Finite Group Databasebeta

← #470393

Subgroups — #470393

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic
3C2×#1921C211yesmaximal, characteristic
4#15194C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11292C212yesmaximal
9#11268C212yesmaximal
10#11208C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13#2571—24no—
14#2571—24no—
15C2×#400—22no—
16C2×#389—21no—
17C2×#389C2212yes—
18C2×#389C2211yescharacteristic
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382—21no—
22C2×#382C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#2099—24no—
26#1921—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1445—21no—
41#1444—21no—
42#1441—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C42—21no—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—22no—
67#400—216no—
68#389—28no—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—22no—
73#389—22no—
74#389—42no—
75#384—22no—
76#384D414yes—
77#382—28no—
78#382—24no—
79#382—24no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4D414yes—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95D412yes—
95#340—24no—
96#340—24no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—42no—
104#336—24no—
105#336—22no—
106#336—22no—
107#336—42no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8#4112yes—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D414yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4#4114yes—
185C2×C4⋊C4#4112yes—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—44no—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198#107—28no—
199#95—28no—
200#95—28no—
201#95—28no—
202#95—24no—
203#95—24no—
204#95—44no—
205#95—42no—
206#95—42no—
207#95—81no—
208#95—82no—
209#95—82no—
210C24C22×D411yescharacteristic
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4#12012yes—
266C22×C4#12012yes—
267C22×C4#13512yes—
268C22×C4#13712yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4112yes—
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C42−1+412yes—
276C22×C42−1+412yes—
277C22×C42−1+412yes—
278C22×C42−1+411yescharacteristic
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—48no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—88no—
324C4⋊C4—88no—
325C42—22no—
326C42—22no—
327C42—24no—
328C42—48no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—42no—
333C23C2×#12011yescharacteristic
334C23#53612yes—
335C23#54611yescharacteristic
336C23#54711yescharacteristic
337C23#54812yes—
338C23C2×#13511yescharacteristic
339C23C2×#13711yescharacteristic
340C23#57512yes—
341C23#57612yes—
342C23#57711yescharacteristic
343C23#57811yescharacteristic
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4#45214yes—
402C2×C4#52511yescharacteristic
403C2×C4#53211yescharacteristic
404C2×C4#53611yescharacteristic
405C2×C4#53611yescharacteristic
406C2×C4#53611yescharacteristic
407C2×C4#53612yes—
408C2×C4#53711yescharacteristic
409C2×C4#53912yes—
410C2×C4#54112yes—
411C2×C4#54112yes—
412C2×C4#54112yes—
413C2×C4#54712yes—
414C2×C4#54811yescharacteristic
415C2×C4#55011yescharacteristic
416C2×C4#55212yes—
417C2×C4#55212yes—
418C2×C4#56312yes—
419C2×C4#56412yes—
420C8—42no—
421C8—81no—
422C8—82no—
423C22#240711yescharacteristic
424C22#243211yescharacteristic
425C22#261411yescharacteristic
426C22#269012yes—
427C22#271912yes—
428C22C2×#45211yescharacteristic
429C22#300511yescharacteristic
430C22#300811yescharacteristic
431C22#301311yescharacteristic
432C22#323212yes—
433C22#323612yes—
434C22#324112yes—
435C22#324312yes—
436C22#330411yescharacteristic
437C22#331912yes—
438C22#332811yescharacteristic
439C22#334211yescharacteristic
440C22#334712yes—
441C22#335611yescharacteristic
442C22#336112yes—
443C22#337512yes—
444C22#337612yes—
445C22#337712yes—
446C4—24no—
447C4—42no—
448C4—84no—
449C4—82no—
450C4—82no—
451C4—82no—
452C4—82no—
453C4—82no—
454C4—84no—
455C4—161no—
456C4#261111yescharacteristic
457C4#263411yescharacteristic
458C4#266411yescharacteristic
459C4#271912yes—
460C4#272112yes—
461C4#282211yescharacteristic
462C2#1668311yescharacteristic
463C2#2412311yescharacteristic
464C2#2412911yescharacteristic
465C2#2564811yescharacteristic
466C2#2565611yescharacteristic
467C2#2964311yescharacteristic
468C2#2964611yescharacteristic
469C2#3146712yes—
470C2#3148512yes—
471C2#3232312yes—
472C2#3234112yes—
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×#20992
C2×#19212
#151942
#113572
#113462
#113282
#112922
#112682
#112082
#108072