Finite Group Databasebeta

← #464517

Subgroups — #464517

Order 512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic
3C2×#2059C212yesmaximal
4C2×#1918C211yesmaximal, characteristic
5#15149C211yesmaximal, characteristic
6#11385C212yesmaximal
7#11371C211yesmaximal, characteristic
8#11346C211yesmaximal, characteristic
9#11327C212yesmaximal
10#11293C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×#125C2212yes—
13C22×#107C2211yescharacteristic
14C22×#106C2212yes—
15#2552—24no—
16#2552—24no—
17C2×#398—22no—
18C2×#398—21no—
19C2×#389—21no—
20C2×#389C2212yes—
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#384—21no—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#2084—24no—
28#2059—28no—
29#1918—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1442—22no—
43#1441—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2211yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C2×#125—24no—
50C2×#125—24no—
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—42no—
54C2×#125—42no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×#107—22no—
74C2×#107—22no—
75C2×#107D414yes—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—42no—
79#398—216no—
80#398—44no—
81#389—28no—
82#389—24no—
83#389—24no—
84#389—24no—
85#389—22no—
86#389—22no—
87#389—42no—
88#384—28no—
89#384—22no—
90#384D414yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95D412yes—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—42no—
111#340Q818yes—
112#336—24no—
113#336—22no—
114#336—22no—
115#336—42no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8C2×D412yes—
143C22×C8C2×Q814yes—
144#125—48no—
145#125—48no—
146#125—48no—
147#125—48no—
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—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×Q814yes—
202C2×C4⋊C4C2×Q814yes—
203C2×C4⋊C4#4112yes—
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#107—28no—
218#106—48no—
219#95—28no—
220#95—24no—
221#95—24no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—42no—
226#95—42no—
227#95—81no—
228C24C22×D411yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—42no—
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—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4#12012yes—
288C22×C4#12012yes—
289C22×C4#13512yes—
290C22×C4AGL(1,ℤ/8)12yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×Q812yes—
295C22×C4C2×#4111yescharacteristic
296C22×C4H(5,𝔽2)11yescharacteristic
297C22×C4H(5,𝔽2)11yescharacteristic
298C22×C42−1+412yes—
299C22×C42−1+412yes—
300C22×C42−1+412yes—
301C2×C8—22no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—44no—
330C4⋊C4—48no—
331C4⋊C4—416no—
332C4⋊C4—416no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—84no—
344C4⋊C4—88no—
345C4⋊C4—88no—
346C4⋊C4—84no—
347C4⋊C4#12114yes—
348C4⋊C4C4⋊Q814yes—
349C42—22no—
350C42—22no—
351C42—24no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—48no—
357C23C2×#12011yescharacteristic
358C23#53612yes—
359C23D4211yescharacteristic
360C23#54611yescharacteristic
361C23D4×Q812yes—
362C23C2×#13511yescharacteristic
363C23C2×AGL(1,ℤ/8)11yescharacteristic
364C23#57512yes—
365C23#57712yes—
366C23#57711yescharacteristic
367C23#57811yescharacteristic
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—44no—
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—42no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#44914yes—
430C2×C4C2×#12111yescharacteristic
431C2×C4C2×C4⋊Q811yescharacteristic
432C2×C4#53511yescharacteristic
433C2×C4#53611yescharacteristic
434C2×C4#53611yescharacteristic
435C2×C4#53611yescharacteristic
436C2×C4#53612yes—
437C2×C4#53612yes—
438C2×C4#54112yes—
439C2×C4#54412yes—
440C2×C4#54412yes—
441C2×C4D4211yescharacteristic
442C2×C4#54611yescharacteristic
443C2×C4D4×Q812yes—
444C2×C4D4×Q812yes—
445C2×C4#55712yes—
446C2×C4#56312yes—
447C2×C4#56312yes—
448C8—42no—
449C8—44no—
450C8—81no—
451C22#240811yescharacteristic
452C22#243411yescharacteristic
453C22#261411yescharacteristic
454C22#267712yes—
455C22#271312yes—
456C22C2×#44911yescharacteristic
457C22#300511yescharacteristic
458C22#300911yescharacteristic
459C22#301311yescharacteristic
460C22#323212yes—
461C22#323412yes—
462C22#324212yes—
463C22#324412yes—
464C22#327112yes—
465C22#327211yescharacteristic
466C22#328311yescharacteristic
467C22#330311yescharacteristic
468C22#331712yes—
469C22#333011yescharacteristic
470C22#338112yes—
471C22#338412yes—
472C22#338812yes—
473C22#339012yes—
474C4—24no—
475C4—44no—
476C4—42no—
477C4—44no—
478C4—82no—
479C4—82no—
480C4—82no—
481C4—84no—
482C4—84no—
483C4—161no—
484C4#254611yescharacteristic
485C4#262111yescharacteristic
486C4#266011yescharacteristic
487C4#267712yes—
488C4#271312yes—
489C4#282111yescharacteristic
490C2#1650711yescharacteristic
491C2#2423711yescharacteristic
492C2#2425411yescharacteristic
493C2#2583511yescharacteristic
494C2#2584611yescharacteristic
495C2#2964011yescharacteristic
496C2#2964811yescharacteristic
497C2#3095212yes—
498C2#3097512yes—
499C2#3214112yes—
500C2#3215912yes—
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×#20842
C2×#20592
C2×#19182
#151492
#113852
#113712
#113462
#113272
#112932
#108062