Finite Group Databasebeta

← #392289

Subgroups — #392289

Order 512.

2078 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C42×D4C211yesmaximal, characteristic
2C2×#1831C211yesmaximal, characteristic, complemented
3C2×#1802C212yesmaximal
4#14687C211yesmaximal, characteristic, complemented
5#10849C212yesmaximal, complemented
6#10806C214yesmaximal, complemented
7#10788C214yesmaximal
8C22×C4×D4C2212yes—
9C22×C4×D4C2211yescharacteristic
10C23×C42—21no—
11C22×C4⋊C8C2212yes—
12C22×#102C2214yes—
13C2×#384C2212yes—
14C2×#382—21no—
15C2×#382—21no—
16C42×D4—24no—
17C42×D4C418yescomplemented
18C2×C4×C4⋊C4C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C22⋊C4—21no—
21C2×C4×C22⋊C4—22no—
22C2×C43C2211yescharacteristic
23#1831—24no—
24#1802—28no—
25C2×#340C2218yes—
26C2×#340C2214yes—
27C2×#336C2214yes—
28C4⋊#95—22no—
29C4⋊#95C2214yes—
30C23×D4C2311yescharacteristic
31C24×C4—22no—
32C24×C4—21no—
33C23×C8C2314yes—
34C2×C4×D4—28no—
35C2×C4×D4—28no—
36C2×C4×D4—24no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4C2×C418yes—
41C2×C4×D4C2×C414yes—
42C2×C4×D4D414yes—
43C2×C4×D4Q814yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2314yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—21no—
51C22×C42—22no—
52C22×C42—21no—
53C22×C42—28no—
54C22×C42—44no—
55C22×C42—44no—
56C22×C42D411yescharacteristic
57C22×C42D412yes—
58C22×C42D411yescharacteristic
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×C4⋊C8—24no—
62C2×C4⋊C8—24no—
63C2×C4⋊C8—24no—
64C2×#102—28no—
65C2×#102—216no—
66#384—24no—
67#384—24no—
68#382—44no—
69#382—44no—
70C4×C4⋊C4—24no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4C2×C418yes—
73C4×C4⋊C4C2×C414yes—
74C4×C22⋊C4—28no—
75C4×C22⋊C4—216no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95—24no—
79C2×#95—24no—
80C43C2×C414yes—
81C43D414yes—
82#340—28no—
83#340—216no—
84#336—28no—
85C25—21no—
86C22×D4—21no—
87C22×D4—21no—
88C22×D4—21no—
89C22×D4—24no—
90C22×D4C22×C412yes—
91C22×D4C2×D412yes—
92C22×D4C2×Q812yes—
93C22×D4#4118yes—
94C23×C4—216no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—28no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—48no—
103C23×C4—44no—
104C23×C4—42no—
105C23×C4—44no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2411yescharacteristic
114C22×C8—24no—
115C22×C8—24no—
116C22×C8—28no—
117C22×C8—28no—
118C22×C8—24no—
119C4×D4—232no—
120C4×D4—216no—
121C4×D4—216no—
122C4×D4—28no—
123C4×D4—28no—
124C4×D4—28no—
125C4×D4C4⋊C4116yes—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4C22×C412yes—
141C2×C4⋊C4C22×C414yes—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×Q812yes—
144C2×C4⋊C4#4118yes—
145C2×C22⋊C4—216no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—28no—
155C2×C42—24no—
156C2×C42—44no—
157C2×C42—44no—
158C2×C42—42no—
159C2×C42—44no—
160C2×C42—42no—
161C2×C42—432no—
162C2×C42C22⋊C414yes—
163C2×C42C22⋊C418yes—
164C2×C42C22⋊C414yes—
165C2×C42D814yes—
166C2×C42SD1614yes—
167C2×C42C22×C414yes—
168C2×C42C22×C412yes—
169C2×C42C2×D414yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×Q812yes—
173C2×C42#4112yes—
174C2×C42#4112yes—
175C4⋊C8—48no—
176#102—416no—
177#95—28no—
178#95—24no—
179#95—48no—
180#95—48no—
181C24—22no—
182C24—22no—
183C24—22no—
184C24—28no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24—41no—
189C24—44no—
190C24C22×D411yescharacteristic
191C2×D4—22no—
192C2×D4—24no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—28no—
196C2×D4—28no—
197C2×D4—28no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4C2×C4⋊C414yes—
201C2×D4#11718yes—
202C2×D4#12418yes—
203C2×D4#12518yes—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—232no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—216no—
218C22×C4—432no—
219C22×C4—432no—
220C22×C4—416no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—44no—
