Finite Group Databasebeta

← #382852

Subgroups — #382852

Order 512.

2050 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×C4×#127C211yesmaximal, characteristic
2C2×#1839C211yesmaximal, characteristic, complemented
3#14687C211yesmaximal, characteristic, complemented
4#11357C212yesmaximal, complemented
5#11049C214yesmaximal, complemented
6#11008C212yesmaximal
7#10788C214yesmaximal, complemented
8C22×#127C2211yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4—21no—
11C22×C4×D4C2212yes—
12C22×#102C2218yes—
13C4×#127—24no—
14C4×#127C418yescomplemented
15C2×#390—21no—
16C2×#390—21no—
17C2×#390C2212yes—
18C2×#384C2211yescharacteristic
19C2×#384C2212yes—
20C2×#382—21no—
21C2×#382—21no—
22C2×C43C2211yescharacteristic
23#1839—24no—
24C2×#340C2218yes—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C4⋊#95—21no—
29C4⋊#95—21no—
30C4⋊#95C2212yes—
31C4⋊#95C2212yes—
32C23×D4—21no—
33C23×D4—21no—
34C23×D4C2312yes—
35C24×C4—22no—
36C23×C8C2314yes—
37C2×#127—28no—
38C2×#127—22no—
39C2×#127C2×C414yes—
40C2×C4×D4—28no—
41C2×C4×D4—28no—
42C2×C4×D4—24no—
43C2×C4×D4—48no—
44C2×C4×D4—48no—
45C2×C4×D4—44no—
46C2×C4×D4—44no—
47C2×C4×D4C2×C418yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2314yes—
53C22×C4⋊C4C2312yes—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#102—216no—
65C2×#102—48no—
66C2×#102—48no—
67#390—28no—
68#390—28no—
69#390—24no—
70#390C2×C418yes—
71#384—22no—
72#384—44no—
73#384C2×C414yes—
74#382—44no—
75#382—44no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C43C2×C414yes—
81C43D414yes—
82#340—48no—
83#336—24no—
84#336—24no—
85#336—24no—
86C25—22no—
87C22×D4—28no—
88C22×D4—28no—
89C22×D4—24no—
90C22×D4—24no—
91C22×D4—22no—
92C22×D4—24no—
93C22×D4—28no—
94C22×D4—24no—
95C22×D4—24no—
96C22×D4—44no—
97C22×D4—44no—
98C22×D4—42no—
99C22×D4—42no—
100C22×D4—42no—
101C22×D4—42no—
102C22×D4C22×C414yes—
103C22×D4#4118yes—
104C23×C4—22no—
105C23×C4—28no—
106C23×C4—24no—
107C23×C4—48no—
108C23×C4—48no—
109C23×C4—44no—
110C23×C4—44no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C22×C8—24no—
119C22×C8—28no—
120C22×C8—28no—
121C22×C8—24no—
122C22×C8—24no—
123#127—216no—
124#127—216no—
125C4×D4—416no—
126C4×D4—416no—
127C4×D4—432no—
128C4×D4—432no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4C22×C414yes—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—28no—
150C2×C22⋊C4—48no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—48no—
154C2×C42—24no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—42no—
162C2×C42C22⋊C414yes—
163C2×C42C22⋊C414yes—
164C2×C42C22⋊C414yes—
165C2×C42C22⋊C414yes—
166C2×C42D814yes—
167C2×C42SD1614yes—
168C2×C42C22×C412yes—
169C2×C42C22×C412yes—
170C2×C42C22×C412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174#102—432no—
175#95—44no—
176#95—44no—
177#95—44no—
178#95—44no—
179C24—24no—
180C24—28no—
181C24—44no—
182C24—42no—
183C24—42no—
184C24—42no—
185C24—42no—
186C24—44no—
187C24—42no—
188C24—42no—
189C24—44no—
190C24C22×D411yescharacteristic
191C2×D4—216no—
192C2×D4—216no—
193C2×D4—24no—
194C2×D4—28no—
195C2×D4—28no—
196C2×D4—28no—
197C2×D4—216no—
198C2×D4—216no—
199C2×D4—48no—
200C2×D4—416no—
201C2×D4—416no—
202C2×D4—416no—
203C2×D4—416no—
204C2×D4—48no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—48no—
208C2×D4—48no—
209C2×D4—48no—
210C2×D4—48no—
211C2×D4#11718yes—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—416no—
