Finite Group Databasebeta

← #200782

Subgroups — #200782

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#60249C211yesmaximal, characteristic
3#17746C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5#17664C211yesmaximal, characteristic
6#9420C218yesmaximal, complemented
7#7992C211yesmaximal, characteristic
8#7992C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3165—22no—
12#3165—22no—
13#3165C414yes—
14#3165C2214yes—
15C2×#501—22no—
16C2×#500C412yes—
17C2×#498C412yes—
18#2987C2214yes—
19#2986—22no—
20#2985—22no—
21#2985—22no—
22Q8×M16—22no—
23Q8×M16—22no—
24#2921—22no—
25#2920C2211yescharacteristic
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#428—22no—
30C2×C4×M16C2211yescharacteristic
31#1558C2214yes—
32#1556C2214yes—
33#1549—22no—
34#1548—22no—
35#1548—22no—
36#1546—22no—
37#1537C2214yes—
38#1534—22no—
39#1526—22no—
40#1291C2214yes—
41#1291C2214yes—
42C23×Q8—21no—
43C22×M16—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533D412yes—
50#533D412yes—
51#533C2312yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2312yes—
66C2×#125D412yes—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×#117D412yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#501—28no—
76#501—28no—
77#500—24no—
78#500C2×C418yes—
79#498—24no—
80#498C2×C418yes—
81#446—24no—
82#446—24no—
83#446—28no—
84#446—28no—
85C8×Q8—24no—
86C8×Q8—24no—
87#433—28no—
88#433—28no—
89#432—21no—
90#432—21no—
91#432—21no—
92#432—21no—
93#432C2312yes—
94#432C2312yes—
95#432C2312yes—
96#432C2312yes—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431D412yes—
104#431D412yes—
105#428—28no—
106#428—28no—
107C2×#107—24no—
108C2×#106—24no—
109#423—24no—
110#423—21no—
111#423—21no—
112#423D412yes—
113#423D412yes—
114#407—21no—
115#407—21no—
116#407—21no—
117#407—21no—
118C4×M16—24no—
119C4×M16—21no—
120C4×M16—21no—
121C4×M16—21no—
122C4×M16—21no—
123C4×M16—21no—
124C4×M16—21no—
125C4×M16C2×C412yes—
126C4×M16C2312yes—
127C2×#97—21no—
128C2×C4×C8C2×C412yes—
129#333—28no—
130#332—28no—
131#330—28no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—41no—
140C22×Q8—84no—
141C23×C4—41no—
142C23×C4C2×D411yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4#4111yescharacteristic
145C2×M16—28no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—41no—
151C2×M16—41no—
152C22×C8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8C22⋊C412yes—
173C4⋊Q8C22⋊C412yes—
174C4⋊Q8C22×C414yes—
175C4⋊Q8C22×C412yes—
176C4⋊Q8C22×C412yes—
177#125—22no—
178#125—22no—
179#125C22⋊C414yes—
180#125C2×D414yes—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197#117—22no—
198#117—22no—
199#117C22⋊C414yes—
200#117C2×D414yes—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C22⋊C4—22no—
220C2×C42—22no—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233#107—216no—
234#106—216no—
235C4⋊C8—28no—
236C4⋊C8—28no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—41no—
242C4⋊C8—41no—
243C4⋊C8C2×D414yes—
244C4⋊C8C2×D414yes—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—41no—
252C22⋊C8—41no—
253#97—28no—
254#97—22no—
255#97—22no—
256#97—22no—
257#97—22no—
258#97—22no—
259#97—22no—
260C4×C8—28no—
261C4×C8—22no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—41no—
265C4×C8—41no—
266C4×C8C22×C414yes—
267C24#12311yescharacteristic
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—82no—
277C2×Q8—82no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8—84no—
283C2×Q8—84no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—84no—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12311yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311M16—28no—
312M16—28no—
313M16—44no—
314M16—44no—
315M16—82no—
316M16—82no—
317C2×C8—28no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4C2×C22⋊C414yes—
342C4⋊C4C2×C22⋊C414yes—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C42—24no—
346C42—24no—
347C42C2×C22⋊C411yescharacteristic
348C42C2×C22⋊C411yescharacteristic
349C42C4×D411yescharacteristic
350C42C4×D411yescharacteristic
351C42C4×D411yescharacteristic
352C42C4×D411yescharacteristic
353C42#12011yescharacteristic
354C42#12011yescharacteristic
355C42#12111yescharacteristic
356C42#12111yescharacteristic
357C42#12111yescharacteristic
358C42#12111yescharacteristic
359C42C23×C411yescharacteristic
360C42C22×D411yescharacteristic
361C42C22×D411yescharacteristic
362C42C22×D411yescharacteristic
363C23—22no—
364C23—41no—
365C23—41no—
366C23#38611yescharacteristic
367C23#38611yescharacteristic
368C23C2×#12311yescharacteristic
369Q8—44no—
370Q8—44no—
371Q8—82no—
372Q8—82no—
373Q8—84no—
374Q8—84no—
375Q8—84no—
376Q8—84no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4#38611yescharacteristic
403C2×C4#38611yescharacteristic
404C2×C4#38611yescharacteristic
405C2×C4#38611yescharacteristic
406C2×C4#38611yescharacteristic
407C2×C4#38611yescharacteristic
408C2×C4#45614yes—
409C2×C4C22×C22⋊C411yescharacteristic
410C2×C4C2×C4×D411yescharacteristic
411C2×C4C2×C4×D411yescharacteristic
412C2×C4C2×#12011yescharacteristic
413C2×C4C2×#12111yescharacteristic
414C2×C4C2×#12111yescharacteristic
415C8—28no—
416C8—44no—
417C8—44no—
418C8—82no—
419C8—82no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—44no—
429C22#188612yes—
430C22C2×#38611yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—44no—
440C4—44no—
441C4—84no—
442C4#188612yes—
443C4#190212yes—
444C4#190312yes—
445C4C2×#45612yes—
446C2—22no—
447C2—41no—
448C2—41no—
449C2C2×#188611yescharacteristic
450C2C2×#190211yescharacteristic
451C2C2×#190311yescharacteristic
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. #9420C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#602492
#177462
#177422
#176642
#94202
#79922