
7. #10820C2semidirect
8. #10806C2semidirect
9. #10806C2semidirect
10. #10797C2semidirect
23. 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
#139182
#138202
#111892
#110662
#108842
#108202
#108062
#107972
#107882