Finite Group Databasebeta

← #393694

Subgroups — #393694

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1839C211yesmaximal, characteristic
2#14688C211yesmaximal, characteristic
3C42×C4⋊C4C211yesmaximal, characteristic
4#10908C212yesmaximal
5#10908C212yesmaximal
6#10908C212yesmaximal
7#10907C212yesmaximal
8#10907C212yesmaximal
9#10907C212yesmaximal
10C22×#107C2212yes—
11C22×#106C2212yes—
12C2×#383—21no—
13C2×C4×C4⋊C4C2211yescharacteristic
14C2×C4×C4⋊C4C2212yes—
15C2×C4×C4⋊C4C2212yes—
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C43—21no—
18C2×C43C2211yescharacteristic
19#1839C418yescomplemented
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C4⋊#95C2211yescharacteristic
31C4⋊#95C2212yes—
32C4⋊#95C2212yes—
33C4⋊#95C2211yescharacteristic
34#1434—21no—
35#1434—21no—
36#1434—21no—
37C4×#95—21no—
38C4×#95—21no—
39C4×#95—21no—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4C2312yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C42—22no—
47C22×C42—21no—
48C22×C42—21no—
49C22×C42—22no—
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×#107—24no—
58C2×#107—24no—
59C2×#107—24no—
60C2×#106—24no—
61C2×#106—24no—
62C2×#106—24no—
63#383—44no—
64C4×C4⋊C4—24no—
65C4×C4⋊C4—24no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—24no—
69C4×C4⋊C4—24no—
70C2×#95—21no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—22no—
82C43—44no—
83C43C2×C414yes—
84C43Q814yes—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#336C2×C414yes—
92#336C2×C418yes—
93#336C2×C418yes—
94#336C2×C414yes—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2411yescharacteristic
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8C22×C414yes—
119C22×C8C22×C414yes—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
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#4114yes—
139C2×C4⋊C4#4114yes—
140C2×C4⋊C4#4114yes—
141C2×C4⋊C4#4114yes—
142C2×C4⋊C4#4114yes—
143C2×C4⋊C4#4114yes—
144C2×C42—24no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—28no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—28no—
159C2×C42—42no—
160C2×C42—44no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42—44no—
164C2×C42—44no—
165C2×C42—44no—
166C2×C42—42no—
167C2×C42—42no—
168C2×C42C4⋊C414yes—
169C2×C42C4⋊C414yes—
170C2×C42C4⋊C414yes—
171C2×C42C4⋊C414yes—
172C2×C42C22×C412yes—
173C2×C42C22×C412yes—
174C2×C42C22×C412yes—
175C2×C42C2×Q812yes—
176C2×C42C2×Q812yes—
177C2×C42C2×Q812yes—
178#107—48no—
179#106—48no—
180#95—24no—
181#95—24no—
182#95—28no—
183#95—28no—
184#95—44no—
185#95—42no—
186#95—42no—
187#95—44no—
188#95—44no—
189#95—42no—
190#95—44no—
191#95—42no—
192C24C22×D411yescharacteristic
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
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#13514yes—
249C22×C4AGL(1,ℤ/8)12yes—
250C22×C4AGL(1,ℤ/8)12yes—
251C22×C4#13712yes—
252C22×C4#13712yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×Q811yescharacteristic
255C22×C4C2×#4112yes—
256C22×C4C2×#4112yes—
257C22×C4C2×#4112yes—
258C22×C4C2×#4112yes—
259C22×C4C2×#4112yes—
260C22×C4C2×#4112yes—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4#12618yes—
290C4⋊C4#12618yes—
291C42—24no—
292C42—22no—
293C42—22no—
294C42—24no—
295C42—22no—
296C42—22no—
297C42—24no—
298C42—24no—
299C42—28no—
300C42—28no—
301C42—28no—
302C42—44no—
303C42—416no—
304C42—48no—
305C42—48no—
306C42—416no—
307C23C22×C4⋊C411yescharacteristic
308C23#52511yescharacteristic
309C23#52511yescharacteristic
310C23#52512yes—
311C23#52512yes—
312C23C2×#13512yes—
313C23C2×AGL(1,ℤ/8)11yescharacteristic
314C23C2×AGL(1,ℤ/8)11yescharacteristic
315C23C2×#13711yescharacteristic
316C23C2×#13711yescharacteristic
317C23#57512yes—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—44no—
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—82no—
374C2×C4—82no—
375C2×C4#42714yes—
376C2×C4#42814yes—
377C2×C4#42814yes—
378C2×C4C4×#4112yes—
379C2×C4C4×#4112yes—
380C2×C4C4×#4111yescharacteristic
381C2×C4C4×#4111yescharacteristic
382C2×C4C2×#12612yes—
383C2×C4C2×#12612yes—
384C2×C4#52912yes—
385C2×C4#52912yes—
386C2×C4#52912yes—
387C2×C4#52912yes—
388C2×C4#52912yes—
389C2×C4#52912yes—
390C2×C4#53312yes—
391C2×C4#53311yescharacteristic
392C2×C4#53311yescharacteristic
393C2×C4#53312yes—
394C8—44no—
395C8—44no—
396C22#229611yescharacteristic
397C22#229611yescharacteristic
398C22#229611yescharacteristic
399C22#285212yes—
400C22#285212yes—
401C22C2×#42711yescharacteristic
402C22C2×#42811yescharacteristic
403C22C2×#42811yescharacteristic
404C22#291111yescharacteristic
405C22#310212yes—
406C22#310212yes—
407C22#310512yes—
408C22#310612yes—
409C22#310712yes—
410C22#310812yes—
411C22#310911yescharacteristic
412C22#311011yescharacteristic
413C22#311211yescharacteristic
414C22#311311yescharacteristic
415C22#311412yes—
416C22#311512yes—
417C22#311612yes—
418C22#311712yes—
419C4—24no—
420C4—24no—
421C4—24no—
422C4—48no—
423C4—48no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—162no—
428C4—162no—
429C4C4×#12612yes—
430C4C4×#12612yes—
431C4#286312yes—
432C4#286312yes—
433C2#1471011yescharacteristic
434C2#2136011yescharacteristic
435C2#2136111yescharacteristic
436C2#2136211yescharacteristic
437C2#2136311yescharacteristic
438C2#2136411yescharacteristic
439C2#2136511yescharacteristic
440C2#3308012yes—
441C2#3308112yes—
442C2#3308412yes—
443C2#3308512yes—
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
19. #1839C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18392
#146882
C42×C4⋊C42
#109082
#109072