Finite Group Databasebeta

← #416114

Subgroups — #416114

Order 512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic
3C2×#1776C211yesmaximal, characteristic
4#14150C211yesmaximal, characteristic
5#11327C212yesmaximal
6#11325C212yesmaximal
7#11220C212yesmaximal
8#11208C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10843C211yesmaximal, characteristic
11#10835C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2571—24no—
18#2571—24no—
19C2×#400—22no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#382—21no—
25C2×#382C2212yes—
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1948—24no—
31#1776—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1444—21no—
42#1442—21no—
43#1436—22no—
44C4⋊#95C2212yes—
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C42—21no—
61C22×C42—21no—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—22no—
67C2×#106—22no—
68C2×#106—22no—
69C2×#106—22no—
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—42no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8Q814yes—
76C2×#97—22no—
77C2×#97—22no—
78C2×#97Q814yes—
79#400—216no—
80#389—28no—
81#389—24no—
82#389—24no—
83#389—24no—
84#389—42no—
85#384—22no—
86#384D414yes—
87#382—28no—
88#382—24no—
89#382—24no—
90#382—22no—
91#382Q814yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4D414yes—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95D412yes—
109#340—24no—
110#340—24no—
111#340—22no—
112#340—22no—
113#340—42no—
114#340—42no—
115#340—42no—
116#336—24no—
117#336—42no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8C2×Q812yes—
146C22×C8C2×Q814yes—
147C22×C8C2×Q812yes—
148C22×C8#4112yes—
149C22×C8#4112yes—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×Q814yes—
198C2×C4⋊C4#4112yes—
199C2×C4⋊C4#4112yes—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—44no—
208C2×C42—42no—
209C2×C42—82no—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×Q812yes—
213C2×C42C2×Q812yes—
214C2×C42C2×Q812yes—
215C2×C42#4112yes—
216#107—28no—
217#106—28no—
218#106—48no—
219C4⋊C8—28no—
220#97—28no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—44no—
229#95—44no—
230#95—42no—
231#95—82no—
232C24C22×D411yescharacteristic
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4#12012yes—
286C22×C4#12012yes—
287C22×C4C22⋊Q812yes—
288C22×C4C22⋊Q812yes—
289C22×C4C4⋊Q812yes—
290C22×C4C4⋊Q812yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×Q811yescharacteristic
294C22×C4C22×Q811yescharacteristic
295C22×C4C22×Q811yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4H(5,𝔽2)11yescharacteristic
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C42−1+411yescharacteristic
304C22×C42−1+411yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8#12514yes—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—84no—
354C4⋊C4—88no—
355C4⋊C4—88no—
356C42—48no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—84no—
362C42C22⋊Q814yes—
363C42C22⋊Q814yes—
364C42C4⋊Q814yes—
365C42C4⋊Q814yes—
366C23C2×#12011yescharacteristic
367C23C2×C22⋊Q811yescharacteristic
368C23C2×C4⋊Q811yescharacteristic
369C23#54611yescharacteristic
370C23#54711yescharacteristic
371C23#54811yescharacteristic
372C23#55011yescharacteristic
373C23#57512yes—
374C23#57512yes—
375C23#57712yes—
376C23#57712yes—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#39514yes—
434C2×C4#39514yes—
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4C2×C22⋊Q811yescharacteristic
437C2×C4#52511yescharacteristic
438C2×C4#52511yescharacteristic
439C2×C4C2×#12511yescharacteristic
440C2×C4#52911yescharacteristic
441C2×C4#52911yescharacteristic
442C2×C4C2×C4⋊Q811yescharacteristic
443C2×C4C2×C4⋊Q811yescharacteristic
444C2×C4#53311yescharacteristic
445C2×C4#53711yescharacteristic
446C2×C4#53711yescharacteristic
447C2×C4#54011yescharacteristic
448C2×C4#54111yescharacteristic
449C2×C4#54411yescharacteristic
450C2×C4#54411yescharacteristic
451C2×C4#54611yescharacteristic
452C2×C4#54711yescharacteristic
453C2×C4#54811yescharacteristic
454C2×C4#55011yescharacteristic
455C2×C4#55411yescharacteristic
456C2×C4#55411yescharacteristic
457C2×C4#55411yescharacteristic
458C2×C4#55411yescharacteristic
459C2×C4#55711yescharacteristic
460C2×C4#55711yescharacteristic
461C2×C4#55711yescharacteristic
462C2×C4#55711yescharacteristic
463C8—24no—
464C8—42no—
465C8—44no—
466C22C2×#39511yescharacteristic
467C22#240711yescharacteristic
468C22#240711yescharacteristic
469C22#245311yescharacteristic
470C22#245411yescharacteristic
471C22#255011yescharacteristic
472C22#255211yescharacteristic
473C22#300511yescharacteristic
474C22#300511yescharacteristic
475C22#300911yescharacteristic
476C22#301011yescharacteristic
477C22#308012yes—
478C22#308412yes—
479C22#316312yes—
480C22#316612yes—
481C22#331112yes—
482C22#332112yes—
483C22#334812yes—
484C22#335012yes—
485C22#337112yes—
486C22#337612yes—
487C22#340112yes—
488C22#340112yes—
489C4—24no—
490C4—24no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4—82no—
496C4—82no—
497C4—84no—
498C4—84no—
499C4#245311yescharacteristic
500C4#245411yescharacteristic
501C4#255011yescharacteristic
502C4#255211yescharacteristic
503C4#258411yescharacteristic
504C4#258411yescharacteristic
505C4#262911yescharacteristic
506C4#285111yescharacteristic
507C2#1521811yescharacteristic
508C2#2355311yescharacteristic
509C2#2355611yescharacteristic
510C2#2413111yescharacteristic
511C2#2413311yescharacteristic
512C2#2419711yescharacteristic
513C2#2420211yescharacteristic
514C2#2652512yes—
515C2#2654312yes—
516C2#2880812yes—
517C2#2884012yes—
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×#25712
C2×#19482
C2×#17762
#141502
#113272
#113252
#112202
#112082
#108502
#108432
#108352