Finite Group Databasebeta

← #414923

Subgroups — #414923

Order 512.

1022 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#21337C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#21324C211yesmaximal, characteristic
5#21306C211yesmaximal, characteristic
6C4×#430C211yesmaximal, characteristic
7#11274C214yesmaximal
8#11253C214yesmaximal
9#10855C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3165—22no—
13#3165—22no—
14#3165—22no—
15#3165—22no—
16C2×#501—22no—
17C2×#500—21no—
18C2×#498—21no—
19C2×#430C2212yes—
20C2×#428C2214yes—
21C2×#423C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2285C2211yescharacteristic
26#1998—22no—
27#1997—22no—
28#1997—22no—
29#1997—21no—
30#1997—21no—
31#1993—22no—
32#1990—22no—
33#1986—22no—
34#1986—22no—
35#1778—22no—
36C4×#108—22no—
37C2×#344C2214yes—
38#1742C2214yes—
39C2×#343C2212yes—
40C2×#343C2212yes—
41#1738C2212yes—
42#1736C2214yes—
43#1733C2212yes—
44#1733C2212yes—
45C23×Q8—21no—
46C22×M16C2312yes—
47C22×M16C2311yescharacteristic
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8D411yescharacteristic
70C2×C4⋊Q8D411yescharacteristic
71C2×C4⋊Q8D412yes—
72C2×C4⋊Q8D412yes—
73C2×#125D412yes—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×#117C2312yes—
79C2×#117C2312yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2312yes—
82C22×C42C2311yescharacteristic
83#501—28no—
84#501—28no—
85#500—24no—
86#500—24no—
87#498—24no—
88#498—24no—
89#430—24no—
90#430Q818yescomplemented
91C2×#108—24no—
92#428—28no—
93#428—28no—
94C2×#107—24no—
95C2×#106—24no—
96#423—21no—
97#423—21no—
98#423—21no—
99#423—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C4×M16—21no—
103C4×M16—21no—
104C4×M16—21no—
105C4×M16—21no—
106C4×M16—21no—
107C4×M16—21no—
108C4×M16—21no—
109C4×M16—21no—
110C4×M16D412yes—
111C4×M16D412yes—
112C4×M16D412yes—
113C4×M16D412yes—
114C2×#97—21no—
115C2×#97—21no—
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120#384—21no—
121#382—21no—
122#380—21no—
123#380—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126#344—28no—
127#343—24no—
128#343—24no—
129#343—24no—
130#343—24no—
131#341—24no—
132#340—28no—
133#339—24no—
134#339—24no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—41no—
140C22×Q8—41no—
141C22×Q8—41no—
142C22×Q8—41no—
143C22×Q8—84no—
144C23×C4—22no—
145C23×C4#4111yescharacteristic
146C23×C4#4111yescharacteristic
147C23×C4C2411yescharacteristic
148C2×M16—22no—
149C2×M16—21no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—21no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—24no—
164C2×M16—24no—
165C2×M16C2×D412yes—
166C2×M16C2×D414yes—
167C2×M16C2×D412yes—
168C2×M16C2×Q814yes—
169C2×M16C2×Q814yes—
170C2×M16C2×Q814yes—
171C2×M16#4114yes—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C4⋊Q8—41no—
196C4⋊Q8—41no—
197C4⋊Q8—41no—
198C4⋊Q8—41no—
199C4⋊Q8—41no—
200C4⋊Q8—41no—
201#125—22no—
202#125—22no—
203#125—22no—
204#125—22no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×Q8—41no—
220C4×Q8—41no—
221#117—22no—
222#117—22no—
223#117—22no—
224#117—22no—
225#117—24no—
226#117—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42#4111yescharacteristic
264#108—216no—
265#107—216no—
266#106—216no—
267C4⋊C8—41no—
268C4⋊C8—41no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271#97—22no—
272#97—22no—
273#97—22no—
274#97—22no—
275#97—41no—
276#97—41no—
277C4×C8—22no—
278C4×C8—22no—
279C4×C8—22no—
280C4×C8—22no—
281C4×C8—41no—
282C4×C8—41no—
283#95—41no—
