Finite Group Databasebeta

← #470378

Subgroups — #470378

Order 512.

1188 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic
3C2×#1927C211yesmaximal, characteristic
4#15193C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11207C212yesmaximal
9#11067C212yesmaximal
10#11050C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×#125—21no—
13C22×#106C2211yescharacteristic
14C22×#103C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17#2552—24no—
18#2552—24no—
19C2×#398—22no—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#384C2212yes—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2072—24no—
30#1927—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—22no—
42#1445—21no—
43#1444—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#125—28no—
50C2×#125—44no—
51C2×#125—44no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C42—21no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C2×#106—22no—
69C2×#106—42no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—22no—
76C2×#103—42no—
77C2×#103—42no—
78#398—216no—
79#391—24no—
80#391—22no—
81#391—22no—
82#391—42no—
83#389—28no—
84#389—24no—
85#389—24no—
86#389—24no—
87#389—42no—
88#384—24no—
89#384—24no—
90#384—22no—
91#384D414yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4D414yes—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95D412yes—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—22no—
114#336—42no—
115#336—42no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8#4112yes—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8#4112yes—
154#125—432no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4SD1614yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4#4114yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217#106—44no—
218#103—28no—
219#103—48no—
220#103—44no—
221#95—28no—
222#95—24no—
223#95—24no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—42no—
228#95—42no—
229#95—42no—
230#95—81no—
231C24C22×D411yescharacteristic
232C2×Q8—22no—
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4#12012yes—
305C22×C4#12012yes—
306C22×C4C2×SD1614yes—
307C22×C4C2×SD1612yes—
308C22×C4AGL(1,ℤ/8)12yes—
309C22×C4#13714yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4112yes—
315C22×C4H(5,𝔽2)12yes—
316C22×C4H(5,𝔽2)11yescharacteristic
317C22×C42−1+412yes—
318C22×C42−1+412yes—
319C22×C42−1+411yescharacteristic
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—28no—
338C4⋊C4—28no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—416no—
348C4⋊C4—416no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—48no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—84no—
361C4⋊C4—88no—
362C4⋊C4—88no—
363C42—22no—
364C42—22no—
365C42—24no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—48no—
371C23C2×#12011yescharacteristic
372C23#53612yes—
373C23#54612yes—
374C23#54811yescharacteristic
375C23#55011yescharacteristic
376C23C22×SD1611yescharacteristic
377C23C2×AGL(1,ℤ/8)11yescharacteristic
378C23C2×#13712yes—
379C23#57611yescharacteristic
380C23#57711yescharacteristic
381C23#57812yes—
382Q8—42no—
383Q8—42no—
384Q8—44no—
385Q8—81no—
386Q8—81no—
387Q8—82no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4#45014yes—
448C2×C4#52511yescharacteristic
449C2×C4#53211yescharacteristic
450C2×C4#53511yescharacteristic
451C2×C4#53611yescharacteristic
452C2×C4#53611yescharacteristic
453C2×C4#53612yes—
454C2×C4#53711yescharacteristic
455C2×C4#53912yes—
456C2×C4#53912yes—
457C2×C4#53912yes—
458C2×C4#54112yes—
459C2×C4#54611yescharacteristic
460C2×C4#54711yescharacteristic
461C2×C4#55012yes—
462C2×C4#55212yes—
463C2×C4#55212yes—
464C2×C4#56212yes—
465C2×C4#56312yes—
466C8—42no—
467C8—81no—
468C8—82no—
469C22#240711yescharacteristic
470C22#243111yescharacteristic
471C22#261411yescharacteristic
472C22#269012yes—
473C22#269412yes—
474C22C2×#45011yescharacteristic
475C22#300411yescharacteristic
476C22#301111yescharacteristic
477C22#301211yescharacteristic
478C22#322912yes—
479C22#323812yes—
480C22#324212yes—
481C22#324312yes—
482C22#329812yes—
483C22#331412yes—
484C22#331612yes—
485C22#333012yes—
486C22#336211yescharacteristic
487C22#336811yescharacteristic
488C22#337412yes—
489C22#339211yescharacteristic
490C22#339812yes—
491C22#339911yescharacteristic
492C4—24no—
493C4—42no—
494C4—44no—
495C4—44no—
496C4—82no—
497C4—84no—
498C4—82no—
499C4—84no—
500C4—82no—
501C4—161no—
502C4#261111yescharacteristic
503C4#263311yescharacteristic
504C4#266411yescharacteristic
505C4#269412yes—
506C4#272112yes—
507C4#282011yescharacteristic
508C2#1668211yescharacteristic
509C2#2418811yescharacteristic
510C2#2420311yescharacteristic
511C2#2557111yescharacteristic
512C2#2557911yescharacteristic
513C2#2964111yescharacteristic
514C2#2964911yescharacteristic
515C2#3145612yes—
516C2#3147212yes—
517C2#3157212yes—
518C2#3158612yes—
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×#25522
C2×#20722
C2×#19272
#151932
#113732
#113462
#113272
#112072
#110672
#110502
#108062