Finite Group Databasebeta

← #470391

Subgroups — #470391

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic
3C2×#1921C211yesmaximal, characteristic
4#15194C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11293C212yesmaximal
9#11267C212yesmaximal
10#11207C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×#125—21no—
13C22×#107C2211yescharacteristic
14#2552—24no—
15#2552—24no—
16C2×#398—22no—
17C2×#389—21no—
18C2×#389C2212yes—
19C2×#389C2211yescharacteristic
20C2×#389C2212yes—
21C2×#384C2212yes—
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#2100—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
48C2×#125—28no—
49C2×#125—48no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C42—21no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—22no—
71#398—216no—
72#389—28no—
73#389—24no—
74#389—24no—
75#389—24no—
76#389—22no—
77#389—22no—
78#389—42no—
79#384—24no—
80#384—24no—
81#384—22no—
82#384D414yes—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4D414yes—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95D412yes—
96#340—24no—
97#340—24no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#336—24no—
106#336—22no—
107#336—22no—
108#336—42no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8#4112yes—
137#125—432no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
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—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—81no—
183C2×C4⋊C4—81no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4Q1614yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4#4114yes—
193C2×C4⋊C4#4112yes—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—44no—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206#107—28no—
207#95—28no—
208#95—24no—
209#95—24no—
210#95—44no—
211#95—42no—
212#95—42no—
213#95—81no—
214#95—82no—
215#95—82no—
216C24C22×D411yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
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—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4#12012yes—
270C22×C4#12012yes—
271C22×C4C2×Q1614yes—
272C22×C4C2×Q1612yes—
273C22×C4AGL(1,ℤ/8)14yes—
274C22×C4#13712yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4112yes—
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C42−1+412yes—
282C22×C42−1+412yes—
283C22×C42−1+412yes—
284C22×C42−1+411yescharacteristic
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—48no—
315C4⋊C4—416no—
316C4⋊C4—416no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—84no—
330C4⋊C4—88no—
331C4⋊C4—84no—
332C4⋊C4—88no—
333C42—22no—
334C42—22no—
335C42—24no—
336C42—48no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C23C2×#12011yescharacteristic
342C23#53612yes—
343C23#54611yescharacteristic
344C23#54711yescharacteristic
345C23#54812yes—
346C23C22×Q1611yescharacteristic
347C23C2×AGL(1,ℤ/8)12yes—
348C23C2×#13711yescharacteristic
349C23#57611yescharacteristic
350C23#57712yes—
351C23#57711yescharacteristic
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—81no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4#45114yes—
410C2×C4#52511yescharacteristic
411C2×C4#53211yescharacteristic
412C2×C4#53611yescharacteristic
413C2×C4#53611yescharacteristic
414C2×C4#53611yescharacteristic
415C2×C4#53612yes—
416C2×C4#53711yescharacteristic
417C2×C4#53912yes—
418C2×C4#54112yes—
419C2×C4#54112yes—
420C2×C4#54112yes—
421C2×C4#54712yes—
422C2×C4#54811yescharacteristic
423C2×C4#55011yescharacteristic
424C2×C4#55212yes—
425C2×C4#55212yes—
426C2×C4#56312yes—
427C2×C4#56412yes—
428C8—42no—
429C8—81no—
430C8—82no—
431C22#240711yescharacteristic
432C22#243211yescharacteristic
433C22#261411yescharacteristic
434C22#269012yes—
435C22#271912yes—
436C22C2×#45111yescharacteristic
437C22#300311yescharacteristic
438C22#301011yescharacteristic
439C22#301211yescharacteristic
440C22#323012yes—
441C22#323912yes—
442C22#324112yes—
443C22#324412yes—
444C22#329911yescharacteristic
445C22#330612yes—
446C22#333311yescharacteristic
447C22#333811yescharacteristic
448C22#334512yes—
449C22#335611yescharacteristic
450C22#336312yes—
451C22#336612yes—
452C22#336812yes—
453C22#337212yes—
454C4—24no—
455C4—44no—
456C4—44no—
457C4—42no—
458C4—82no—
459C4—84no—
460C4—82no—
461C4—82no—
462C4—84no—
463C4—161no—
464C4#261111yescharacteristic
465C4#263411yescharacteristic
466C4#266411yescharacteristic
467C4#271912yes—
468C4#272112yes—
469C4#282211yescharacteristic
470C2#1668311yescharacteristic
471C2#2412211yescharacteristic
472C2#2412811yescharacteristic
473C2#2565111yescharacteristic
474C2#2565511yescharacteristic
475C2#2964211yescharacteristic
476C2#2964711yescharacteristic
477C2#3146012yes—
478C2#3147412yes—
479C2#3231612yes—
480C2#3233012yes—
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×#21002
C2×#19212
#151942
#113572
#113462
#113282
#112932
#112672
#112072
#108072