Finite Group Databasebeta

← #467153

Subgroups — #467153

Order 512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic
3C2×#1920C211yesmaximal, characteristic
4#15200C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11385C212yesmaximal
7#11357C211yesmaximal, characteristic
8#11346C211yesmaximal, characteristic
9#11292C212yesmaximal
10#11220C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×#125C2212yes—
13C22×#106C2211yescharacteristic
14#2552—24no—
15#2552—24no—
16C2×#398—22no—
17C2×#389—21no—
18C2×#389C2212yes—
19C2×#389C2211yescharacteristic
20C2×#384—21no—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#2100—24no—
25#1920—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1445—21no—
40#1442—22no—
41#1442—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×#125—24no—
49C2×#125—24no—
50C2×#125—24no—
51C2×#125—24no—
52C2×#125—42no—
53C2×#125—42no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#106—22no—
72C2×#106—22no—
73C2×#106Q814yes—
74#398—216no—
75#389—28no—
76#389—24no—
77#389—24no—
78#389—22no—
79#389—22no—
80#384—28no—
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—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95D412yes—
97#340—24no—
98#340—24no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#336—22no—
107#336—22no—
108#336—42no—
109#336—42no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8C2×Q812yes—
138#125—48no—
139#125—48no—
140#125—48no—
141#125—48no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—81no—
184C2×C4⋊C4—81no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×Q812yes—
192C2×C4⋊C4C2×Q812yes—
193C2×C4⋊C4#4112yes—
194C2×C4⋊C4#4114yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208#106—28no—
209#95—28no—
210#95—24no—
211#95—24no—
212#95—44no—
213#95—42no—
214#95—42no—
215#95—81no—
216#95—82no—
217#95—82no—
218#95—82no—
219C24C22×D411yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4#12012yes—
277C22×C4#12012yes—
278C22×C4#13512yes—
279C22×C4#13712yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×Q811yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4112yes—
285C22×C4H(5,𝔽2)12yes—
286C22×C42−1+412yes—
287C22×C42−1+412yes—
288C22×C42−1+411yescharacteristic
289C22×C42−1+411yescharacteristic
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—44no—
317C4⋊C4—48no—
318C4⋊C4—416no—
319C4⋊C4—416no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—44no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C4⋊C4—88no—
335C4⋊C4C22⋊Q814yes—
336C4⋊C4#12414yes—
337C42—22no—
338C42—22no—
339C42—24no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—48no—
345C23C2×#12011yescharacteristic
346C23#53612yes—
347C23#54612yes—
348C23#54811yescharacteristic
349C23D4×Q811yescharacteristic
350C23C2×#13511yescharacteristic
351C23C2×#13711yescharacteristic
352C23#57512yes—
353C23#57611yescharacteristic
354C23#57712yes—
355C23#57711yescharacteristic
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—44no—
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—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#45214yes—
416C2×C4C2×C22⋊Q811yescharacteristic
417C2×C4C2×#12411yescharacteristic
418C2×C4#53511yescharacteristic
419C2×C4#53611yescharacteristic
420C2×C4#53611yescharacteristic
421C2×C4#53611yescharacteristic
422C2×C4#53612yes—
423C2×C4#54112yes—
424C2×C4#54112yes—
425C2×C4#54212yes—
426C2×C4#54412yes—
427C2×C4#54612yes—
428C2×C4#54811yescharacteristic
429C2×C4D4×Q811yescharacteristic
430C2×C4#55612yes—
431C2×C4#55712yes—
432C2×C4#56312yes—
433C2×C4#56312yes—
434C8—42no—
435C8—81no—
436C8—82no—
437C22#240911yescharacteristic
438C22#243511yescharacteristic
439C22#261411yescharacteristic
440C22#271312yes—
441C22#271612yes—
442C22C2×#45211yescharacteristic
443C22#300511yescharacteristic
444C22#301011yescharacteristic
445C22#301211yescharacteristic
446C22#323212yes—
447C22#323612yes—
448C22#324112yes—
449C22#324412yes—
450C22#330312yes—
451C22#330912yes—
452C22#331712yes—
453C22#333112yes—
454C22#336111yescharacteristic
455C22#337112yes—
456C22#337211yescharacteristic
457C22#338111yescharacteristic
458C22#338511yescharacteristic
459C22#338812yes—
460C4—24no—
461C4—44no—
462C4—44no—
463C4—42no—
464C4—82no—
465C4—82no—
466C4—82no—
467C4—84no—
468C4—161no—
469C4—162no—
470C4#256811yescharacteristic
471C4#262811yescharacteristic
472C4#264211yescharacteristic
473C4#271312yes—
474C4#271612yes—
475C4#282111yescharacteristic
476C2#1656011yescharacteristic
477C2#2433111yescharacteristic
478C2#2434011yescharacteristic
479C2#2588611yescharacteristic
480C2#2589311yescharacteristic
481C2#2964311yescharacteristic
482C2#2964711yescharacteristic
483C2#3214112yes—
484C2#3216112yes—
485C2#3223712yes—
486C2#3225512yes—
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×#19202
#152002
#113852
#113572
#113462
#112922
#112202
#108062