Finite Group Databasebeta

← #470340

Subgroups — #470340

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic
3C2×#1927C211yesmaximal, characteristic
4#15192C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11325C212yesmaximal
8#11219C212yesmaximal
9#11067C212yesmaximal
10#11052C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2211yescharacteristic
14C22×#103C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17#2571—24no—
18#2571—24no—
19C2×#400—22no—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#382—21no—
26C2×#382C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2095—24no—
30#1927—24no—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1445—21no—
41#1443—22no—
42#1442—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
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×C42—21no—
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C2×#107—24no—
63C2×#107—24no—
64C2×#107—42no—
65C2×#106—22no—
66C2×#106—42no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—22no—
73C2×#103—42no—
74C2×#103—42no—
75#400—216no—
76#391—24no—
77#391—22no—
78#391—22no—
79#391—42no—
80#389—28no—
81#389—24no—
82#389—24no—
83#384—22no—
84#384D414yes—
85#382—28no—
86#382—24no—
87#382—24no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4D414yes—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95D412yes—
105#340—24no—
106#340—22no—
107#340—42no—
108#340—42no—
109#340—42no—
110#336—24no—
111#336—22no—
112#336—22no—
113#336—42no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8#4112yes—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8#4112yes—
151C22×C8#4114yes—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—81no—
185C2×C4⋊C4—81no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D414yes—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—24no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204#107—48no—
205#106—44no—
206#103—28no—
207#103—48no—
208#103—44no—
209#95—28no—
210#95—28no—
211#95—28no—
212#95—24no—
213#95—24no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—42no—
218#95—42no—
219#95—42no—
220#95—81no—
221#95—82no—
222C24C22×D411yescharacteristic
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—24no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
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—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4#12012yes—
298C22×C4#12012yes—
299C22×C4#13512yes—
300C22×C4#13712yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4112yes—
306C22×C4H(5,𝔽2)12yes—
307C22×C4H(5,𝔽2)11yescharacteristic
308C22×C42−1+412yes—
309C22×C42−1+412yes—
310C22×C42−1+411yescharacteristic
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—44no—
335C4⋊C4—48no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—44no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—84no—
350C4⋊C4—88no—
351C42—22no—
352C42—22no—
353C42—24no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—48no—
359C23C2×#12011yescharacteristic
360C23#53612yes—
361C23#54611yescharacteristic
362C23#54711yescharacteristic
363C23#55012yes—
364C23C2×#13511yescharacteristic
365C23C2×#13711yescharacteristic
366C23#57512yes—
367C23#57611yescharacteristic
368C23#57711yescharacteristic
369C23#57812yes—
370Q8—42no—
371Q8—42no—
372Q8—44no—
373Q8—81no—
374Q8—81no—
375Q8—82no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4#45214yes—
436C2×C4#52511yescharacteristic
437C2×C4#53211yescharacteristic
438C2×C4#53511yescharacteristic
439C2×C4#53611yescharacteristic
440C2×C4#53611yescharacteristic
441C2×C4#53612yes—
442C2×C4#53711yescharacteristic
443C2×C4#53912yes—
444C2×C4#53912yes—
445C2×C4#53912yes—
446C2×C4#54112yes—
447C2×C4#54612yes—
448C2×C4#54811yescharacteristic
449C2×C4#55011yescharacteristic
450C2×C4#55212yes—
451C2×C4#55212yes—
452C2×C4#56212yes—
453C2×C4#56312yes—
454C8—42no—
455C8—44no—
456C8—81no—
457C22#240711yescharacteristic
458C22#243111yescharacteristic
459C22#261411yescharacteristic
460C22#269412yes—
461C22#272112yes—
462C22C2×#45211yescharacteristic
463C22#300511yescharacteristic
464C22#301111yescharacteristic
465C22#301211yescharacteristic
466C22#323212yes—
467C22#323612yes—
468C22#324212yes—
469C22#324312yes—
470C22#330511yescharacteristic
471C22#331312yes—
472C22#332911yescharacteristic
473C22#334011yescharacteristic
474C22#334912yes—
475C22#335711yescharacteristic
476C22#339312yes—
477C22#339812yes—
478C22#340112yes—
479C22#340312yes—
480C4—24no—
481C4—42no—
482C4—82no—
483C4—84no—
484C4—82no—
485C4—82no—
486C4—82no—
487C4—82no—
488C4—161no—
489C4—162no—
490C4#261111yescharacteristic
491C4#263311yescharacteristic
492C4#266411yescharacteristic
493C4#269012yes—
494C4#269412yes—
495C4#282011yescharacteristic
496C2#1668211yescharacteristic
497C2#2412111yescharacteristic
498C2#2413511yescharacteristic
499C2#2563211yescharacteristic
500C2#2563911yescharacteristic
501C2#2964311yescharacteristic
502C2#2964911yescharacteristic
503C2#3157912yes—
504C2#3159512yes—
505C2#3238312yes—
506C2#3239912yes—
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×#20952
C2×#19272
#151922
#113572
#113462
#113252
#112192
#110672
#110522
#108062