Finite Group Databasebeta

← #368937

Subgroups — #368937

Order 512.

1260 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#21332C211yesmaximal, characteristic
4#21324C211yesmaximal, characteristic
5#21306C211yesmaximal, characteristic
6#11213C214yesmaximal
7#10903C214yesmaximal
8#10837C211yesmaximal, characteristic
9C4×#344C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3165—22no—
13#3165—22no—
14#3165—22no—
15#3165—22no—
16C2×#501C414yescomplemented
17C2×#500C412yescomplemented
18C2×#498C412yescomplemented
19C2×#428C2214yes—
20C2×C4×M16C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2369—22no—
23#2365—22no—
24#2345—22no—
25Q8×C22⋊C4—22no—
26#2331—22no—
27#2316C2212yes—
28C4×C4⋊Q8—22no—
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31#2274C2212yes—
32C4×#117C2211yescharacteristic
33#1996—22no—
34#1996—22no—
35#1993—22no—
36#1990—22no—
37#1986—22no—
38#1986—22no—
39C2×#344C2212yes—
40C2×#344C2212yes—
41#1740C2212yes—
42#1740C2212yes—
43#1736C2214yes—
44#1733C2214yes—
45C2×H(3,ℤ/4)C2214yes—
46C23×Q8—21no—
47C22×M16C2312yes—
48C22×M16C2312yes—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2×C412yes—
66C2×C4⋊Q8C2×C412yes—
67C2×C4⋊Q8D412yes—
68C2×C4⋊Q8D412yes—
69C2×C4⋊Q8D412yes—
70C2×C4⋊Q8D412yes—
71C2×#125C2×C412yes—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C4×Q8—24no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×#117C2312yes—
78C2×#117C2312yes—
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2312yes—
82C22×C4⋊C4—21no—
83C22×C4⋊C4C2312yes—
84C22×C42C2311yescharacteristic
85#501—28no—
86#501—28no—
87#500—24no—
88#500—24no—
89#498—24no—
90#498—24no—
91#428—28no—
92#428—28no—
93C2×#107C2×C418yes—
94C2×#106C2×C418yes—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16—21no—
102C4×M16—21no—
103C2×#97—21no—
104C2×#97C2×C412yes—
105C2×C4×C8—21no—
106C2×C4×C8C2×C412yes—
107#391—24no—
108#391—28no—
109#389—24no—
110#389—24no—
111#389—24no—
112#387—28no—
113#384—21no—
114#384—21no—
115#382—21no—
116#382—21no—
117#380—21no—
118#380—21no—
119#380—24no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D412yes—
131C4×C22⋊C4—24no—
132C4×C22⋊C4D412yes—
133C4×C22⋊C4D412yes—
134#376—21no—
135C43—21no—
136#344—24no—
137#344Q818yescomplemented
138#342—24no—
139#342—24no—
140#340—28no—
141#339—28no—
142H(3,ℤ/4)—28no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—84no—
152C23×C4—22no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16C2×Q814yes—
166C2×M16C2×Q814yes—
167C2×M16#4114yes—
168C22×C8—22no—
169C22×C8C22×C414yes—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8C22⋊C414yes—
191C4⋊Q8C22⋊C414yes—
192C4⋊Q8C22⋊C414yes—
193C4⋊Q8C22⋊C414yes—
194#125—22no—
195#125—22no—
196#125—22no—
197#125—22no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—88no—
215#117—22no—
216#117—24no—
217#117—24no—
218#117—22no—
219#117—22no—
220#117—22no—
221#117—22no—
222#117—22no—
223#117—22no—
224#117—24no—
225#117—24no—
226#117C2×Q814yes—
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—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4C2×D412yes—
258C2×C22⋊C4C2×D412yes—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—28no—
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C2×D414yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42#4111yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42#4111yescharacteristic
279#107—216no—
280#106—216no—
281#97—22no—
282#97—22no—
283#97—41no—
284#97—41no—
285C4×C8—22no—
286C4×C8—22no—
287C4×C8—41no—
288C4×C8—41no—
289#95—28no—
290#95—28no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—41no—
296#95—41no—
297C24C22×D411yescharacteristic
