Finite Group Databasebeta

← #414467

Subgroups — #414467

Order 512.

1112 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2083C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4C2×#1941C211yesmaximal, characteristic
5#15242C211yesmaximal, characteristic
6#15230C211yesmaximal, characteristic
7#11325C212yesmaximal
8#11325C212yesmaximal
9#10862C212yesmaximal
10#10842C212yesmaximal
11C8×#95C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#107C2212yes—
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C22×C4×C8C2212yes—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23C2×#382C2212yes—
24#2083—24no—
25#2057—24no—
26#1990—24no—
27#1941—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1446—22no—
37#1445—22no—
38#1441—22no—
39#1441—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C42C2311yescharacteristic
63C22×C42C2312yes—
64C2×#107—24no—
65C2×#107—24no—
66C2×#107—24no—
67C2×#107—24no—
68C2×#107—42no—
69C2×#107—42no—
70C2×#106—22no—
71C2×#106—22no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—42no—
75C2×C4×C8—42no—
76C2×C4×C8—42no—
77C2×C4×C8—42no—
78C2×C4×C8D414yes—
79C2×C4×C8Q818yescomplemented
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#384—22no—
85#384—22no—
86#382—24no—
87#382—24no—
88#382—42no—
89#382—42no—
90C2×#95—24no—
91C2×#95—24no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95D412yes—
97C2×#95D412yes—
98#340—24no—
99#340—24no—
100#340—22no—
101#340—22no—
102#340—24no—
103#340—24no—
104#336D414yes—
105#336Q818yescomplemented
106C22×Q8—44no—
107C22×Q8—42no—
108C22×Q8—44no—
109C22×Q8—82no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C2×D414yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8C2×D414yes—
131C22×C8C2×D412yes—
132C22×C8C2×D412yes—
133C22×C8C2×Q814yes—
134C22×C8C2×Q814yes—
135C22×C8C2×Q814yes—
136C22×C8C2×Q814yes—
137C4⋊Q8—28no—
138C4⋊Q8—48no—
139C4⋊Q8—44no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4#4114yes—
180C2×C4⋊C4#4112yes—
181C2×C4⋊C4#4112yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×Q814yes—
191C2×C42C2×Q814yes—
192#107—48no—
193#107—48no—
194#106—28no—
195#106—44no—
196C4×C8—44no—
197C4×C8—48no—
198#95—24no—
199#95—24no—
200#95—48no—
201#95—48no—
202#95—42no—
203#95—42no—
204#95—84no—
205#95—84no—
206C24C22×D411yescharacteristic
207C2×Q8—48no—
208C2×Q8—816no—
209C2×Q8—84no—
210C2×Q8—84no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4#12712yes—
250C22×C4#12712yes—
251C22×C4C4⋊Q818yes—
252C22×C4#13514yes—
253C22×C4#13514yes—
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×Q812yes—
257C22×C4C22×Q812yes—
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4112yes—
260C22×C4C2×#4111yescharacteristic
261C22×C4H(5,𝔽2)12yes—
262C22×C4H(5,𝔽2)11yescharacteristic
263C22×C42−1+411yescharacteristic
264C2×C8—22no—
265C2×C8—24no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8C4⋊Q814yes—
279C2×C8C4⋊Q814yes—
280C2×C8C4⋊Q818yes—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—48no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C42—22no—
309C42—24no—
310C42—22no—
311C42—24no—
312C42—24no—
313C42—24no—
314C42—24no—
315C23C2×#12711yescharacteristic
316C23C2×C4⋊Q812yes—
317C23#54612yes—
318C23#54611yescharacteristic
319C23#54811yescharacteristic
320C23C2×#13512yes—
321C23C2×#13512yes—
322C23#57612yes—
323C23#57812yes—
324Q8—816no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—28no—
328C2×C4—28no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—28no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—48no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—84no—
352C2×C4—82no—
353C2×C4—84no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4#49514yes—
367C2×C4#49514yes—
368C2×C4#52511yescharacteristic
369C2×C4#52511yescharacteristic
370C2×C4C2×C4⋊Q811yescharacteristic
371C2×C4C2×C4⋊Q811yescharacteristic
372C2×C4C2×C4⋊Q812yes—
373C2×C4#53312yes—
374C2×C4#53312yes—
375C2×C4#53411yescharacteristic
376C2×C4#53711yescharacteristic
377C2×C4#54312yes—
378C2×C4#54412yes—
379C2×C4#54611yescharacteristic
380C2×C4#54612yes—
381C2×C4#54811yescharacteristic
382C2×C4#55412yes—
383C2×C4#55412yes—
384C2×C4#55412yes—
385C2×C4#55712yes—
386C8—24no—
387C8—24no—
388C8—28no—
389C22#253911yescharacteristic
390C22#254011yescharacteristic
391C22#254912yes—
392C22#255012yes—
393C22#286811yescharacteristic
394C22C2×#49511yescharacteristic
395C22C2×#49511yescharacteristic
396C22#315511yescharacteristic
397C22#315611yescharacteristic
398C22#316014yes—
399C22#316414yes—
400C22#329714yes—
401C22#330014yes—
402C22#330212yes—
403C22#330512yes—
404C22#336112yes—
405C22#336412yes—
406C4—24no—
407C4—24no—
408C4—48no—
409C4—84no—
410C4—82no—
411C4—82no—
412C4—161no—
413C4—161no—
414C4—162no—
415C4#239911yescharacteristic
416C4#254912yes—
417C4#255012yes—
418C4#258311yescharacteristic
419C4#258411yescharacteristic
420C4#265911yescharacteristic
421C2#1517311yescharacteristic
422C2#2833111yescharacteristic
423C2#2833511yescharacteristic
424C2#2839911yescharacteristic
425C2#2840311yescharacteristic
426C2#2877314yes—
427C2#2879314yes—
428C2#3331311yescharacteristic
429C2#3331911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
79. C2×C4×C8Q8semidirect
105. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20832
C2×#20572
C2×#19902
C2×#19412
#152422
#152302
#113252
#108622
#108422
C8×#952