Finite Group Databasebeta

← #412565

Subgroups — #412565

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2064C212yesmaximal
2C2×#1950C211yesmaximal, characteristic
3#21145C211yesmaximal, characteristic
4#20801C211yesmaximal, characteristic
5#20647C211yesmaximal, characteristic
6C2×#1805C211yesmaximal, characteristic
7C2×C4×#101C211yesmaximal, characteristic
8#11249C212yesmaximal
9#11216C212yesmaximal
10#11145C212yesmaximal
11#10858C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2212yes—
15C2×#428C2212yes—
16C2×#424C2212yes—
17C2×#423C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C2×#419C2212yes—
20C22×#103C2212yes—
21C22×#101C2211yescharacteristic
22C22×#101C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#382C2211yescharacteristic
27#2064—28no—
28#1950—24no—
29#1940—22no—
30#1939—22no—
31#1847—22no—
32#1846—21no—
33#1846—21no—
34#1805—22no—
35#1805—22no—
36#1805—21no—
37#1805—21no—
38#1803—22no—
39C4×#101—24no—
40C2×#343C2212yes—
41C2×#343C2211yescharacteristic
42C2×#343C2211yescharacteristic
43C2×#343C2212yes—
44C2×#341C2212yes—
45#1736C2212yes—
46C2×#340C2212yes—
47#1729C2212yes—
48C23×Q8C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C23×C8C2312yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8Q814yes—
63C2×#117C2312yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C2×#108—24no—
70C2×#108—24no—
71C2×#108—24no—
72#428—24no—
73#428—24no—
74#428—42no—
75#428—42no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#106—22no—
79C2×#106—22no—
80#424—24no—
81#424—42no—
82#423—22no—
83#423—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8D412yes—
90C2×C4⋊C8D412yes—
91#419—24no—
92#419—42no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—42no—
98C2×#101—22no—
99C2×#101—22no—
100C2×#101—22no—
101C2×#101—22no—
102C2×#101D414yes—
103C2×#101Q814yes—
104C4×M16—22no—
105C4×M16—22no—
106C4×M16—22no—
107C4×M16D414yes—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112#384D412yes—
113#382—21no—
114#382—21no—
115#382—21no—
116#382—21no—
117#382D411yescharacteristic
118#382D411yescharacteristic
119#380—21no—
120#380—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C2×#95—21no—
124C2×#95—21no—
125#343—24no—
126#343—22no—
127#343—22no—
128#343—42no—
129#341—24no—
130#340—24no—
131#340—22no—
132#340—22no—
133#336—24no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8C2×D412yes—
146C22×Q8C2×Q812yes—
147C22×Q8C2×Q812yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4#4111yescharacteristic
152C23×C4#4111yescharacteristic
153C23×C4C2411yescharacteristic
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16C2×D412yes—
177C2×M16C2×D412yes—
178C2×M16C2×D412yes—
179C2×M16C2×D412yes—
180C2×M16C2×Q812yes—
181C2×M16C2×Q812yes—
182C2×M16#4112yes—
183C2×M16#4112yes—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8C2×D414yes—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×Q812yes—
228C2×C22⋊C4—22no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42—42no—
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#108—48no—
243#107—44no—
244#107—44no—
245#106—44no—
246#106—44no—
247C4⋊C8—24no—
248C4⋊C8—24no—
249C4⋊C8—48no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252#103—44no—
253#103—44no—
254#103—48no—
255#101—28no—
256#101—28no—
257#97—24no—
258#97—42no—
259C4×C8—24no—
260C4×C8—42no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269C24C2×#4111yescharacteristic
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—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8C4⋊Q814yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4C22⋊Q812yes—
323C22×C4C22⋊Q812yes—
324C22×C4#12612yes—
325C22×C4#12612yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×Q811yescharacteristic
331C22×C4C22×Q811yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C42−1+411yescharacteristic
339C22×C42−1+411yescharacteristic
340M16—22no—
341M16—22no—
342M16—22no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—41no—
351M16—41no—
352M16—41no—
353M16—41no—
354M16—42no—
355M16—42no—
356M16—42no—
357M16—42no—
358M16#12114yes—
359M16C22⋊Q814yes—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—22no—
369C2×C8—24no—
370C2×C8—22no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—44no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—82no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C42—42no—
401C42—42no—
402C42—44no—
403C42—44no—
404C42#12112yes—
405C42#12112yes—
406C42#12112yes—
407C42#12112yes—
408C42C22⋊Q812yes—
409C42C22⋊Q812yes—
410C42C22⋊Q812yes—
411C42C22⋊Q812yes—
412C23—22no—
413C23—22no—
414C23C2×#12111yescharacteristic
415C23C2×C22⋊Q811yescharacteristic
416C23C2×#12611yescharacteristic
417C23#53211yescharacteristic
418C23#53311yescharacteristic
419C23#54011yescharacteristic
420C23#54111yescharacteristic
421Q8—24no—
422Q8—24no—
423Q8—42no—
424Q8—42no—
425Q8—42no—
426Q8—42no—
427Q8—42no—
428Q8—42no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
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—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—162no—
458C2×C4—162no—
459C2×C4#39612yes—
460C2×C4#39612yes—
461C2×C4#39612yes—
462C2×C4#39612yes—
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×#12111yescharacteristic
465C2×C4C2×#12111yescharacteristic
466C2×C4C2×C22⋊Q811yescharacteristic
467C2×C4C2×C22⋊Q811yescharacteristic
468C2×C4C2×C22⋊Q811yescharacteristic
469C2×C4C2×C4⋊Q811yescharacteristic
470C2×C4#53311yescharacteristic
471C2×C4#53711yescharacteristic
472C2×C4#53711yescharacteristic
473C2×C4#54411yescharacteristic
474C2×C4#54411yescharacteristic
475C2×C4D4211yescharacteristic
476C2×C4D4211yescharacteristic
477C2×C4#54611yescharacteristic
478C2×C4#54711yescharacteristic
479C2×C4#54711yescharacteristic
480C2×C4#54711yescharacteristic
481C2×C4#54811yescharacteristic
482C2×C4D4×Q811yescharacteristic
483C2×C4D4×Q811yescharacteristic
484C2×C4#55011yescharacteristic
485C2×C4#55211yescharacteristic
486C2×C4#55311yescharacteristic
487C2×C4#55411yescharacteristic
488C2×C4#55411yescharacteristic
489C2×C4#55511yescharacteristic
490C2×C4#55611yescharacteristic
491C8—24no—
492C8—24no—
493C8—44no—
494C8—42no—
495C8—42no—
496C8—82no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22C2×#39611yescharacteristic
506C22#242811yescharacteristic
507C22#242911yescharacteristic
508C22#249611yescharacteristic
509C22#249711yescharacteristic
510C22#262911yescharacteristic
511C22#285711yescharacteristic
512C22#306712yes—
513C22#306912yes—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—44no—
519C4—82no—
520C4—82no—
521C4—82no—
522C4—162no—
523C4#241611yescharacteristic
524C4#241711yescharacteristic
525C4#243911yescharacteristic
526C4#244011yescharacteristic
527C4#244011yescharacteristic
528C4#244111yescharacteristic
529C4#253711yescharacteristic
530C4#253811yescharacteristic
531C2—21no—
532C2—21no—
533C2—21no—
534C2—21no—
535C2#1513111yescharacteristic
536C2#2360111yescharacteristic
537C2#2360311yescharacteristic
538C2#2540912yes—
539C2#2549412yes—
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
C2×#20642
C2×#19502
#211452
#208012
#206472
C2×#18052
C2×C4×#1012
#112492
#112162
#111452
#108582