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—48no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×Q8—84no—
312C2×Q8—84no—
313C2×Q8—84no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—84no—
326C22×C4C4×D411yescharacteristic
327C22×C4C4×D411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4#12011yescharacteristic
331C22×C4#12011yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12111yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×Q812yes—
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4C2×#4111yescharacteristic
346C22×C42−1+412yes—
347C22×C42−1+412yes—
348M16—24no—
349M16—24no—
350M16—24no—
351M16—24no—
352M16—24no—
353M16—24no—
354M16—42no—
355M16—42no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8C4×Q818yescomplemented
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—44no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—88no—
390C4⋊C4C2×C22⋊C418yes—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4C4⋊Q814yes—
401C42—24no—
402C42—28no—
403C42—44no—
404C42—44no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—44no—
409C42—88no—
410C42C2×C22⋊C411yescharacteristic
411C42C2×C22⋊C411yescharacteristic
412C42C2×C22⋊C411yescharacteristic
413C42C2×C22⋊C411yescharacteristic
414C42C4×D411yescharacteristic
415C42C4×D411yescharacteristic
416C42C4×D411yescharacteristic
417C42C4×D411yescharacteristic
418C42#12111yescharacteristic
419C42#12111yescharacteristic
420C42#12111yescharacteristic
421C42#12111yescharacteristic
422C42#12311yescharacteristic
423C42#12311yescharacteristic
424C42#12311yescharacteristic
425C42#12311yescharacteristic
426C23—41no—
427C23—41no—
428C23C2×C4×D411yescharacteristic
429C23C2×#12011yescharacteristic
430C23C2×#12111yescharacteristic
431C23C2×C4⋊Q811yescharacteristic
432C23#53211yescharacteristic
433C23#53611yescharacteristic
434C23#53611yescharacteristic
435Q8—48no—
436Q8—48no—
437Q8—84no—
438Q8—84no—
439Q8—82no—
440Q8—82no—
441Q8—82no—
442Q8—82no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4#38611yescharacteristic
476C2×C4#38611yescharacteristic
477C2×C4#38611yescharacteristic
478C2×C4#38611yescharacteristic
479C2×C4#38611yescharacteristic
480C2×C4#38611yescharacteristic
481C2×C4#38611yescharacteristic
482C2×C4#38611yescharacteristic
483C2×C4C22×C22⋊C411yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×C4×Q812yes—
486C2×C4#51912yes—
487C2×C4C2×#12111yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4#53312yes—
490C2×C4#54112yes—
491C2×C4#54712yes—
492C2×C4#54812yes—
493C2×C4#54812yes—
494C2×C4D4×Q812yes—
495C2×C4D4×Q812yes—
496C2×C4D4×Q812yes—
497C2×C4#55612yes—
498C2×C4#55712yes—
499C8—28no—
500C8—44no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22C2×#38611yescharacteristic
512C22C4×C4⋊Q811yescharacteristic
513C22#232711yescharacteristic
514C22#243211yescharacteristic
515C22#243411yescharacteristic
516C22#254511yescharacteristic
517C22#254611yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—44no—
525C4—48no—
526C4—82no—
527C4—84no—
528C4—82no—
529C4—82no—
530C4—162no—
531C4#190412yes—
532C4#190512yes—
533C4Q8×C22⋊C412yes—
534C4#236512yes—
535C4#242912yes—
536C4#244612yes—
537C2—21no—
538C2—21no—
539C2—21no—
540C2—21no—
541C2—21no—
542C2—21no—
543C2#1402411yescharacteristic
544C2C2×#190411yescharacteristic
545C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×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
16. C2×#501C4semidirect
17. C2×#500C4semidirect
18. C2×#498C4semidirect
137. #344Q8semidirect
361. C2×C8C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#342162
#213322
#213242
#213062
#112132
#109032
#108372
C4×#3442