Finite Group Databasebeta

← #416749

Subgroups — #416749

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2C2×#1949C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic
4#13955C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11384C212yesmaximal
7#11220C212yesmaximal
8#11219C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2212yes—
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#97C2211yescharacteristic
18#2551—24no—
19#2551—24no—
20C2×#395—22no—
21C2×#389—21no—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1949—24no—
29#1836—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1442—21no—
41#1442—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—22no—
45C23×Q8—22no—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4⋊Q8—28no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—48no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×#107—22no—
74C2×#107—24no—
75C2×#106—22no—
76C2×#106—24no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8Q814yescomplemented
80C2×#97—22no—
81C2×#97—22no—
82C2×#97Q814yescomplemented
83#395—216no—
84#389—28no—
85#389—24no—
86#389—24no—
87#384—22no—
88#384D414yes—
89#382—22no—
90#382—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4D414yes—
96C4×C4⋊C4Q814yescomplemented
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107#340—24no—
108#340—24no—
109#340—22no—
110#340—22no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#336—42no—
116#336—42no—
117C22×Q8—28no—
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8—82no—
123C22×Q8—82no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8C2×Q814yes—
150C22×C8C2×Q812yes—
151C22×C8C2×Q812yes—
152C4⋊Q8—48no—
153C4⋊Q8—48no—
154C4⋊Q8—432no—
155C4⋊Q8—48no—
156C4⋊Q8—48no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
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—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—81no—
193C2×C4⋊C4—81no—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×Q814yes—
198C2×C4⋊C4#4114yes—
199C2×C4⋊C4#4114yes—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—44no—
209C2×C42—81no—
210C2×C42—81no—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×Q812yes—
214C2×C42C2×Q812yes—
215C2×C42C2×Q812yes—
216C2×C42#4112yes—
217#107—44no—
218#106—44no—
219C4⋊C8—28no—
220#97—28no—
221#95—28no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—42no—
226#95—42no—
227#95—44no—
228#95—44no—
229#95—82no—
230#95—82no—
231C24C22×D411yescharacteristic
232C2×Q8—416no—
233C2×Q8—416no—
234C2×Q8—88no—
235C2×Q8—88no—
236C2×Q8—88no—
237C2×Q8—88no—
238C2×Q8—88no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
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—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4#12012yes—
292C22×C4#12012yes—
293C22×C4C22⋊Q812yes—
294C22×C4C22⋊Q812yes—
295C22×C4C4⋊Q812yes—
296C22×C4C4⋊Q812yes—
297C22×C4AGL(1,ℤ/8)14yes—
298C22×C4#13714yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×Q811yescharacteristic
302C22×C4C22×Q811yescharacteristic
303C22×C4C22×Q811yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4112yes—
306C22×C4C2×#4112yes—
307C22×C4H(5,𝔽2)12yes—
308C22×C42−1+412yes—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—48no—
337C4⋊C4—416no—
338C4⋊C4—416no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—84no—
350C4⋊C4#12414yes—
351C4⋊C4#12414yes—
352C42—48no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—84no—
358C42C22⋊Q814yes—
359C42C22⋊Q814yes—
360C42C4⋊Q814yes—
361C42C4⋊Q814yes—
362C23C2×#12011yescharacteristic
363C23C2×C22⋊Q811yescharacteristic
364C23C2×C4⋊Q811yescharacteristic
365C23#54612yes—
366C23#54812yes—
367C23C2×AGL(1,ℤ/8)12yes—
368C23C2×#13712yes—
369C23#57611yescharacteristic
370C23#57711yescharacteristic
371C23#57711yescharacteristic
372C23#57811yescharacteristic
373Q8—832no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4#39514yes—
433C2×C4#39514yes—
434C2×C4C2×C22⋊Q811yescharacteristic
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4C2×#12411yescharacteristic
437C2×C4C2×#12411yescharacteristic
438C2×C4C2×C4⋊Q811yescharacteristic
439C2×C4C2×C4⋊Q811yescharacteristic
440C2×C4#53312yes—
441C2×C4#53312yes—
442C2×C4#53511yescharacteristic
443C2×C4#53611yescharacteristic
444C2×C4#54412yes—
445C2×C4#54412yes—
446C2×C4#54612yes—
447C2×C4#54812yes—
448C2×C4#55412yes—
449C2×C4#55512yes—
450C2×C4#55612yes—
451C2×C4#55712yes—
452C8—42no—
453C8—42no—
454C8—44no—
455C22C2×#39511yescharacteristic
456C22#243511yescharacteristic
457C22#243611yescharacteristic
458C22#243912yes—
459C22#255012yes—
460C22#300311yescharacteristic
461C22#300411yescharacteristic
462C22#301211yescharacteristic
463C22#301311yescharacteristic
464C22#307912yes—
465C22#308111yescharacteristic
466C22#308211yescharacteristic
467C22#316311yescharacteristic
468C22#316411yescharacteristic
469C22#316512yes—
470C22#331012yes—
471C22#331212yes—
472C22#331812yes—
473C22#332012yes—
474C22#336712yes—
475C22#336912yes—
476C22#337312yes—
477C22#337512yes—
478C4—24no—
479C4—24no—
480C4—44no—
481C4—44no—
482C4—44no—
483C4—44no—
484C4—84no—
485C4—84no—
486C4—161no—
487C4—161no—
488C4#243912yes—
489C4#255012yes—
490C4#256311yescharacteristic
491C4#256811yescharacteristic
492C4#266011yescharacteristic
493C4#286211yescharacteristic
494C2#1523011yescharacteristic
495C2#2355211yescharacteristic
496C2#2355711yescharacteristic
497C2#2589111yescharacteristic
498C2#2589411yescharacteristic
499C2#2593911yescharacteristic
500C2#2594211yescharacteristic
501C2#2603612yes—
502C2#2603812yes—
503C2#2880112yes—
504C2#2880312yes—
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

Complements

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

NH ≅ G/Ntype
79. C2×C4⋊C8Q8semidirect
82. C2×#97Q8semidirect
96. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#19492
C2×#18362
#139552
#113842
#112202
#112192
#108492
#108352
#107032