Finite Group Databasebeta

← #465165

Subgroups — #465165

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2043C212yesmaximal
2C2×#1981C211yesmaximal, characteristic
3C2×#1928C211yesmaximal, characteristic
4#15190C211yesmaximal, characteristic
5#13933C211yesmaximal, characteristic
6#11255C211yesmaximal, characteristic
7#11124C212yesmaximal
8#11077C211yesmaximal, characteristic
9#11067C212yesmaximal
10#11010C211yesmaximal, characteristic
11#11010C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2212yes—
14C22×#107C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#389C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#2043—28no—
28#1981—24no—
29#1928—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1444—21no—
36#1444—22no—
37#1444—21no—
38#1442—22no—
39C4⋊#95—21no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42#1434—21no—
43C4×#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×Q8C2312yes—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#107—22no—
72C2×#107—42no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—42no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—22no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103D414yes—
95#391—22no—
96#391—22no—
97#391—42no—
98#391—42no—
99#389—24no—
100#389—42no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D414yes—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95D411yescharacteristic
114C2×#95D412yes—
115C2×#95D411yescharacteristic
116#340—22no—
117#340—42no—
118#336—24no—
119#336—22no—
120#336—42no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—21no—
143C22×Q8C2×D412yes—
144C22×Q8#4114yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8C2×D412yes—
173C4⋊Q8—28no—
174C4⋊Q8—48no—
175C4⋊Q8—44no—
176C4⋊Q8—48no—
177C4⋊Q8—48no—
178C4⋊Q8—48no—
179C4⋊Q8—48no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4#4112yes—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—82no—
223C2×C42C2×D412yes—
224C2×C42#4112yes—
225#107—44no—
226C4⋊C8—44no—
227#103—28no—
228#103—48no—
229#103—44no—
230#103—48no—
231#103—48no—
232#103—48no—
233#103—48no—
234#95—24no—
235#95—24no—
236#95—22no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—82no—
244#95—82no—
245#95—81no—
246#95Q1614yes—
247C24C22×D411yescharacteristic
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4C2×Q1614yes—
349C22×C4C2×Q1612yes—
350C22×C4#13714yes—
351C22×C4#13712yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4111yescharacteristic
357C22×C4H(5,𝔽2)12yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C4H(5,𝔽2)12yes—
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C42−1+411yescharacteristic
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—416no—
387C4⋊C4—416no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—88no—
397C4⋊C4#12114yes—
398C42—22no—
399C42—22no—
400C42—24no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—84no—
406C23C2×#12111yescharacteristic
407C23#53512yes—
408C23D4211yescharacteristic
409C23#54612yes—
410C23#54811yescharacteristic
411C23C22×Q1611yescharacteristic
412C23C2×#13712yes—
413C23C2×#13711yescharacteristic
414C23#57711yescharacteristic
415C23#57712yes—
416C23#57811yescharacteristic
417Q8—42no—
418Q8—44no—
419Q8—44no—
420Q8—44no—
421Q8—44no—
422Q8—42no—
423Q8—44no—
424Q8—82no—
425Q8—81no—
426Q8—82no—
427Q8—81no—
428Q8—82no—
429Q8—82no—
430Q8—82no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4#46214yes—
489C2×C4C2×#12111yescharacteristic
490C2×C4#52511yescharacteristic
491C2×C4#53511yescharacteristic
492C2×C4#53511yescharacteristic
493C2×C4#53511yescharacteristic
494C2×C4#53512yes—
495C2×C4#53711yescharacteristic
496C2×C4#53812yes—
497C2×C4#53912yes—
498C2×C4#54012yes—
499C2×C4#54012yes—
500C2×C4#54612yes—
501C2×C4#54711yescharacteristic
502C2×C4#55112yes—
503C2×C4#55311yescharacteristic
504C2×C4#55312yes—
505C2×C4H(3,𝔽4)12yes—
506C2×C4#56212yes—
507C8—42no—
508C8—81no—
509C8—82no—
510C22#241111yescharacteristic
511C22#241611yescharacteristic
512C22#261911yescharacteristic
513C22#267012yes—
514C22#271112yes—
515C22C2×#46211yescharacteristic
516C22#303911yescharacteristic
517C22#304211yescharacteristic
518C22#304611yescharacteristic
519C22#320912yes—
520C22#321812yes—
521C22#322112yes—
522C22#322512yes—
523C22#326612yes—
524C22#327811yescharacteristic
525C22D4×Q1611yescharacteristic
526C22#329912yes—
527C22#332012yes—
528C22#332012yes—
529C22#332812yes—
530C22#336311yescharacteristic
531C22#337212yes—
532C22#337411yescharacteristic
533C4—24no—
534C4—44no—
535C4—44no—
536C4—44no—
537C4—42no—
538C4—82no—
539C4—84no—
540C4—82no—
541C4—162no—
542C4—161no—
543C4#253511yescharacteristic
544C4#261911yescharacteristic
545C4#263611yescharacteristic
546C4#270612yes—
547C4#275312yes—
548C4#281911yescharacteristic
549C2#1651611yescharacteristic
550C2#2448211yescharacteristic
551C2#2449611yescharacteristic
552C2#2473211yescharacteristic
553C2#2474811yescharacteristic
554C2#2979911yescharacteristic
555C2#2980411yescharacteristic
556C2#3053012yes—
557C2#3054212yes—
558C2#3208412yes—
559C2#3209812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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×#20432
C2×#19812
C2×#19282
#151902
#139332
#112552
#111242
#110772
#110672
#110102