Finite Group Databasebeta

← #416735

Subgroups — #416735

Order 512.

1188 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#1836C211yesmaximal, characteristic
3C2×#1835C211yesmaximal, characteristic
4#14276C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11384C212yesmaximal
7#11208C212yesmaximal
8#11207C212yesmaximal
9#10843C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2660—24no—
17#2660—24no—
18C2×#398—22no—
19C2×#395—22no—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#376C2211yescharacteristic
28#1836—24no—
29#1835—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1441—22no—
41#1441—22no—
42C4⋊#95C2212yes—
43C23×Q8—21no—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—42no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—24no—
53C2×#125—42no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#97—22no—
71C2×#97—22no—
72C2×#97Q814yescomplemented
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8Q814yescomplemented
76#398—216no—
77#395—216no—
78#389—24no—
79#389—24no—
80#384—22no—
81#384—22no—
82#384—22no—
83#384—22no—
84#384D414yes—
85#384Q814yescomplemented
86#382—22no—
87#382—22no—
88#382—22no—
89#382—22no—
90#382—22no—
91#382D414yes—
92#376—22no—
93#376—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95D412yes—
101C2×#95D412yes—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108#340—42no—
109#340—42no—
110#336—24no—
111#336—24no—
112C22×Q8—24no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—82no—
116C22×Q8—82no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C2×Q814yes—
141C22×C8C2×Q814yes—
142C4⋊Q8—48no—
143C4⋊Q8—48no—
144#125—48no—
145#125—48no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
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—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×Q814yes—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×Q812yes—
204C2×C42C2×Q812yes—
205C2×C42C2×Q812yes—
206C2×C42#4114yes—
207C2×C42#4114yes—
208C2×C42#4112yes—
209#97—28no—
210C4×C8—28no—
211#95—28no—
212#95—28no—
213#95—24no—
214#95—24no—
215#95—44no—
216#95—44no—
217#95—82no—
218#95—82no—
219C24C22×D411yescharacteristic
220C2×Q8—48no—
221C2×Q8—88no—
222C2×Q8—88no—
223C2×Q8—84no—
224C2×Q8—84no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
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—24no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4#12014yes—
278C22×C4C22⋊Q814yes—
279C22×C4#12414yes—
280C22×C4#12412yes—
281C22×C4#12412yes—
282C22×C4C4⋊Q814yes—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×Q811yescharacteristic
286C22×C4C22×Q811yescharacteristic
287C22×C4C22×Q811yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4112yes—
290C22×C4C2×#4112yes—
291C22×C4H(5,𝔽2)12yes—
292C22×C42−1+412yes—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
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—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—48no—
314C4⋊C4—48no—
315C4⋊C4—48no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—88no—
327C4⋊C4—88no—
328C4⋊C4—88no—
329C4⋊C4—88no—
330C42—24no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42C22⋊Q814yes—
335C42C22⋊Q814yes—
336C42C4⋊Q814yes—
337C42C4⋊Q814yes—
338C23C2×#12011yescharacteristic
339C23C2×C22⋊Q811yescharacteristic
340C23C2×#12411yescharacteristic
341C23C2×#12411yescharacteristic
342C23C2×C4⋊Q811yescharacteristic
343C23#53511yescharacteristic
344C23#53611yescharacteristic
345C23#57612yes—
346C23#57712yes—
347C23#57712yes—
348C23#57812yes—
349Q8—816no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4#39518yes—
408C2×C4C2×C22⋊Q811yescharacteristic
409C2×C4C2×C22⋊Q811yescharacteristic
410C2×C4C2×C4⋊Q811yescharacteristic
411C2×C4C2×C4⋊Q811yescharacteristic
412C2×C4#53312yes—
413C2×C4#53312yes—
414C2×C4#54412yes—
415C2×C4#54412yes—
416C2×C4#54612yes—
417C2×C4#54612yes—
418C2×C4#54812yes—
419C2×C4#54812yes—
420C2×C4#55412yes—
421C2×C4#55512yes—
422C2×C4#55612yes—
423C2×C4#55712yes—
424C8—44no—
425C8—44no—
426C22C2×#39511yescharacteristic
427C22#243511yescharacteristic
428C22#243611yescharacteristic
429C22#256311yescharacteristic
430C22#256811yescharacteristic
431C22#266011yescharacteristic
432C22#286211yescharacteristic
433C22#301212yes—
434C22#301312yes—
435C22#308112yes—
436C22#308212yes—
437C22#313612yes—
438C22#313612yes—
439C22#313712yes—
440C22#313712yes—
441C22#316312yes—
442C22#316412yes—
443C22#322011yescharacteristic
444C22#322111yescharacteristic
445C22#322211yescharacteristic
446C22#322311yescharacteristic
447C22#324312yes—
448C22#324412yes—
449C4—28no—
450C4—42no—
451C4—42no—
452C4—44no—
453C4—82no—
454C4—82no—
455C4—82no—
456C4—82no—
457C4—84no—
458C4—162no—
459C4#243912yes—
460C4#243912yes—
461C4#255012yes—
462C4#255012yes—
463C2#1523011yescharacteristic
464C2#2355712yes—
465C2#2586711yescharacteristic
466C2#2586811yescharacteristic
467C2#2591612yes—
468C2#2915211yescharacteristic
469C2#2915311yescharacteristic
470C2#2922912yes—
471C2#3031311yescharacteristic
472C2#3031411yescharacteristic
473C2#3329212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
72. C2×#97Q8semidirect
75. C2×C4×C8Q8semidirect
85. #384Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#26602
C2×#18362
C2×#18352
#142762
#113852
#113842
#112082
#112072
#108432
#108412
#106942