228C22×C4—416no—
229C22×C4—48no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4—44no—
236C22×C4—48no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—48no—
240C22×C4C2×C22⋊C414yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C414yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4#12112yes—
245C22×C4#12112yes—
246C22×C4C22⋊Q812yes—
247C22×C4C22⋊Q812yes—
248C22×C4C2×D814yes—
249C22×C4C2×D812yes—
250C22×C4C2×SD1614yes—
251C22×C4C2×SD1612yes—
252C22×C4AGL(1,ℤ/8)12yes—
253C22×C4#13712yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×Q811yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4114yes—
261C22×C4C2×#4114yes—
262C2×C8—44no—
263C2×C8—48no—
264C2×C8—44no—
265C2×C8—48no—
266C2×C8—44no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—28no—
272C4⋊C4—44no—
273C4⋊C4—48no—
274C4⋊C4—44no—
275C4⋊C4—88no—
276C4⋊C4—88no—
277C4⋊C4C2×C4⋊C414yes—
278C4⋊C4#11718yes—
279C22⋊C4—232no—
280C22⋊C4—216no—
281C42—24no—
282C42—24no—
283C42—28no—
284C42—28no—
285C42—48no—
286C42—416no—
287C42—48no—
288C42#102116yescomplemented
289C42C2×C4⋊C414yes—
290C42#11714yes—
291C42#12114yes—
292C42C22⋊Q814yes—
293C23—28no—
294C23—28no—
295C23—24no—
296C23—28no—
297C23—44no—
298C23—48no—
299C23—48no—
300C23—48no—
301C23—48no—
302C23—42no—
303C23—42no—
304C23—44no—
305C23—42no—
306C23—42no—
307C23—44no—
308C23—42no—
309C23—42no—
310C23C22×C22⋊C411yescharacteristic
311C23C2×#12111yescharacteristic
312C23C2×C22⋊Q811yescharacteristic
313C23#52514yes—
314C23C22×D811yescharacteristic
315C23C22×SD1611yescharacteristic
316C23C2×AGL(1,ℤ/8)11yescharacteristic
317C23C2×#13711yescharacteristic
318C23#57514yes—
319D4—28no—
320D4—28no—
321D4—28no—
322D4—24no—
323D4#38218yescomplemented
324C2×C4—24no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—28no—
336C2×C4—464no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—432no—
350C2×C4—88no—
351C2×C4—84no—
352C2×C4—88no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4#38014yes—
356C2×C4#38014yes—
357C2×C4C2×#10218yes—
358C2×C4C2×#10214yes—
359C2×C4#41714yes—
360C2×C4#45914yes—
361C2×C4#46114yes—
362C2×C4#47414yes—
363C2×C4#47614yes—
364C2×C4C22×C4⋊C411yescharacteristic
365C2×C4C2×#11712yes—
366C2×C4C2×#11711yescharacteristic
367C2×C4C2×#11712yes—
368C2×C4C2×#12111yescharacteristic
369C2×C4C2×C22⋊Q811yescharacteristic
370C2×C4#52514yes—
371C2×C4C2×#12412yes—
372C2×C4C2×#12512yes—
373C2×C4#52914yes—
374C2×C4#52914yes—
375C2×C4#53212yes—
376C2×C4#53312yes—
377C8—84no—
378C22—28no—
379C22—48no—
380C22—48no—
381C22—48no—
382C22—48no—
383C22—48no—
384C22—44no—
385C22—44no—
386C22—48no—
387C22C2×#38011yescharacteristic
388C22#228012yes—
389C22#285012yes—
390C22#285112yes—
391C22C22×#10211yescharacteristic
392C22C2×#41711yescharacteristic
393C22#289612yes—
394C22C2×#45911yescharacteristic
395C22C2×#46111yescharacteristic
396C22#304012yes—
397C22C2×#47411yescharacteristic
398C22C2×#47611yescharacteristic
399C22#308012yes—
400C22#310014yes—
401C22#310114yes—
402C22#311514yes—
403C22#311614yes—
404C4—24no—
405C4—28no—
406C4—44no—
407C4—44no—
408C4—48no—
409C4—164no—
410C4#180218yes—
411C4C2×#38211yescharacteristic
412C4#228312yes—
413C4#228511yescharacteristic
414C4#285012yes—
415C4#285112yes—
416C2—48no—
417C2#1468211yescharacteristic
418C2C2×#180211yescharacteristic
419C2#2064911yescharacteristic
420C2#2073212yes—
421C2#2074312yes—
422C2#3302412yes—
423C2#3302612yes—
424C2#3306612yes—
425C2#3306812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. C2×#1831C2semidirect
4. #14687C2semidirect
5. #10849C2semidirect
6. #10806C2semidirect
17. C42×D4C4semidirect
288. C42#102semidirect
323. D4#382semidirect

Maximal subgroups

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

subgroupindex
C2×C42×D42
C2×#18312
C2×#18022
#146872
#108492
#108062
#107882