Finite Group Databasebeta

← #411535

Subgroups — #411535

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1920C212yesmaximal
2#14712C211yesmaximal, characteristic
3#13755C211yesmaximal, characteristic
4#11357C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11323C211yesmaximal, characteristic
7#11315C212yesmaximal
8#11267C211yesmaximal, characteristic
9#11257C212yesmaximal
10#10885C212yesmaximal
11#10795C211yesmaximal, characteristic
12C22×#106C2212yes—
13C2×#389—21no—
14C2×#389—21no—
15C2×#389C2212yes—
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×#376—21no—
21C2×C43C2211yescharacteristic
22#1920—28no—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C4⋊#95—21no—
35C4⋊#95—21no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42#1434—21no—
43C4×#95—21no—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#106—24no—
67C2×#106—24no—
68C2×#106—24no—
69#389—24no—
70#389—24no—
71#389—44no—
72#389—44no—
73#384—24no—
74#384—22no—
75#384—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4D414yes—
80#376—44no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C43—22no—
95C43—22no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336—24no—
104#336—24no—
105#336—22no—
106#336—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8#4114yes—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4#4112yes—
175C2×C4⋊C4#4112yes—
176C2×C4⋊C4#4114yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42SD1614yes—
205C2×C42C2×D412yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209#106—216no—
210#95—24no—
211#95—24no—
212#95—42no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—42no—
217#95—42no—
218#95—42no—
219#95—42no—
220#95—42no—
221#95—42no—
222#95—44no—
223C24C22×D411yescharacteristic
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12312yes—
299C22×C4#12312yes—
300C22×C4C2×SD1612yes—
301C22×C4C2×SD1612yes—
302C22×C4C2×SD1612yes—
303C22×C4#13512yes—
304C22×C4AGL(1,ℤ/8)12yes—
305C22×C4AGL(1,ℤ/8)12yes—
306C22×C4#13712yes—
307C22×C4#13712yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C2×#4112yes—
310C22×C4C2×#4112yes—
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C42−1+412yes—
317C22×C42−1+412yes—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—48no—
339C4⋊C4—48no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—84no—
351C4⋊C4—88no—
352C4⋊C4—84no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—24no—
358C42—24no—
359C42—28no—
360C42—44no—
361C42—44no—
362C42—48no—
363C42—42no—
364C42—42no—
365C42—48no—
366C42—44no—
367C23C2×#12111yescharacteristic
368C23C2×#12311yescharacteristic
369C23#52511yescharacteristic
370C23#52511yescharacteristic
371C23#53211yescharacteristic
372C23#53611yescharacteristic
373C23#53611yescharacteristic
374C23C22×SD1611yescharacteristic
375C23C2×#13511yescharacteristic
376C23C2×AGL(1,ℤ/8)11yescharacteristic
377C23C2×AGL(1,ℤ/8)11yescharacteristic
378C23C2×#13711yescharacteristic
379C23C2×#13711yescharacteristic
380C23#57511yescharacteristic
381C23#57511yescharacteristic
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4#46114yes—
464C2×C4#48214yes—
465C2×C4#48314yes—
466C2×C4#52512yes—
467C2×C4#52911yescharacteristic
468C2×C4#52912yes—
469C2×C4#52911yescharacteristic
470C2×C4#53211yescharacteristic
471C2×C4#53211yescharacteristic
472C2×C4#53612yes—
473C2×C4#54112yes—
474C2×C4#55212yes—
475C2×C4#55212yes—
476C2×C4#55212yes—
477C2×C4#55212yes—
478C2×C4#55212yes—
479C2×C4#55212yes—
480C2×C4#55612yes—
481C2×C4#55612yes—
482C8—44no—
483C8—82no—
484C22#239211yescharacteristic
485C22#248411yescharacteristic
486C22#249911yescharacteristic
487C22#259911yescharacteristic
488C22#261111yescharacteristic
489C22#262111yescharacteristic
490C22#285411yescharacteristic
491C22C2×#46111yescharacteristic
492C22#303511yescharacteristic
493C22#303711yescharacteristic
494C22#303811yescharacteristic
495C22C2×#48311yescharacteristic
496C22C2×#48211yescharacteristic
497C22#309411yescharacteristic
498C22#309511yescharacteristic
499C22#310111yescharacteristic
500C22#310311yescharacteristic
501C22#310611yescharacteristic
502C22#310811yescharacteristic
503C22#311011yescharacteristic
504C22#311311yescharacteristic
505C22#311411yescharacteristic
506C22#311511yescharacteristic
507C22#316711yescharacteristic
508C22#317211yescharacteristic
509C22#318011yescharacteristic
510C22#318211yescharacteristic
511C22#322711yescharacteristic
512C22#322711yescharacteristic
513C22#323111yescharacteristic
514C22#323211yescharacteristic
515C22#323611yescharacteristic
516C22#323811yescharacteristic
517C22#323811yescharacteristic
518C22#324011yescharacteristic
519C4—24no—
520C4—24no—
521C4—24no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—82no—
526C4—84no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—162no—
531C4#252812yes—
532C4#253312yes—
533C4#260412yes—
534C4#261112yes—
535C2#1512211yescharacteristic
536C2#2367711yescharacteristic
537C2#2368411yescharacteristic
538C2#2761111yescharacteristic
539C2#2761611yescharacteristic
540C2#2787511yescharacteristic
541C2#2788311yescharacteristic
542C2#2939211yescharacteristic
543C2#2939511yescharacteristic
544C2#2957411yescharacteristic
545C2#2958111yescharacteristic
546C2#2986811yescharacteristic
547C2#2987111yescharacteristic
548C2#3310811yescharacteristic
549C2#3311311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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×#19202
#147122
#137552
#113572
#113232
#113152
#112672
#112572
#108852
#107952