Finite Group Databasebeta

← #464831

Subgroups — #464831

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic, complemented
2C2×#2092C212yesmaximal
3C2×#2058C211yesmaximal, characteristic, complemented
4C2×#2057C211yesmaximal, characteristic, complemented
5C2×#2023C211yesmaximal, characteristic
6C2×#2023C211yesmaximal, characteristic
7#14700C211yesmaximal, characteristic
8#11372C212yesmaximal, complemented
9#11325C212yesmaximal, complemented
10#11051C212yesmaximal
11#10795C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18#2862—28no—
19C2×#391—22no—
20C2×#391—22no—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388—22no—
26C2×#382C2212yes—
27C2×#382C2212yes—
28C2×C43C2211yescharacteristic
29#2092—28no—
30#2058—24no—
31#2057—24no—
32#2023—24no—
33#2023—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1437—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8—21no—
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8—42no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C42D412yes—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—42no—
78C2×#106—24no—
79C2×#106—24no—
80C2×#106—42no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—42no—
86C2×#102—42no—
87#391—216no—
88#391—216no—
89#390—24no—
90#390—22no—
91#390—22no—
92#390—22no—
93#390Q814yescomplemented
94#389—22no—
95#389—22no—
96#388—48no—
97#382—24no—
98#382—24no—
99#382—24no—
100#382D418yes—
101C2×#95—24no—
102C2×#95—24no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C43—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#336—24no—
112#336—24no—
113#336D414yes—
114#336D414yes—
115C25—21no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4C2×Q812yes—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8C2×D414yes—
159C22×C8C2×D414yes—
160C4⋊Q8—48no—
161C4⋊Q8—48no—
162C4⋊Q8—48no—
163C4⋊Q8—48no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×Q812yes—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42C2×D414yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×Q812yes—
225C2×C42#4112yes—
226#107—48no—
227#106—48no—
228#102—48no—
229#102—48no—
230#95—216no—
231#95—216no—
232#95—44no—
233#95—44no—
234#95—42no—
235C24—22no—
236C24—41no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24C22×D411yescharacteristic
247C2×Q8—48no—
248C2×Q8—48no—
249C2×Q8—416no—
250C2×Q8—416no—
251C2×Q8—84no—
252C2×Q8—84no—
253C2×Q8—88no—
254C2×Q8—88no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4#12014yes—
317C22×C4C22⋊Q814yes—
318C22×C4#12712yes—
319C22×C4#12712yes—
320C22×C4#13514yes—
321C22×C4AGL(1,ℤ/8)14yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×Q811yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4H(5,𝔽2)12yes—
328C22×C42−1+412yes—
329C22×C42−1+411yescharacteristic
330C22×C42−1+411yescharacteristic
331C22×C42−1+412yes—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—84no—
365C22⋊C4—416no—
366C22⋊C4—88no—
367C22⋊C4—84no—
368C22⋊C4—84no—
369C42—24no—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—48no—
376C42—48no—
377C23—44no—
378C23—44no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23C2×#12011yescharacteristic
401C23C2×C22⋊Q811yescharacteristic
402C23C2×#12711yescharacteristic
403C23#53511yescharacteristic
404C23#53611yescharacteristic
405C23#53611yescharacteristic
406C23#53611yescharacteristic
407C23C2×#13512yes—
408C23C2×AGL(1,ℤ/8)12yes—
409C23#57512yes—
410C23#57512yes—
411Q8—816no—
412Q8—816no—
413D4—44no—
414D4—44no—
415D4—82no—
416D4—82no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—48no—
443C2×C4—48no—
444C2×C4—42no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4#49514yes—
465C2×C4#49614yes—
466C2×C4#53612yes—
467C2×C4#53612yes—
468C2×C4#53612yes—
469C2×C4#53612yes—
470C2×C4#54212yes—
471C2×C4#54412yes—
472C2×C4D4212yes—
473C2×C4D4212yes—
474C2×C4#54711yescharacteristic
475C2×C4#54711yescharacteristic
476C2×C4#54812yes—
477C2×C4D4×Q812yes—
478C2×C4D4×Q811yescharacteristic
479C2×C4D4×Q811yescharacteristic
480C2×C4#56312yes—
481C2×C4#56312yes—
482C8—44no—
483C8—44no—
484C22—84no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—84no—
489C22—161no—
490C22—162no—
491C22—161no—
492C22—161no—
493C22—161no—
494C22—162no—
495C22#240911yescharacteristic
496C22#243311yescharacteristic
497C22#254211yescharacteristic
498C22#261411yescharacteristic
499C22#262811yescharacteristic
500C22#265811yescharacteristic
501C22#282111yescharacteristic
502C22#300212yes—
503C22#300612yes—
504C22#307612yes—
505C22#307712yes—
506C22C2×#49511yescharacteristic
507C22C2×#49611yescharacteristic
508C22#315311yescharacteristic
509C22#315311yescharacteristic
510C22#321112yes—
511C22#321312yes—
512C22#322812yes—
513C22#322812yes—
514C22#323312yes—
515C22#323512yes—
516C22#324012yes—
517C22#324012yes—
518C4—24no—
519C4—24no—
520C4—44no—
521C4—44no—
522C4—82no—
523C4—84no—
524C4—84no—
525C4—161no—
526C4—161no—
527C4#267412yes—
528C4#267412yes—
529C4#267712yes—
530C4#267712yes—
531C2—162no—
532C2#1651111yescharacteristic
533C2#2430412yes—
534C2#2572411yescharacteristic
535C2#2573111yescharacteristic
536C2#2851011yescharacteristic
537C2#2852111yescharacteristic
538C2#2964412yes—
539C2#2998212yes—
540C2#3027811yescharacteristic
541C2#3028611yescharacteristic
542C2#3284012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2862C2semidirect
3. C2×#2058C2semidirect
4. C2×#2057C2semidirect
8. #11372C2semidirect
9. #11325C2semidirect
11. #10795C2semidirect
93. #390Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
C2×#20922
C2×#20582
C2×#20572
C2×#20232
#147002
#113722
#113252
#110512
#107952