Finite Group Databasebeta

← #375146

Subgroups — #375146

Order 512.

2190 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#466C211yesmaximal, characteristic
2C2×#1874C212yesmaximal, complemented
3C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
4C22×#340C211yesmaximal, characteristic, complemented
5#14137C212yesmaximal, complemented
6#11379C212yesmaximal, complemented
7#11022C212yesmaximal, complemented
8#10925C214yesmaximal, complemented
9C23×D8C2211yescharacteristic
10C24×C8C2211yescharacteristic
11C22×#121C2212yes—
12C23×C4⋊C4C2212yes—
13C2×#466—24no—
14C2×#466—24no—
15C2×#466C418yescomplemented
16C22×#107C2211yescharacteristic
17C22×#102C2214yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C4×C8C2212yes—
21C2×#390C2214yes—
22C2×#386—24no—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1874—28no—
25C8×C22⋊C4—24no—
26C2×#340—24no—
27C2×#340C2212yes—
28C2×#340C2214yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31#1439—24no—
32C4⋊#95C2214yes—
33C23×D4—22no—
34C23×D4C2312yes—
35C24×C4—22no—
36C24×C4C2311yescharacteristic
37C22×D8—22no—
38C22×D8—22no—
39C22×D8—44no—
40C22×D8C2×C414yescomplemented
41C23×C8—44no—
42C23×C8C2×C414yes—
43C23×C8D414yes—
44C23×C8D414yes—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×#121—24no—
49C2×#121—24no—
50C2×#121—24no—
51C2×#121—24no—
52C2×#121—24no—
53C2×#121C2×C418yes—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—28no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—44no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2314yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—28no—
63C22×C22⋊C4D412yes—
64C22×C42C2312yes—
65#466—232no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107C2×C414yescomplemented
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—28no—
72C2×#102—28no—
73C2×#102—44no—
74C2×#102C2×C418yescomplemented
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8D414yes—
78C2×C4×C8—24no—
79C2×C4×C8—44no—
80#390—28no—
81#390—44no—
82#386—416no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—22no—
85C2×#95—28no—
86C2×#95D412yes—
87#340—28no—
88#340—24no—
89#340—24no—
90#336—24no—
91C25—22no—
92C25C2×D411yescharacteristic
93C22×D4—28no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—24no—
99C22×D4—24no—
100C22×D4—22no—
101C22×D4—24no—
102C22×D4—24no—
103C22×D4—44no—
104C22×D4—44no—
105C22×D4—48no—
106C22×D4—48no—
107C22×D4C22×C414yes—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—22no—
112C23×C4—48no—
113C23×C4—44no—
114C23×C4—48no—
115C23×C4—84no—
116C23×C4C22×C412yes—
117C23×C4C2×D414yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4#4112yes—
123C23×C4#4112yes—
124C23×C4C2411yescharacteristic
125C2×D8—28no—
126C2×D8—48no—
127C2×D8—48no—
128C2×D8—48no—
129C22×C8—24no—
130C22×C8—24no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—24no—
134C22×C8—24no—
135C22×C8—48no—
136C22×C8—48no—
137C22×C8—48no—
138C22×C8C22⋊C4116yescomplemented
139C22×C8C22×C412yes—
140C22×C8C22×C412yes—
141C22×C8C2×D414yes—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146#121—216no—
147#121—216no—
148#121—48no—
149#121—48no—
150C2×C4⋊C4—216no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—416no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4C22×C414yes—
166C2×C4⋊C4#4118yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—416no—
173C2×C22⋊C4—416no—
174C2×C22⋊C4—88no—
175C2×C22⋊C4D818yescomplemented
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—24no—
179C2×C42—24no—
180#107—28no—
181#102—216no—
182#102—416no—
183C22⋊C8—28no—
184C4×C8—48no—
185#95—24no—
186#95—416no—
187C24—24no—
188C24—24no—
189C24—24no—
190C24—42no—
191C24—42no—
192C24—44no—
193C24—44no—
194C24—44no—
195C24—44no—
196C24—44no—
197C24—44no—
198C24C4×D412yes—
199C24#12112yes—
200C24#12712yes—
201C24C2×D814yes—
202C24#13514yes—
203C24C22×D411yescharacteristic
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—28no—
207C2×D4—28no—
208C2×D4—416no—
