Finite Group Databasebeta

← #200854

Subgroups — #200854

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#34371C211yesmaximal, characteristic
3C2×#1556C211yesmaximal, characteristic
4#17596C211yesmaximal, characteristic
5#17596C211yesmaximal, characteristic
6#9427C218yesmaximal, complemented
7#7972C211yesmaximal, characteristic
8#7972C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3165—22no—
12#3165—22no—
13#3165C414yescomplemented
14#3165C2214yes—
15C2×#501—22no—
16C2×#500C412yescomplemented
17C2×#498C412yescomplemented
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#428—22no—
21C2×C4×M16C2211yescharacteristic
22#2381C2214yes—
23#2365—22no—
24#2345—22no—
25Q8×C22⋊C4—22no—
26#2331—22no—
27#2327—22no—
28#2313—22no—
29#2283C2211yescharacteristic
30#2283C2211yescharacteristic
31#1556—22no—
32#1556C2214yes—
33C2×#332—22no—
34#1520—22no—
35#1520—22no—
36#1515C2214yes—
37#1515C2214yes—
38#1506—22no—
39#1506—22no—
40#1288C2214yes—
41#1288C2214yes—
42C23×Q8—21no—
43C22×M16—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533C2×C412yes—
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⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8C2312yes—
66C2×#125—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C4×Q8—24no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×#117—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D412yes—
75C22×C42C2311yescharacteristic
76#501—28no—
77#501—28no—
78#500—24no—
79#500C2×C418yes—
80#498—24no—
81#498C2×C418yes—
82#432—21no—
83#432—21no—
84#432—21no—
85#432—21no—
86#432—21no—
87#432—21no—
88#432C2312yes—
89#432C2312yes—
90#428—28no—
91#428—28no—
92C2×#107—24no—
93C2×#106—24no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C4×M16—21no—
99C4×M16—21no—
100C4×M16C2×C412yes—
101C4×M16C2312yes—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8C2×C412yes—
106#391—24no—
107#391—28no—
108#389—24no—
109#387—28no—
110#387—28no—
111#387—28no—
112#383C2312yes—
113#383C2312yes—
114#382—21no—
115#382—21no—
116#381—21no—
117#381—21no—
118#380—24no—
119#380D412yes—
120#380D412yes—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C22⋊C4—24no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127#376D412yes—
128#376D412yes—
129#332—28no—
130#328—28no—
131#328—28no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—84no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C2×M16—28no—
146C2×M16—22no—
147C2×M16—22no—
148C22×C8—22no—
149C22×C8—41no—
150C22×C8—41no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8C22⋊C414yes—
170C4⋊Q8C22⋊C414yes—
171C4⋊Q8C22×C412yes—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C22×C414yes—
174C4⋊Q8C2×D414yes—
175#125—22no—
176#125—22no—
177#125—41no—
178#125—41no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—88no—
196#117—22no—
197#117—22no—
198#117—41no—
199#117—41no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
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—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4C22⋊C414yes—
219C2×C4⋊C4C22⋊C414yes—
220C2×C4⋊C4C2×D414yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—41no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—28no—
227C2×C42—41no—
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42C2411yescharacteristic
240#107—216no—
241#106—216no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—41no—
249C22⋊C8—41no—
250#97—22no—
251#97—22no—
252#97—22no—
253#97—22no—
254#97—41no—
255#97—41no—
256C4×C8—22no—
257C4×C8C22×C414yes—
258#95—28no—
259#95—28no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95C2×D414yes—
267#95C2×D414yes—
268C24#12011yescharacteristic
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—48no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—82no—
281C2×Q8—82no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—84no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—84no—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—28no—
315M16—28no—
316C2×C8—28no—
317C2×C8—24no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—82no—
321C2×C8—82no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
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—44no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—88no—
344C4⋊C4C2×C22⋊C418yes—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C42—44no—
350C42—44no—
351C42—42no—
352C42—42no—
353C42—88no—
354C42C2×C22⋊C411yescharacteristic
355C42C2×C22⋊C411yescharacteristic
356C42C4×D411yescharacteristic
357C42C4×D411yescharacteristic
358C42C4×D411yescharacteristic
359C42C4×D411yescharacteristic
360C42#12111yescharacteristic
361C42#12111yescharacteristic
362C42#12111yescharacteristic
363C42#12111yescharacteristic
364C42#12311yescharacteristic
365C42#12311yescharacteristic
366C42C23×C411yescharacteristic
367C42C22×D411yescharacteristic
368C42C2×#4111yescharacteristic
369C42C2×#4111yescharacteristic
370C23—41no—
371C23—41no—
372C23#38611yescharacteristic
373C23#38611yescharacteristic
374C23#45614yes—
375C23C2×#12011yescharacteristic
376Q8—48no—
377Q8—48no—
378Q8—84no—
379Q8—84no—
380Q8—82no—
381Q8—82no—
382Q8—82no—
383Q8—82no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—28no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—48no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4#38611yescharacteristic
417C2×C4#38611yescharacteristic
418C2×C4#38611yescharacteristic
419C2×C4#38611yescharacteristic
420C2×C4#38611yescharacteristic
421C2×C4#38611yescharacteristic
422C2×C4C22×C22⋊C411yescharacteristic
423C2×C4C2×C4×D411yescharacteristic
424C2×C4C2×C4×D411yescharacteristic
425C2×C4C2×#12111yescharacteristic
426C2×C4C2×#12111yescharacteristic
427C2×C4C2×#12311yescharacteristic
428C8—28no—
429C8—82no—
430C8—82no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—44no—
438C22#188612yes—
439C22#188612yes—
440C22C2×#38611yescharacteristic
441C22C2×#45612yes—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—48no—
453C4—84no—
454C4—84no—
455C4—82no—
456C4—82no—
457C4#190412yes—
458C4#190512yes—
459C2—22no—
460C2—22no—
461C2—22no—
462C2C2×#188611yescharacteristic
463C2C2×#190411yescharacteristic
464C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #9427C2semidirect
13. #3165C4semidirect
16. C2×#500C4semidirect
17. C2×#498C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#343712
C2×#15562
#175962
#94272
#79722