Finite Group Databasebeta

← #415127

Subgroups — #415127

Order 512.

1190 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3C2×#1939C211yesmaximal, characteristic
4C2×C4×#106C211yesmaximal, characteristic
5#14161C211yesmaximal, characteristic
6#13910C211yesmaximal, characteristic
7#11227C214yesmaximal
8#11220C214yesmaximal
9#10849C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#106C2212yes—
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4×C8C2211yescharacteristic
15C2×#389C2214yes—
16C2×#384—21no—
17C2×#384C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1990—24no—
22#1987—24no—
23#1939—24no—
24C4×#106—24no—
25C2×#340C2214yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2214yes—
30C2×#336C2212yes—
31#1446—22no—
32#1444—22no—
33C4⋊#95—21no—
34C4⋊#95—21no—
35C4⋊#95—21no—
36C4⋊#95C2214yes—
37C23×Q8—21no—
38C23×C8C2312yes—
39C23×C8C2311yescharacteristic
40C23×C8C2311yescharacteristic
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8Q814yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42—21no—
58C22×C42—21no—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×#106—24no—
62C2×#106—24no—
63C2×#106Q818yescomplemented
64C2×C4⋊C8—24no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—44no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8D414yes—
72#389—28no—
73#389—44no—
74#384—22no—
75#384—22no—
76#384—44no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4D414yes—
83C2×#95—21no—
84C2×#95—24no—
85C2×#95—21no—
86C2×#95—24no—
87C2×#95—24no—
88C2×#95D412yes—
89#340—24no—
90#340—24no—
91#340—28no—
92#340—24no—
93#340—44no—
94#336—24no—
95C22×Q8—24no—
96C22×Q8—42no—
97C22×Q8—42no—
98C22×Q8—82no—
99C22×Q8—82no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—24no—
120C22×C8—24no—
121C22×C8C2×D412yes—
122C22×C8C2×D412yes—
123C22×C8C2×Q814yes—
124C4⋊Q8—28no—
125C4⋊Q8—28no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—48no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—84no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4SD1614yes—
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2×Q814yes—
164C2×C4⋊C4C2×Q814yes—
165C2×C4⋊C4C2×Q814yes—
166C2×C4⋊C4#4114yes—
167C2×C42—24no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×Q812yes—
180C2×C42#4112yes—
181C2×C42#4112yes—
182C2×C42#4112yes—
183#106—216no—
184C4⋊C8—48no—
185C4⋊C8—44no—
186C4×C8—28no—
187#95—24no—
188#95—24no—
189#95—48no—
190#95—42no—
191#95—84no—
192#95—84no—
193C24C22×D411yescharacteristic
194C2×Q8—48no—
195C2×Q8—88no—
196C2×Q8—88no—
197C2×Q8—84no—
198C2×Q8—84no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4#12112yes—
235C22×C4#12112yes—
236C22×C4#12112yes—
237C22×C4#12112yes—
238C22×C4C22⋊Q812yes—
239C22×C4C22⋊Q812yes—
240C22×C4C2×SD1614yes—
241C22×C4C2×SD1612yes—
242C22×C4#13512yes—
243C22×C4#13712yes—
244C22×C4#13712yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×Q811yescharacteristic
248C22×C4C22×Q812yes—
249C22×C4C2×#4111yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4112yes—
253C22×C4H(5,𝔽2)12yes—
254C22×C42−1+412yes—
255C2×C8—22no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—44no—
268C2×C8—44no—
269C2×C8C4⋊Q814yes—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—28no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—48no—
283C4⋊C4—44no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—816no—
288C4⋊C4—84no—
289C4⋊C4—84no—
290C4⋊C4C22⋊Q818yes—
291C42—42no—
292C42—42no—
293C42—44no—
294C42—48no—
295C42—84no—
296C42#12114yes—
297C42C22⋊Q814yes—
298C42C22⋊Q814yes—
299C42#12614yes—
300C23C2×#12111yescharacteristic
301C23C2×#12111yescharacteristic
302C23C2×C22⋊Q811yescharacteristic
303C23#54612yes—
304C23D4×Q812yes—
305C23C22×SD1611yescharacteristic
306C23C2×#13511yescharacteristic
307C23C2×#13711yescharacteristic
308C23C2×#13711yescharacteristic
309C23#57612yes—
310C23#57712yes—
311Q8—816no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—28no—
317C2×C4—28no—
318C2×C4—24no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—84no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—84no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4#39614yes—
351C2×C4#39614yes—
352C2×C4#46114yes—
353C2×C4#46414yes—
354C2×C4#46514yes—
355C2×C4#47014yes—
356C2×C4#47514yes—
357C2×C4#47914yes—
358C2×C4C2×#12111yescharacteristic
359C2×C4C2×C22⋊Q811yescharacteristic
360C2×C4C2×C22⋊Q811yescharacteristic
361C2×C4C2×C22⋊Q812yes—
362C2×C4C2×#12611yescharacteristic
363C2×C4C2×C4⋊Q811yescharacteristic
364C2×C4#53211yescharacteristic
365C2×C4#53312yes—
366C2×C4#53511yescharacteristic
367C2×C4#53611yescharacteristic
368C2×C4#53712yes—
369C2×C4#54212yes—
370C2×C4#55212yes—
371C2×C4#55412yes—
372C2×C4#55712yes—
373C2×C4#55712yes—
374C2×C4Q8212yes—
375C2×C4#55912yes—
376C8—24no—
377C8—42no—
378C8—82no—
379C22C2×#39611yescharacteristic
380C22#241712yes—
381C22#245112yes—
382C22#254411yescharacteristic
383C22#254611yescharacteristic
384C22C2×#46111yescharacteristic
385C22C2×#46411yescharacteristic
386C22#304512yes—
387C22C2×#46511yescharacteristic
388C22C2×#47011yescharacteristic
389C22#305711yescharacteristic
390C22#305811yescharacteristic
391C22C2×#47511yescharacteristic
392C22C2×#47911yescharacteristic
393C22#308112yes—
394C22#329512yes—
395C22#330512yes—
396C22#332912yes—
397C22#333112yes—
398C22Q8×SD1612yes—
399C22#338112yes—
400C22#338512yes—
401C22#338512yes—
402C4—24no—
403C4—44no—
404C4—48no—
405C4—84no—
406C4—82no—
407C4—84no—
408C4—162no—
409C4#206618yes—
410C4#251112yes—
411C4#256511yescharacteristic
412C4#256711yescharacteristic
413C4#258212yes—
414C4#262011yescharacteristic
415C4#285711yescharacteristic
416C2#1518411yescharacteristic
417C2C2×#206611yescharacteristic
418C2#2361211yescharacteristic
419C2#2481412yes—
420C2#2481512yes—
421C2#2644212yes—
422C2#2644512yes—
423C2#2859911yescharacteristic
424C2#2860811yescharacteristic
425C2#2867711yescharacteristic
426C2#2868211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
63. C2×#106Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19902
C2×#19872
C2×#19392
C2×C4×#1062
#141612
#139102
#112272
#112202
#108492