209C2×D4—432no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—44no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—816no—
225C2×D4—88no—
226C2×D4—88no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—28no—
235C22×C4—28no—
236C22×C4—28no—
237C22×C4—24no—
238C22×C4—28no—
239C22×C4—24no—
240C22×C4—28no—
241C22×C4—24no—
242C22×C4—28no—
243C22×C4—416no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—48no—
250C22×C4—44no—
251C22×C4—48no—
252C22×C4—832no—
253C22×C4—88no—
254C22×C4—88no—
255C22×C4C2×C22⋊C414yes—
256C22×C4C4×D414yes—
257C22×C4C4×D412yes—
258C22×C4#12114yes—
259C22×C4#12112yes—
260C22×C4#12414yes—
261C22×C4#12712yes—
262C22×C4C2×D818yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C22×C4C2×#4114yes—
269C22×C4C2×#4111yescharacteristic
270C22×C4H(5,𝔽2)14yes—
271D8—416no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—28no—
280C2×C8—416no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—44no—
284C2×C8C2×C22⋊C414yes—
285C2×C8C2×C22⋊C414yes—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—28no—
289C4⋊C4—28no—
290C4⋊C4—416no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C22⋊C4—216no—
296C22⋊C4—416no—
297C22⋊C4—832no—
298C42—42no—
299C42—42no—
300C42—44no—
301C23—24no—
302C23—24no—
303C23—24no—
304C23—216no—
305C23—216no—
306C23—24no—
307C23—24no—
308C23—24no—
309C23—48no—
310C23—48no—
311C23—48no—
312C23—44no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23—82no—
317C23—88no—
318C23—88no—
319C23—88no—
320C23—88no—
321C23—84no—
322C23—82no—
323C23—82no—
324C23—84no—
325C23—84no—
326C23#39014yes—
327C23C4×D818yes—
328C23#46618yes—
329C23#49314yes—
330C23#49514yes—
331C23C2×C4×D411yescharacteristic
332C23C2×#12111yescharacteristic
333C23C2×#12711yescharacteristic
334C23#54614yes—
335C23C22×D812yes—
336C23C2×#13512yes—
337C23#57714yes—
338D4—44no—
339D4—48no—
340D4—44no—
341D4—832no—
342D4—82no—
343D4—82no—
344D4—84no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—28no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—24no—
353C2×C4—216no—
354C2×C4—28no—
355C2×C4—48no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—48no—
359C2×C4—416no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—832no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—88no—
368C2×C4—84no—
369C2×C4#39014yes—
370C2×C4C22×C22⋊C411yescharacteristic
371C2×C4C2×C4×D411yescharacteristic
372C2×C4C4×#4112yes—
373C2×C4#51812yes—
374C2×C4C2×#12111yescharacteristic
375C2×C4#52512yes—
376C2×C4C2×#12411yescharacteristic
377C2×C4#53212yes—
378C2×C4#53412yes—
379C2×C4#53412yes—
380C2×C4#54614yes—
381C2×C4#55314yes—
382C2×C4#55314yes—
383C8—24no—
384C8—24no—
385C8—44no—
386C22—24no—
387C22—216no—
388C22—216no—
389C22—28no—
390C22—28no—
391C22—24no—
392C22—84no—
393C22—816no—
394C22—84no—
395C22—84no—
396C22—168no—
397C22—162no—
398C22—162no—
399C22—162no—
400C22—162no—
401C22#197018yes—
402C22C2×#39011yescharacteristic
403C22#236412yes—
404C22#253912yes—
405C22#254312yes—
406C22C2×C4×D812yes—
407C22#295212yes—
408C22C2×#46612yes—
409C22#306012yes—
410C22C2×#49311yescharacteristic
411C22C2×#49511yescharacteristic
412C22#315412yes—
413C22#329418yes—
414C22#330118yes—
415C4—24no—
416C4—48no—
417C4—88no—
418C4—84no—
419C4#229711yescharacteristic
420C4#230811yescharacteristic
421C4#236412yes—
422C4#254312yes—
423C4#259112yes—
424C2—28no—
425C2—164no—
426C2#1415211yescharacteristic
427C2C2×#197011yescharacteristic
428C2#2122911yescharacteristic
429C2#2312714yes—
430C2#2832912yes—
431C2#2833412yes—
432C2#2855114yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#466C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1874C2semidirect
3. C2×C8×C22⋊C4C2semidirect
4. C22×#340C2semidirect
5. #14137C2semidirect
6. #11379C2semidirect
7. #11022C2semidirect
8. #10925C2semidirect
15. C2×#466C4semidirect
40. C22×D8C2×C4semidirect
68. C2×#107C2×C4semidirect
74. C2×#102C2×C4semidirect
138. C22×C8C22⋊C4semidirect
175. C2×C22⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C22×#4662
C2×#18742
C2×C8×C22⋊C42
C22×#3402
#141372
#113792
#110222
#109252