Finite Group Databasebeta

← #200986

Subgroups — #200986

Order 512.

1049 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic
3#17751C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#9434C218yesmaximal, complemented
7#8063C211yesmaximal, characteristic
8#8060C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3165—22no—
12#3165—22no—
13#3165C414yes—
14#3165C2214yes—
15C2×#501C414yes—
16C2×#500—21no—
17C2×#498—21no—
18#2986C2214yes—
19#2985—22no—
20#2984—22no—
21#2983—21no—
22#2982—21no—
23#2962—22no—
24#2960—22no—
25#2924—22no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#428—22no—
30C2×#423C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#1561C2214yes—
33#1556—22no—
34#1552—22no—
35#1549—22no—
36#1549—21no—
37#1548—21no—
38#1548—21no—
39#1548C2214yes—
40#1546—21no—
41#1546C2214yes—
42C2×#333—22no—
43#1299C2214yes—
44#1299C2214yes—
45C23×Q8—21no—
46C22×M16—21no—
47C22×M16—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533C2×C412yes—
53#533D412yes—
54#533D412yes—
55#533C2312yes—
56C2×C4⋊Q8—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⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8C2×C412yes—
66C2×C4⋊Q8C2×C412yes—
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8D411yescharacteristic
69C2×C4⋊Q8C2312yes—
70C2×#125C2×C412yes—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×#117D412yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C42C2311yescharacteristic
79#501—28no—
80#501C2×C4116yes—
81#500—24no—
82#500—24no—
83#498—24no—
84#498—24no—
85#446—24no—
86#446—28no—
87#446—24no—
88#446—24no—
89C8×Q8—24no—
90C8×Q8—24no—
91C8×Q8—24no—
92#433—28no—
93#432—24no—
94#432—21no—
95#432—21no—
96#432—21no—
97#432D412yes—
98#431—24no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431D411yescharacteristic
105#431D411yescharacteristic
106#431C2312yes—
107#431C2312yes—
108#428—28no—
109#428—28no—
110C2×#107—24no—
111C2×#106—24no—
112#424—24no—
113#423—21no—
114#423—21no—
115#423—21no—
116#423D412yes—
117#423C2312yes—
118#423C2312yes—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8D412yes—
122#407—21no—
123#407—21no—
124C2×C22⋊C8—21no—
125#405—24no—
126C4×M16—21no—
127C4×M16—21no—
128C4×M16—21no—
129C4×M16—21no—
130C4×M16C2×C412yes—
131C4×M16C2312yes—
132C2×#97C2×C412yes—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135#333—24no—
136#333—24no—
137#332—24no—
138#332—24no—
139#332—28no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—41no—
146C22×Q8—41no—
147C22×Q8—41no—
148C22×Q8—84no—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151C23×C4#4111yescharacteristic
152C23×C4#4111yescharacteristic
153C2×M16—28no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—41no—
160C2×M16—82no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—41no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182C4⋊Q8—41no—
183C4⋊Q8C22⋊C412yes—
184C4⋊Q8C22⋊C412yes—
185C4⋊Q8C22×C414yes—
186C4⋊Q8C2×D412yes—
187C4⋊Q8C2×D412yes—
188#125—22no—
189#125—22no—
190#125C22⋊C414yes—
191#125C22×C414yes—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208#117—22no—
209#117—22no—
210#117C22⋊C414yes—
211#117C2×D414yes—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C412yes—
230C2×C22⋊C4—22no—
231C2×C42—22no—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42C2411yescharacteristic
244#107—216no—
245#106—216no—
246C4⋊C8—28no—
247C4⋊C8—28no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8C2×D414yes—
257C4⋊C8C2×D412yes—
258C4⋊C8C2×D412yes—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—41no—
265C22⋊C8—41no—
266#97—28no—
267#97—22no—
268#97—22no—
269#97—41no—
270#97C22×C414yes—
271C4×C8—28no—
272C4×C8—22no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—22no—
276C4×C8—22no—
277C4×C8—22no—
278C4×C8—41no—
279C24#12311yescharacteristic
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—82no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×Q8—84no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—84no—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12311yescharacteristic
320C22×C4#12311yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325M16—28no—
326M16—28no—
327M16—44no—
328M16—82no—
329M16—82no—
330M16—82no—
331C2×C8—28no—
332C2×C8—24no—
333C2×C8—22no—
334C2×C8—24no—
335C2×C8—22no—
336C2×C8—44no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—82no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4C2×C22⋊C414yes—
359C4⋊C4C2×C22⋊C414yes—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C42—24no—
363C42—24no—
364C42C2×C22⋊C411yescharacteristic
365C42C2×C22⋊C411yescharacteristic
366C42C4×D411yescharacteristic
367C42C4×D411yescharacteristic
368C42C4×D411yescharacteristic
369C42C4×D411yescharacteristic
370C42#12011yescharacteristic
371C42#12011yescharacteristic
372C42#12111yescharacteristic
373C42#12111yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42C23×C411yescharacteristic
377C42C22×D411yescharacteristic
378C42C22×D411yescharacteristic
379C42C22×D411yescharacteristic
380C23—22no—
381C23—22no—
382C23—41no—
383C23#38611yescharacteristic
384C23#38611yescharacteristic
385C23C2×#12311yescharacteristic
386Q8—44no—
387Q8—42no—
388Q8—42no—
389Q8—44no—
390Q8—82no—
391Q8—84no—
392Q8—84no—
393Q8—84no—
394Q8—84no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—82no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4#38611yescharacteristic
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38611yescharacteristic
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#45614yes—
428C2×C4#47112yes—
429C2×C4#47312yes—
430C2×C4C22×C22⋊C411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×#12011yescharacteristic
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C8—28no—
437C8—42no—
438C8—42no—
439C8—44no—
440C8—82no—
441C8—82no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22#188612yes—
451C22#190812yes—
452C22C2×#38611yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—44no—
461C4—44no—
462C4—84no—
463C4#188612yes—
464C4#190812yes—
465C4C2×#45612yes—
466C4C2×#47111yescharacteristic
467C4C2×#47311yescharacteristic
468C4#306812yes—
469C2—22no—
470C2—22no—
471C2—41no—
472C2C2×#188611yescharacteristic
473C2C2×#190811yescharacteristic
474C2#2100511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#123
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9434C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#602502
#177512
#177422
C2×#15482
#94342
#80632
#80602