Finite Group Databasebeta

← #415322

Subgroups — #415322

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3C2×#2584C211yesmaximal, characteristic
4C2×#1989C211yesmaximal, characteristic
5C2×#1987C211yesmaximal, characteristic
6C2×#1985C211yesmaximal, characteristic
7#11328C214yesmaximal
8#11327C214yesmaximal
9#10703C211yesmaximal, characteristic
10C22×C4⋊Q8—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#501—24no—
15C2×#501—24no—
16C2×#501—24no—
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#97C2211yescharacteristic
21#2584—24no—
22#2584—24no—
23#2546—24no—
24#2546—24no—
25C2×#398—21no—
26C2×#398—22no—
27C2×#395—21no—
28C2×#391—21no—
29C2×#389—22no—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#389C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1989—24no—
36#1987—24no—
37#1985—24no—
38C2×#340C2214yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2214yes—
42C2×#336C2214yes—
43C23×Q8—21no—
44C23×Q8D411yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—28no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—48no—
54C2×C4⋊Q8D414yes—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125D414yes—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#501—232no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×#97—22no—
95C2×#97—22no—
96C2×#97D414yes—
97#398—28no—
98#398—216no—
99#395—28no—
100#391—28no—
101#389—216no—
102#389—24no—
103#389—24no—
104#389—24no—
105#389—24no—
106#389—28no—
107#389—28no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4D414yes—
111C4×C4⋊C4D414yes—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—24no—
117#340—28no—
118#340—24no—
119#340—24no—
120#340Q8116yescomplemented
121#336—28no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—24no—
125C22×Q8—22no—
126C22×Q8—24no—
127C22×Q8—42no—
128C22×Q8—44no—
129C22×Q8—44no—
130C22×Q8—42no—
131C22×Q8—42no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—24no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—28no—
149C22×C8C2×D414yes—
150C22×C8C2×Q818yes—
151C4⋊Q8—28no—
152C4⋊Q8—28no—
153C4⋊Q8—432no—
154C4⋊Q8—44no—
155C4⋊Q8—44no—
156#125—28no—
157#125—28no—
158#125—44no—
159#125—44no—
160C4×Q8—44no—
161C4×Q8—44no—
162C4×Q8—44no—
163C4×Q8—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—216no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—84no—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×Q818yes—
212C2×C4⋊C4C2×Q818yes—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—44no—
222C2×C42—82no—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42#4112yes—
229#107—216no—
230#106—216no—
231C4⋊C8—44no—
232#97—28no—
233#95—28no—
234#95—28no—
235#95—216no—
236#95—24no—
237#95—24no—
238C24C22×D411yescharacteristic
239C2×Q8—28no—
240C2×Q8—48no—
241C2×Q8—48no—
242C2×Q8—44no—
243C2×Q8—48no—
244C2×Q8—48no—
245C2×Q8—44no—
246C2×Q8—48no—
247C2×Q8—88no—
248C2×Q8—88no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4#12012yes—
290C22×C4#12012yes—
291C22×C4#12012yes—
292C22×C4#12012yes—
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4AGL(1,ℤ/8)14yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×Q814yes—
302C22×C4C2×#4111yescharacteristic
303C22×C42−1+414yes—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—28no—
307C2×C8—28no—
308C2×C8—42no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8C4⋊Q818yescomplemented
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—416no—
324C4⋊C4—432no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—48no—
332C4⋊C4—44no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—48no—
336C4⋊C4—88no—
337C4⋊C4C4⋊Q818yes—
338C4⋊C4C4⋊Q818yescomplemented
339C42—48no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—84no—
345C42#12014yes—
346C42#12114yes—
347C42#12114yes—
348C42#12714yes—
349C23C2×#12011yescharacteristic
350C23C2×#12011yescharacteristic
351C23C2×#12111yescharacteristic
352C23D4×Q814yes—
353C23C2×AGL(1,ℤ/8)12yes—
354C23#57512yes—
355C23#57712yes—
356C23#57812yes—
357Q8—416no—
358Q8—816no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—84no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—84no—
403C2×C4—82no—
404C2×C4#39214yes—
405C2×C4#39214yes—
406C2×C4C2×#12011yescharacteristic
407C2×C4C2×#12111yescharacteristic
408C2×C4C2×#12111yescharacteristic
409C2×C4C2×#12711yescharacteristic
410C2×C4C2×C4⋊Q812yes—
411C2×C4C2×C4⋊Q812yes—
412C2×C4C2×C4⋊Q812yes—
413C2×C4#53612yes—
414C2×C4#53612yes—
415C2×C4#53612yes—
416C2×C4D4×Q814yes—
417C2×C4D4×Q814yes—
418C2×C4#55714yes—
419C8—28no—
420C8—82no—
421C22C2×#39211yescharacteristic
422C22#243412yes—
423C22#243412yes—
424C22#254612yes—
425C22#300311yescharacteristic
426C22#300511yescharacteristic
427C22#300612yes—
428C22#300911yescharacteristic
429C22#301111yescharacteristic
430C22#301312yes—
431C22#305512yes—
432C22#306312yes—
433C22#338414yes—
434C22#338614yes—
435C22#338814yes—
436C22#339014yes—
437C4—24no—
438C4—24no—
439C4—44no—
440C4—48no—
441C4—48no—
442C4—82no—
443C4—84no—
444C4—82no—
445C4—82no—
446C4#243412yes—
447C4#254612yes—
448C4#254612yes—
449C4#265811yescharacteristic
450C4#286811yescharacteristic
451C2#1519011yescharacteristic
452C2#2339111yescharacteristic
453C2#2339411yescharacteristic
454C2#2584412yes—
455C2#2584612yes—
456C2#2584912yes—
457C2#2585112yes—
458C2#2868514yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
120. #340Q8semidirect
312. C2×C8C4⋊Q8semidirect
338. C4⋊C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#25462
C2×#25842
C2×#19892
C2×#19872
C2×#19852
#113282
#113272
#107032