Finite Group Databasebeta

← #468355

Subgroups — #468355

Order 512.

1132 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2509C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4#15128C211yesmaximal, characteristic
5#11327C212yesmaximal
6#11292C212yesmaximal
7#11227C211yesmaximal, characteristic
8#11220C212yesmaximal
9#11208C212yesmaximal
10#11202C211yesmaximal, characteristic
11#10807C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15#2509—24no—
16#2509—24no—
17C2×#400—21no—
18C2×#400—21no—
19C2×#395—21no—
20C2×#389—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#387—21no—
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2058—24no—
31#1987—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1446—21no—
43#1444—21no—
44#1441—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#106—24no—
73C2×#106—24no—
74C2×#106—42no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—42no—
77#400—28no—
78#400—28no—
79#395—44no—
80#389—28no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—24no—
89#389—42no—
90#389D414yes—
91#387—28no—
92#382—22no—
93#382—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4D414yes—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112#340—24no—
113#340—24no—
114#340—24no—
115#340—42no—
116#340—42no—
117#340—42no—
118#340Q818yescomplemented
119#336—22no—
120#336—42no—
121#336—42no—
122C22×Q8—44no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—82no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8C2×Q814yes—
154C4⋊Q8—28no—
155C4⋊Q8—48no—
156C4⋊Q8—44no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×Q814yes—
208C2×C4⋊C4C2×Q814yes—
209C2×C4⋊C4#4114yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—24no—
223C2×C42—82no—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226#106—48no—
227C4⋊C8—44no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—22no—
236#95—22no—
237#95—44no—
238#95—42no—
239#95—42no—
240#95—44no—
241#95—82no—
242C24C22×D411yescharacteristic
243C2×Q8—48no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—88no—
247C2×Q8—88no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4#12012yes—
312C22×C4#12012yes—
313C22×C4#13512yes—
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4#13712yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×Q812yes—
319C22×C4C2×#4112yes—
320C22×C4H(5,𝔽2)12yes—
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C42−1+411yescharacteristic
323C22×C42−1+411yescharacteristic
324C22×C42−1+411yescharacteristic
325C22×C42−1+412yes—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—48no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C4⋊C4—88no—
369C4⋊C4C4⋊Q814yes—
370C42—22no—
371C42—22no—
372C42—24no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—42no—
380C42—84no—
381C23C2×#12011yescharacteristic
382C23#53611yescharacteristic
383C23#53711yescharacteristic
384C23#54612yes—
385C23D4×Q812yes—
386C23C2×#13511yescharacteristic
387C23C2×AGL(1,ℤ/8)11yescharacteristic
388C23C2×#13711yescharacteristic
389C23#57511yescharacteristic
390C23#57712yes—
391C23#57712yes—
392Q8—816no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4#45214yes—
461C2×C4#52511yescharacteristic
462C2×C4C2×C4⋊Q811yescharacteristic
463C2×C4#53511yescharacteristic
464C2×C4#53611yescharacteristic
465C2×C4#53611yescharacteristic
466C2×C4#53611yescharacteristic
467C2×C4#53711yescharacteristic
468C2×C4#53711yescharacteristic
469C2×C4#54112yes—
470C2×C4#54112yes—
471C2×C4#54412yes—
472C2×C4#54412yes—
473C2×C4#54612yes—
474C2×C4D4×Q812yes—
475C2×C4#55412yes—
476C2×C4#55712yes—
477C2×C4#56212yes—
478C2×C4#56312yes—
479C8—44no—
480C8—82no—
481C22#240411yescharacteristic
482C22#243411yescharacteristic
483C22#261611yescharacteristic
484C22#270412yes—
485C22#271312yes—
486C22C2×#45211yescharacteristic
487C22#300611yescharacteristic
488C22#300911yescharacteristic
489C22#301011yescharacteristic
490C22#323311yescharacteristic
491C22#323811yescharacteristic
492C22#324412yes—
493C22#324711yescharacteristic
494C22#325211yescharacteristic
495C22#326312yes—
496C22#330112yes—
497C22#331112yes—
498C22#332012yes—
499C22#332412yes—
500C22#338112yes—
501C22#338412yes—
502C22#338712yes—
503C22#338812yes—
504C4—24no—
505C4—42no—
506C4—44no—
507C4—42no—
508C4—81no—
509C4—82no—
510C4—84no—
511C4—81no—
512C4—82no—
513C4—81no—
514C4—84no—
515C4—82no—
516C4#258411yescharacteristic
517C4#263411yescharacteristic
518C4#266011yescharacteristic
519C4#270412yes—
520C4#271312yes—
521C4#282011yescharacteristic
522C2#1660811yescharacteristic
523C2#2396411yescharacteristic
524C2#2397911yescharacteristic
525C2#2583611yescharacteristic
526C2#2585111yescharacteristic
527C2#2969211yescharacteristic
528C2#2969711yescharacteristic
529C2#3187912yes—
530C2#3189312yes—
531C2#3214412yes—
532C2#3215012yes—
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
118. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25092
C2×#20582
C2×#19872
#151282
#113272
#112922
#112272
#112202
#112082
#112022
#108072