Finite Group Databasebeta

← #464845

Subgroups — #464845

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic, complemented
2C2×#2094C212yesmaximal
3C2×#2058C211yesmaximal, characteristic, complemented
4C2×#2023C211yesmaximal, characteristic
5C2×#1934C211yesmaximal, characteristic
6#15144C211yesmaximal, characteristic
7#15143C211yesmaximal, characteristic, complemented
8#11378C212yesmaximal, complemented
9#11323C212yesmaximal, complemented
10#11121C212yesmaximal
11#10797C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#498—28no—
15C2×#498—24no—
16C22×#106C2212yes—
17C22×#106C2212yes—
18C22×#106C2212yes—
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#395—22no—
24C2×#393—22no—
25C2×#390C2211yescharacteristic
26C2×#390C2212yes—
27C2×#389C2211yescharacteristic
28C2×#384C2212yes—
29#2094—28no—
30#2058—24no—
31#2023—24no—
32#1934—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40#1444—22no—
41#1439—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8—21no—
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—24no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73#498—232no—
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—42no—
82C2×#106D418yes—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—24no—
86C2×#102—22no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—42no—
90C2×#102Q814yescomplemented
91C2×C4×C8—24no—
92C2×C4×C8D414yes—
93#395—48no—
94#393—48no—
95#390—24no—
96#390—22no—
97#390—22no—
98#390—42no—
99#389—22no—
100#389—22no—
101#384—24no—
102#384—42no—
103C2×#95—24no—
104C2×#95—24no—
105C2×#95—21no—
106C2×#95D412yes—
107C2×#95D412yes—
108#340—24no—
109#340—22no—
110#340—22no—
111#340—24no—
112#336—24no—
113#336—24no—
114#336D414yes—
115C25—21no—
116C22×Q8—44no—
117C22×Q8—44no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8—82no—
123C22×Q8—82no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4C2×Q812yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D414yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8C2×D414yes—
155C22×C8C2×D414yes—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8C2×Q812yes—
159C22×C8#4112yes—
160C4⋊Q8—28no—
161C4⋊Q8—28no—
162C4⋊Q8—48no—
163C4⋊Q8—48no—
164C4⋊Q8—44no—
165C4⋊Q8—44no—
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—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
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—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—84no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4C2×Q812yes—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—48no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—84no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218#106—216no—
219#106—216no—
220#106—48no—
221#102—28no—
222#102—48no—
223#102—44no—
224C4×C8—28no—
225#95—24no—
226#95—48no—
227#95—42no—
228#95—48no—
229#95SD1618yes—
230C24—22no—
231C24—41no—
232C24—42no—
233C24—42no—
234C24—41no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24C22×D411yescharacteristic
240C2×Q8—48no—
241C2×Q8—48no—
242C2×Q8—816no—
243C2×Q8—816no—
244C2×Q8—84no—
245C2×Q8—84no—
246C2×Q8—84no—
247C2×Q8—84no—
248C2×D4—22no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4#12712yes—
300C22×C4#12712yes—
301C22×C4C2×SD1618yes—
302C22×C4C2×SD1614yes—
303C22×C4AGL(1,ℤ/8)14yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D412yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4H(5,𝔽2)12yes—
310C22×C42−1+412yes—
311C22×C42−1+411yescharacteristic
312C22×C42−1+411yescharacteristic
313C22×C42−1+412yes—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—22no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8#12014yes—
330C2×C8C22⋊Q814yes—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—48no—
338C4⋊C4—48no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—816no—
352C4⋊C4—84no—
353C22⋊C4—816no—
354C22⋊C4—84no—
355C22⋊C4—84no—
356C42—22no—
357C42—22no—
358C42—24no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—42no—
367C23—82no—
368C23—82no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—82no—
375C23—81no—
376C23—84no—
377C23—84no—
378C23—84no—
379C23—84no—
380C23C2×#12711yescharacteristic
381C23#53612yes—
382C23D4212yes—
383C23#54711yescharacteristic
384C23D4×Q811yescharacteristic
385C23C22×SD1612yes—
386C23C2×AGL(1,ℤ/8)12yes—
387C23#57712yes—
388C23#57712yes—
389Q8—816no—
390Q8—816no—
391D4—42no—
392D4—42no—
393D4—44no—
394D4—81no—
395D4—81no—
396D4—82no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—42no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4#49214yes—
434C2×C4#49614yes—
435C2×C4C2×#12011yescharacteristic
436C2×C4C2×C22⋊Q811yescharacteristic
437C2×C4#53511yescharacteristic
438C2×C4#53611yescharacteristic
439C2×C4#53611yescharacteristic
440C2×C4#53611yescharacteristic
441C2×C4#53612yes—
442C2×C4#53612yes—
443C2×C4#53612yes—
444C2×C4#54212yes—
445C2×C4#54412yes—
446C2×C4D4212yes—
447C2×C4#54711yescharacteristic
448C2×C4#54812yes—
449C2×C4D4×Q812yes—
450C2×C4D4×Q811yescharacteristic
451C2×C4#56312yes—
452C2×C4#56312yes—
453C8—24no—
454C8—24no—
455C8—44no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—88no—
460C22—164no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22#243311yescharacteristic
466C22#254211yescharacteristic
467C22#265811yescharacteristic
468C22#267412yes—
469C22#267712yes—
470C22C2×#49211yescharacteristic
471C22C2×#49611yescharacteristic
472C22#315411yescharacteristic
473C22#315411yescharacteristic
474C22#322714yes—
475C22#324414yes—
476C22#326514yes—
477C22D4×SD1614yes—
478C22#333512yes—
479C22#335412yes—
480C22Q8×SD1612yes—
481C22#338412yes—
482C4—24no—
483C4—24no—
484C4—48no—
485C4—82no—
486C4—84no—
487C4—84no—
488C4—161no—
489C4—161no—
490C4#240911yescharacteristic
491C4#261411yescharacteristic
492C4#262811yescharacteristic
493C4#267412yes—
494C4#267712yes—
495C4#282111yescharacteristic
496C2—162no—
497C2#1651111yescharacteristic
498C2#2577411yescharacteristic
499C2#2578311yescharacteristic
500C2#2853011yescharacteristic
501C2#2854111yescharacteristic
502C2#3027711yescharacteristic
503C2#3029011yescharacteristic
504C2#3077514yes—
505C2#3095414yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#498C2semidirect
3. C2×#2058C2semidirect
7. #15143C2semidirect
8. #11378C2semidirect
9. #11323C2semidirect
11. #10797C2semidirect
90. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4982
C2×#20942
C2×#20582
C2×#20232
C2×#19342
#151442
#151432
#113782
#113232
#111212
#107972