Finite Group Databasebeta

← #470018

Subgroups — #470018

Order 512.

1088 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×#2514C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4#15151C211yesmaximal, characteristic
5#11328C212yesmaximal
6#11268C212yesmaximal
7#11202C211yesmaximal, characteristic
8#11124C212yesmaximal
9#11077C211yesmaximal, characteristic
10#11067C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16#2514—24no—
17#2514—24no—
18C2×#401—21no—
19C2×#400—21no—
20C2×#398—21no—
21C2×#398—21no—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2212yes—
27C2×#387—21no—
28C2×#384C2211yescharacteristic
29C2×#383—21no—
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#2072—24no—
34#1947—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C42—21no—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#107—24no—
66C2×#107—42no—
67C2×#107—42no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—22no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76#401—28no—
77#400—28no—
78#398—44no—
79#398—44no—
80#391—22no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391—22no—
85#391D414yes—
86#389—22no—
87#389—22no—
88#389—42no—
89#389—42no—
90#387—28no—
91#384—22no—
92#384—22no—
93#383—28no—
94#382—22no—
95#382—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4D414yes—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#336—24no—
123#336—22no—
124#336—42no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8C2×D412yes—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8#4114yes—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4#4114yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223#107—48no—
224C4⋊C8—28no—
225#103—48no—
226#103—48no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—22no—
238#95—22no—
239#95—44no—
240#95—42no—
241#95—44no—
242#95—44no—
243#95—42no—
244C24C22×D411yescharacteristic
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4#12012yes—
327C22×C4#12012yes—
328C22×C4#13512yes—
329C22×C4AGL(1,ℤ/8)12yes—
330C22×C4AGL(1,ℤ/8)12yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4112yes—
334C22×C4C2×#4112yes—
335C22×C4H(5,𝔽2)11yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C42−1+412yes—
339C22×C42−1+412yes—
340C22×C42−1+411yescharacteristic
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—88no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C42—22no—
380C42—22no—
381C42—24no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—42no—
389C42—48no—
390C23C2×#12011yescharacteristic
391C23#53511yescharacteristic
392C23#53711yescharacteristic
393C23#54712yes—
394C23#54812yes—
395C23C2×#13511yescharacteristic
396C23C2×AGL(1,ℤ/8)11yescharacteristic
397C23C2×AGL(1,ℤ/8)11yescharacteristic
398C23#57511yescharacteristic
399C23#57812yes—
400C23#57812yes—
401Q8—44no—
402Q8—44no—
403Q8—81no—
404Q8—81no—
405Q8—81no—
406Q8—81no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4#44914yes—
475C2×C4#52511yescharacteristic
476C2×C4#53211yescharacteristic
477C2×C4#53511yescharacteristic
478C2×C4#53511yescharacteristic
479C2×C4#53611yescharacteristic
480C2×C4#53611yescharacteristic
481C2×C4#53711yescharacteristic
482C2×C4#53711yescharacteristic
483C2×C4#53912yes—
484C2×C4#54112yes—
485C2×C4#54112yes—
486C2×C4#54212yes—
487C2×C4#54712yes—
488C2×C4#54712yes—
489C2×C4#55112yes—
490C2×C4#55312yes—
491C2×C4#56312yes—
492C2×C4#56312yes—
493C8—44no—
494C8—82no—
495C22#240611yescharacteristic
496C22#243211yescharacteristic
497C22#261511yescharacteristic
498C22#272312yes—
499C22#272512yes—
500C22C2×#44911yescharacteristic
501C22#300611yescharacteristic
502C22#301111yescharacteristic
503C22#301111yescharacteristic
504C22#321211yescharacteristic
505C22#321711yescharacteristic
506C22#322312yes—
507C22#324711yescharacteristic
508C22#325111yescharacteristic
509C22#326112yes—
510C22#333612yes—
511C22#334312yes—
512C22#334912yes—
513C22#335312yes—
514C22#336412yes—
515C22#336712yes—
516C22#336712yes—
517C22#337712yes—
518C4—24no—
519C4—42no—
520C4—42no—
521C4—81no—
522C4—82no—
523C4—81no—
524C4—82no—
525C4—82no—
526C4—84no—
527C4—82no—
528C4—81no—
529C4—162no—
530C4#260311yescharacteristic
531C4#263711yescharacteristic
532C4#266411yescharacteristic
533C4#271812yes—
534C4#272212yes—
535C4#282111yescharacteristic
536C2#1667311yescharacteristic
537C2#2409411yescharacteristic
538C2#2410711yescharacteristic
539C2#2565011yescharacteristic
540C2#2566111yescharacteristic
541C2#2966211yescharacteristic
542C2#2967411yescharacteristic
543C2#3243412yes—
544C2#3244812yes—
545C2#3248312yes—
546C2#3249712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25142
C2×#20722
C2×#19472
#151512
#113282
#112682
#112022
#111242
#110772
#110672
#108062