Finite Group Databasebeta

← #468173

Subgroups — #468173

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic
3C2×#1919C211yesmaximal, characteristic
4#15158C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11380C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11328C212yesmaximal
9#11293C212yesmaximal
10#11220C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2211yescharacteristic
17#2546—24no—
18#2546—24no—
19C2×#398—22no—
20C2×#395—21no—
21C2×#391—21no—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#2057—24no—
28#1919—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1441—22no—
42#1441—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8D411yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4⋊Q8—28no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8—48no—
55C2×C4⋊Q8D414yes—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—42no—
59C2×C4×Q8—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D412yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#107—24no—
77C2×#107—42no—
78C2×#107—42no—
79C2×#106—22no—
80C2×#106—22no—
81C2×#106Q814yescomplemented
82#398—216no—
83#395—44no—
84#391—28no—
85#389—24no—
86#389—24no—
87#389—24no—
88#389—24no—
89#389—22no—
90#389—22no—
91#389—42no—
92#389—42no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104#340—24no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—24no—
109#340—42no—
110#340—42no—
111#340Q818yescomplemented
112#336—22no—
113#336—22no—
114#336—42no—
115#336—42no—
116C22×Q8—22no—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—24no—
120C22×Q8—22no—
121C22×Q8—42no—
122C22×Q8—44no—
123C22×Q8—44no—
124C22×Q8—42no—
125C22×Q8—42no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8C2×Q814yes—
153C22×C8C2×Q812yes—
154C4⋊Q8—48no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4⋊Q8—432no—
158C4×Q8—48no—
159C4×Q8—44no—
160C4×Q8—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×Q814yes—
208C2×C4⋊C4C2×Q812yes—
209C2×C4⋊C4C2×Q812yes—
210C2×C4⋊C4C2×Q814yes—
211C2×C4⋊C4#4112yes—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—44no—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224#107—48no—
225#106—28no—
226#95—28no—
227#95—28no—
228#95—24no—
229#95—44no—
230#95—44no—
231#95—42no—
232#95—42no—
233#95—81no—
234#95—82no—
235C24C22×D411yescharacteristic
236C2×Q8—28no—
237C2×Q8—48no—
238C2×Q8—48no—
239C2×Q8—44no—
240C2×Q8—48no—
241C2×Q8—48no—
242C2×Q8—44no—
243C2×Q8—48no—
244C2×Q8—88no—
245C2×Q8—88no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4#12012yes—
302C22×C4#12012yes—
303C22×C4#13512yes—
304C22×C4AGL(1,ℤ/8)14yes—
305C22×C4AGL(1,ℤ/8)12yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×Q811yescharacteristic
309C22×C4C22×Q812yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4H(5,𝔽2)12yes—
312C22×C4H(5,𝔽2)12yes—
313C22×C42−1+412yes—
314C22×C42−1+411yescharacteristic
315C22×C42−1+411yescharacteristic
316C2×C8—22no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—416no—
348C4⋊C4—416no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—88no—
359C4⋊C4—84no—
360C4⋊C4C22⋊Q814yes—
361C4⋊C4C4⋊Q814yes—
362C42—24no—
363C42—22no—
364C42—22no—
365C42—48no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C23C2×#12011yescharacteristic
371C23#53412yes—
372C23#54811yescharacteristic
373C23D4×Q811yescharacteristic
374C23D4×Q812yes—
375C23C2×#13511yescharacteristic
376C23C2×AGL(1,ℤ/8)12yes—
377C23C2×AGL(1,ℤ/8)11yescharacteristic
378C23#57711yescharacteristic
379C23#57811yescharacteristic
380C23#57812yes—
381Q8—416no—
382Q8—816no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
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—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4#44914yes—
443C2×C4C2×C22⋊Q811yescharacteristic
444C2×C4C2×C4⋊Q811yescharacteristic
445C2×C4#53412yes—
446C2×C4#53511yescharacteristic
447C2×C4#53511yescharacteristic
448C2×C4#53611yescharacteristic
449C2×C4#53611yescharacteristic
450C2×C4#54212yes—
451C2×C4#54412yes—
452C2×C4#54412yes—
453C2×C4#54412yes—
454C2×C4#54811yescharacteristic
455C2×C4D4×Q812yes—
456C2×C4D4×Q811yescharacteristic
457C2×C4#55412yes—
458C2×C4Q8212yes—
459C2×C4#56312yes—
460C2×C4#56312yes—
461C8—42no—
462C8—44no—
463C8—81no—
464C22#240911yescharacteristic
465C22#243411yescharacteristic
466C22#261311yescharacteristic
467C22#272912yes—
468C22#273412yes—
469C22C2×#44911yescharacteristic
470C22#300311yescharacteristic
471C22#301111yescharacteristic
472C22#301311yescharacteristic
473C22#318812yes—
474C22#319212yes—
475C22#320112yes—
476C22#320212yes—
477C22#336411yescharacteristic
478C22#336712yes—
479C22#336811yescharacteristic
480C22#338111yescharacteristic
481C22#338312yes—
482C22#338412yes—
483C22#338611yescharacteristic
484C22#338612yes—
485C22#338612yes—
486C22#338612yes—
487C4—24no—
488C4—44no—
489C4—44no—
490C4—42no—
491C4—44no—
492C4—82no—
493C4—84no—
494C4—84no—
495C4—82no—
496C4—161no—
497C4#256711yescharacteristic
498C4#262811yescharacteristic
499C4#266011yescharacteristic
500C4#272912yes—
501C4#273412yes—
502C4#282111yescharacteristic
503C2#1660311yescharacteristic
504C2#2432811yescharacteristic
505C2#2434211yescharacteristic
506C2#2584011yescharacteristic
507C2#2584411yescharacteristic
508C2#2959411yescharacteristic
509C2#2960511yescharacteristic
510C2#3257912yes—
511C2#3258912yes—
512C2#3268812yes—
513C2#3269312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
81. C2×#106Q8semidirect
111. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#20572
C2×#19192
#151582
#113842
#113802
#113282
#112932
#112202
#108072