Finite Group Databasebeta

← #375050

Subgroups — #375050

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C212yesmaximal
2C2×#1831C211yesmaximal, characteristic
3C4×#382C211yesmaximal, characteristic
4C4×#382C211yesmaximal, characteristic
5#11325C212yesmaximal
6#11268C212yesmaximal
7#10908C212yesmaximal
8#10907C212yesmaximal
9#10850C212yesmaximal
10C22×#107C2212yes—
11C22×#106C2212yes—
12C22×C4⋊C8C2212yes—
13C2×#382C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×C4×C4⋊C4C2212yes—
18C2×C4×C4⋊C4C2212yes—
19C2×C43C2211yescharacteristic
20#1949—28no—
21#1831C418yescomplemented
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C4⋊#95C2212yes—
32C4⋊#95C2212yes—
33#1434—21no—
34#1434—21no—
35#1434—21no—
36#1434—21no—
37C4×#95—21no—
38C4×#95—21no—
39C4×#95—21no—
40C4×#95—21no—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C42—22no—
48C22×C42—22no—
49C22×C42D411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#107—24no—
57C2×#107—24no—
58C2×#107—42no—
59C2×#106—24no—
60C2×#106—24no—
61C2×#106—42no—
62C2×C4⋊C8—24no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8C2×C418yes—
65#382—24no—
66#382—24no—
67#382—22no—
68#382—22no—
69#382D414yes—
70#382Q814yes—
71C4×C4⋊C4—24no—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—42no—
74C4×C4⋊C4—42no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C43C2×C414yes—
90C43Q814yes—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—42no—
96#340—42no—
97#336C2×C418yes—
98#336C2×C414yes—
99#336C2×C414yes—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2411yescharacteristic
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8C22×C414yes—
124C22×C8C22×C414yes—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
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—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C2×D414yes—
146C2×C4⋊C4C2×Q814yes—
147C2×C4⋊C4#4114yes—
148C2×C4⋊C4#4114yes—
149C2×C42—24no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—42no—
155C2×C42—42no—
156C2×C42—42no—
157C2×C42—44no—
158C2×C42—44no—
159C2×C42—44no—
160C2×C42—44no—
161C2×C42—44no—
162C2×C42—44no—
163C2×C42C4⋊C414yes—
164C2×C42C4⋊C414yes—
165C2×C42C4⋊C414yes—
166C2×C42C4⋊C414yes—
167C2×C42C22×C412yes—
168C2×C42C22×C412yes—
169C2×C42C22×C412yes—
170C2×C42C2×D412yes—
171C2×C42C2×Q812yes—
172C2×C42C2×Q812yes—
173C2×C42C2×Q812yes—
174C2×C42C2×Q812yes—
175C2×C42#4112yes—
176C2×C42#4112yes—
177#107—48no—
178#106—48no—
179C4⋊C8—216no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—44no—
194C24C22×D411yescharacteristic
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
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—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4C2×C4⋊C412yes—
243C22×C4C2×C4⋊C412yes—
244C22×C4C2×C4⋊C412yes—
245C22×C4C2×C4⋊C412yes—
246C22×C4C2×C4⋊C412yes—
247C22×C4C2×C4⋊C412yes—
248C22×C4#12112yes—
249C22×C4#12112yes—
250C22×C4C22⋊Q812yes—
251C22×C4C22⋊Q812yes—
252C22×C4#13514yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×Q811yescharacteristic
256C22×C4C22×Q811yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4112yes—
260C22×C4C2×#4112yes—
261C22×C4H(5,𝔽2)12yes—
262C22×C42−1+412yes—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C42—24no—
294C42—24no—
295C42—24no—
296C42—24no—
297C42—28no—
298C42—44no—
299C42—44no—
300C42—44no—
301C42—88no—
302C42—88no—
303C42C4×D414yes—
304C42C4×Q814yes—
305C42#12514yes—
306C42C4⋊Q814yes—
307C23C22×C4⋊C411yescharacteristic
308C23C2×#12111yescharacteristic
309C23C2×C22⋊Q811yescharacteristic
310C23#52511yescharacteristic
311C23#52511yescharacteristic
312C23#53511yescharacteristic
313C23#53611yescharacteristic
314C23C2×#13512yes—
315C23#57512yes—
316C23#57512yes—
317C23#57512yes—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4#38914yes—
374C2×C4#38914yes—
375C2×C4#42714yes—
376C2×C4C2×C4×D411yescharacteristic
377C2×C4C2×C4×Q811yescharacteristic
378C2×C4C4×#4112yes—
379C2×C4#52012yes—
380C2×C4C2×#12511yescharacteristic
381C2×C4C2×C4⋊Q811yescharacteristic
382C2×C4#53312yes—
383C2×C4#54412yes—
384C2×C4#54612yes—
385C2×C4#54812yes—
386C2×C4#55212yes—
387C2×C4#55312yes—
388C2×C4#55412yes—
389C2×C4#55612yes—
390C2×C4#55712yes—
391C2×C4#55912yes—
392C8—44no—
393C8—44no—
394C22C2×#38911yescharacteristic
395C22#229611yescharacteristic
396C22#231711yescharacteristic
397C22#253511yescharacteristic
398C22#253611yescharacteristic
399C22#258011yescharacteristic
400C22#258111yescharacteristic
401C22C2×#42711yescharacteristic
402C22#291111yescharacteristic
403C22#291111yescharacteristic
404C22#291111yescharacteristic
405C22#303512yes—
406C22#304012yes—
407C22#307612yes—
408C22#308012yes—
409C22#310212yes—
410C22#310212yes—
411C22#311112yes—
412C22#311112yes—
413C22#321112yes—
414C22#321912yes—
415C22#323212yes—
416C22#324012yes—
417C4—24no—
418C4—24no—
419C4—24no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—84no—
424C4—84no—
425C4—84no—
426C4—84no—
427C4#236712yes—
428C4#237012yes—
429C4#255012yes—
430C4#257012yes—
431C2#1415011yescharacteristic
432C2#2116911yescharacteristic
433C2#2117211yescharacteristic
434C2#2136011yescharacteristic
435C2#2136611yescharacteristic
436C2#2198311yescharacteristic
437C2#2198911yescharacteristic
438C2#2816312yes—
439C2#2820812yes—
440C2#2926112yes—
441C2#2927912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1831C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19492
C2×#18312
C4×#3822
#113252
#112682
#109082
#109072
#108502