231C22×C4—416no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—416no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—816no—
245C22×C4—816no—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×D812yes—
253C22×C4C2×D812yes—
254C22×C4C2×D812yes—
255C22×C4C2×SD1612yes—
256C22×C4C2×SD1612yes—
257C22×C4C2×SD1612yes—
258C22×C4AGL(1,ℤ/8)12yes—
259C22×C4AGL(1,ℤ/8)14yes—
260C22×C4#13712yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C2×#4114yes—
264C22×C4H(5,𝔽2)14yes—
265C22×C4H(5,𝔽2)12yes—
266C22×C42−1+412yes—
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—48no—
271C2×C8—48no—
272C4⋊C4—28no—
273C4⋊C4—48no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—48no—
277C4⋊C4—48no—
278C4⋊C4—88no—
279C4⋊C4—88no—
280C22⋊C4—416no—
281C22⋊C4—416no—
282C22⋊C4—416no—
283C42—24no—
284C42—24no—
285C42—24no—
286C42—24no—
287C42—24no—
288C42—24no—
289C42—28no—
290C42—28no—
291C42—44no—
292C42#102116yescomplemented
293C23—28no—
294C23—44no—
295C23—44no—
296C23—48no—
297C23—48no—
298C23—48no—
299C23—48no—
300C23—48no—
301C23—48no—
302C23—44no—
303C23—44no—
304C23—44no—
305C23—44no—
306C23—84no—
307C23—82no—
308C23—82no—
309C23—84no—
310C23—84no—
311C23—84no—
312C23—84no—
313C23—82no—
314C23—82no—
315C23—84no—
316C23C22×C22⋊C411yescharacteristic
317C23#52512yes—
318C23#53512yes—
319C23#53511yescharacteristic
320C23#53611yescharacteristic
321C23C22×D811yescharacteristic
322C23C22×SD1611yescharacteristic
323C23C2×AGL(1,ℤ/8)11yescharacteristic
324C23C2×AGL(1,ℤ/8)12yes—
325C23C2×#13711yescharacteristic
326C23#57512yes—
327D4—216no—
328D4—432no—
329D4—432no—
330D4—48no—
331D4—416no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—832no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4C2×#10214yes—
373C2×C4C2×#10214yes—
374C2×C4C2×#10214yes—
375C2×C4#41714yes—
376C2×C4#41814yes—
377C2×C4C2×#11712yes—
378C2×C4#51812yes—
379C2×C4#52012yes—
380C2×C4#52512yes—
381C2×C4#53512yes—
382C2×C4#53712yes—
383C2×C4#53818yes—
384C2×C4#53918yes—
385C8—84no—
386C22—48no—
387C22—48no—
388C22—48no—
389C22—84no—
390C22—84no—
391C22—88no—
392C22—88no—
393C22—84no—
394C22—84no—
395C22—88no—
396C22—88no—
397C22#228011yescharacteristic
398C22#230911yescharacteristic
399C22#231111yescharacteristic
400C22#263112yes—
401C22#263312yes—
402C22C22×#10211yescharacteristic
403C22C2×#41811yescharacteristic
404C22C2×#41711yescharacteristic
405C22#289611yescharacteristic
406C22#310012yes—
407C22#310112yes—
408C22#310912yes—
409C22#311012yes—
410C22#320512yes—
411C22#320511yescharacteristic
412C22#320611yescharacteristic
413C22#320812yes—
414C22#321312yes—
415C22#321412yes—
416C22#321612yes—
417C22#322611yescharacteristic
418C22#322711yescharacteristic
419C22#323412yes—
420C4—24no—
421C4—24no—
422C4—24no—
423C4—24no—
424C4—24no—
425C4—42no—
426C4—84no—
427C4—164no—
428C4#231814yes—
429C4#263614yes—
430C2—88no—
431C2#1434511yescharacteristic
432C2#2073211yescharacteristic
433C2#2073811yescharacteristic
434C2#2180311yescharacteristic
435C2#2180911yescharacteristic
436C2#2185911yescharacteristic
437C2#2186511yescharacteristic
438C2#3003812yes—
439C2#3003912yes—
440C2#3007512yes—
441C2#3007612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
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×#1839C2semidirect
3. #14687C2semidirect
4. #11357C2semidirect
5. #11049C2semidirect
7. #10788C2semidirect
14. C4×#127C4semidirect
292. C42#102semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1272
C2×#18392
#146872
#113572
#110492
#110082
#107882