284#95—41no—
285C24C2×#4111yescharacteristic
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—82no—
295C2×Q8—82no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—84no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—84no—
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12311yescharacteristic
316C22×C4#12311yescharacteristic
317C22×C4#12311yescharacteristic
318C22×C4#12311yescharacteristic
319C22×C4#12411yescharacteristic
320C22×C4#12411yescharacteristic
321C22×C4#12411yescharacteristic
322C22×C4#12411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×Q812yes—
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C42−1+412yes—
332C22×C42−1+412yes—
333M16—22no—
334M16—24no—
335M16—24no—
336M16—24no—
337M16—24no—
338M16—24no—
339M16—22no—
340M16—22no—
341M16—24no—
342M16—42no—
343M16—42no—
344M16—42no—
345M16—42no—
346M16—42no—
347M16—44no—
348M16C22⋊Q818yes—
349M16C4⋊Q814yes—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—44no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—44no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42#12011yescharacteristic
393C42#12011yescharacteristic
394C42#12011yescharacteristic
395C42#12011yescharacteristic
396C42#12111yescharacteristic
397C42#12111yescharacteristic
398C42#12111yescharacteristic
399C42#12111yescharacteristic
400C42#12111yescharacteristic
401C42#12111yescharacteristic
402C42#12111yescharacteristic
403C42#12111yescharacteristic
404C42#12311yescharacteristic
405C42#12311yescharacteristic
406C42#12311yescharacteristic
407C42#12311yescharacteristic
408C23—41no—
409C23—41no—
410C23C2×#12111yescharacteristic
411C23C2×#12311yescharacteristic
412C23C2×#12411yescharacteristic
413C23#55612yes—
414C23#55712yes—
415Q8—82no—
416Q8—82no—
417Q8—84no—
418Q8—84no—
419Q8—84no—
420Q8—82no—
421Q8—84no—
422Q8—82no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4#39411yescharacteristic
447C2×C4#39411yescharacteristic
448C2×C4#39411yescharacteristic
449C2×C4#39411yescharacteristic
450C2×C4#39411yescharacteristic
451C2×C4#39411yescharacteristic
452C2×C4#39411yescharacteristic
453C2×C4#39411yescharacteristic
454C2×C4C2×#12011yescharacteristic
455C2×C4C2×#12111yescharacteristic
456C2×C4C2×#12111yescharacteristic
457C2×C4C2×C22⋊Q812yes—
458C2×C4C2×#12311yescharacteristic
459C2×C4C2×C4⋊Q811yescharacteristic
460C2×C4#53211yescharacteristic
461C2×C4#53312yes—
462C2×C4#53612yes—
463C2×C4#53611yescharacteristic
464C2×C4#53611yescharacteristic
465C2×C4#54112yes—
466C2×C4#54812yes—
467C2×C4#54812yes—
468C2×C4D4×Q812yes—
469C2×C4D4×Q812yes—
470C2×C4#55612yes—
471C2×C4#55712yes—
472C8—24no—
473C8—28no—
474C8—42no—
475C8—44no—
476C8—82no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—42no—
484C22—42no—
485C22C2×#39411yescharacteristic
486C22#244612yes—
487C22#254511yescharacteristic
488C22#254611yescharacteristic
489C22#256812yes—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—84no—
497C4—82no—
498C4—82no—
499C4—162no—
500C4#204812yes—
501C4#204912yes—
502C4#240912yes—
503C4#244612yes—
504C4#254511yescharacteristic
505C4#254611yescharacteristic
506C4#262111yescharacteristic
507C4#285711yescharacteristic
508C2—21no—
509C2—22no—
510C2—22no—
511C2—21no—
512C2#1518011yescharacteristic
513C2C2×#204811yescharacteristic
514C2C2×#204911yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
90. #430Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#213372
C2×#19972
#213242
#213062
C4×#4302
#112742
#112532
#108552