Finite Group Databasebeta

← #416262

Subgroups — #416262

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21170C211yesmaximal, characteristic
2#21143C211yesmaximal, characteristic
3C2×#1837C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1801C211yesmaximal, characteristic
6C2×C4×#103C211yesmaximal, characteristic
7#11303C212yesmaximal
8#11216C212yesmaximal
9#11145C212yesmaximal
10#11062C212yesmaximal
11#10847C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#428C2212yes—
14C2×#424C2212yes—
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C2×#419C2212yes—
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#101C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1949—22no—
26#1948—21no—
27#1947—21no—
28#1939—22no—
29#1939—22no—
30#1837—24no—
31#1805—22no—
32#1803—21no—
33#1803—21no—
34#1801—24no—
35C4×#103—22no—
36C4×#103—22no—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#341C2212yes—
41C2×#341C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46#1729C2212yes—
47C23×Q8C2311yescharacteristic
48C22×M16C2312yes—
49C22×M16C2312yes—
50C22×M16C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4×Q8—22no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8Q814yes—
61C2×#117C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#428—24no—
70#428—42no—
71C2×#107—22no—
72C2×#106—22no—
73#424—42no—
74#424—42no—
75#423—22no—
76#423—22no—
77#423—22no—
78#423—22no—
79#423—22no—
80#423D414yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88#419—24no—
89#419—42no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#101—24no—
99C2×#101—24no—
100C2×#101—42no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8D412yes—
104C2×C4×C8D412yes—
105#384D411yescharacteristic
106#384D411yescharacteristic
107#382D412yes—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C2×#95—21no—
117#343—24no—
118#343—24no—
119#343—42no—
120#341—24no—
121#341—24no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#336—24no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8C2×Q812yes—
141C22×Q8#4112yes—
142C22×Q8#4112yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4#4111yescharacteristic
148C23×C4#4111yescharacteristic
149C23×C4C2411yescharacteristic
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16C2×D414yes—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—42no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—42no—
193#117—22no—
194#117—22no—
195#117—22no—
196#117—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×Q812yes—
223C2×C4⋊C4#4112yes—
224C2×C4⋊C4#4112yes—
225C2×C22⋊C4—22no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42—42no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42—82no—
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×Q812yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242#107—44no—
243#106—44no—
244C4⋊C8—24no—
245C4⋊C8—24no—
246C4⋊C8—48no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251#103—28no—
252#103—28no—
253#103—44no—
254#103—44no—
255#101—48no—
256C4×C8—24no—
257C4×C8—24no—
258#95—22no—
259#95—22no—
260#95—41no—
261#95—41no—
262C24C2×#4111yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8#12514yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—81no—
314C22×C4—81no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4C22⋊Q812yes—
324C22×C4C22⋊Q812yes—
325C22×C4#12612yes—
326C22×C4#12612yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×Q811yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C42−1+411yescharacteristic
340C22×C42−1+411yescharacteristic
341M16—24no—
342M16—24no—
343M16—42no—
344M16—42no—
345M16—42no—
346M16—42no—
347M16—42no—
348M16—42no—
349M16—42no—
350M16—42no—
351M16—42no—
352M16—42no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C42—42no—
401C42—42no—
402C42—44no—
403C42—44no—
404C42—82no—
405C42—82no—
406C42#12112yes—
407C42#12112yes—
408C42#12112yes—
409C42#12112yes—
410C42C22⋊Q812yes—
411C42C22⋊Q812yes—
412C42C22⋊Q812yes—
413C42C22⋊Q812yes—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23C2×#12111yescharacteristic
419C23C2×C22⋊Q811yescharacteristic
420C23C2×#12611yescharacteristic
421C23#55211yescharacteristic
422C23#55211yescharacteristic
423C23#55311yescharacteristic
424C23#55311yescharacteristic
425Q8—24no—
426Q8—24no—
427Q8—42no—
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—42no—
432Q8—42no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#39612yes—
472C2×C4#39612yes—
473C2×C4#39612yes—
474C2×C4#39612yes—
475C2×C4#47112yes—
476C2×C4#47312yes—
477C2×C4C2×#12111yescharacteristic
478C2×C4C2×#12111yescharacteristic
479C2×C4C2×C22⋊Q811yescharacteristic
480C2×C4C2×C22⋊Q811yescharacteristic
481C2×C4#52511yescharacteristic
482C2×C4#52511yescharacteristic
483C2×C4C2×#12511yescharacteristic
484C2×C4#52911yescharacteristic
485C2×C4#52911yescharacteristic
486C2×C4#53311yescharacteristic
487C2×C4#53511yescharacteristic
488C2×C4#53611yescharacteristic
489C2×C4#54211yescharacteristic
490C2×C4#54211yescharacteristic
491C2×C4#54411yescharacteristic
492C2×C4#54411yescharacteristic
493C2×C4#54611yescharacteristic
494C2×C4#54611yescharacteristic
495C2×C4#54611yescharacteristic
496C2×C4#54711yescharacteristic
497C2×C4#54811yescharacteristic
498C2×C4#54811yescharacteristic
499C2×C4#54811yescharacteristic
500C2×C4#55011yescharacteristic
501C2×C4#55211yescharacteristic
502C2×C4#55311yescharacteristic
503C2×C4#55511yescharacteristic
504C2×C4#55611yescharacteristic
505C8—42no—
506C8—42no—
507C8—44no—
508C8—82no—
509C8—82no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22C2×#39611yescharacteristic
523C22#244111yescharacteristic
524C22#244211yescharacteristic
525C22#249411yescharacteristic
526C22#249511yescharacteristic
527C22#253511yescharacteristic
528C22#253611yescharacteristic
529C22C2×#47111yescharacteristic
530C22C2×#47311yescharacteristic
531C22#306711yescharacteristic
532C22#306711yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—44no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4#206414yes—
543C4#241411yescharacteristic
544C4#241511yescharacteristic
545C4#242611yescharacteristic
546C4#242711yescharacteristic
547C4#243911yescharacteristic
548C4#244211yescharacteristic
549C4#262811yescharacteristic
550C4#285111yescharacteristic
551C2—21no—
552C2—21no—
553C2—21no—
554C2—21no—
555C2#1522011yescharacteristic
556C2C2×#206411yescharacteristic
557C2#2360111yescharacteristic
558C2#2817511yescharacteristic
559C2#2817611yescharacteristic
560C2#2821811yescharacteristic
561C2#2821911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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
#211702
#211432
C2×#18372
#206472
C2×#18012
C2×C4×#1032
#113032
#112162
#111452
#110622
#108472