Finite Group Databasebeta

← #364065

Subgroups — #364065

Order 512.

2026 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C42×D4C211yesmaximal, characteristic
2C2×#1839C211yesmaximal, characteristic, complemented
3C2×#1745C211yesmaximal, characteristic, complemented
4#11346C212yesmaximal, complemented
5#10964C212yesmaximal
6#10922C214yesmaximal, complemented
7#10712C214yesmaximal, complemented
8C22×C4×D4C411yescharacteristic
9C22×C4×D4C411yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C23×C42C2212yes—
12C22×C22⋊C8C2218yes—
13C22×#97C2212yes—
14C42×D4—24no—
15C42×D4—24no—
16C2×C4×C4⋊C4C411yescharacteristic
17C2×C4×C4⋊C4C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C22⋊C4C412yes—
20C2×C4×C22⋊C4C412yes—
21C2×C4×C22⋊C4C2212yes—
22C2×C43C2211yescharacteristic
23#1839—24no—
24#1745—24no—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2218yes—
31C23×D4C2×C411yescharacteristic
32C24×C4C2×C412yes—
33C24×C4C2×C412yes—
34C24×C4C2312yes—
35C23×C8C2314yes—
36C23×C8C2314yes—
37C2×C4×D4—24no—
38C2×C4×D4—28no—
39C2×C4×D4—28no—
40C2×C4×D4—24no—
41C2×C4×D4—24no—
42C2×C4×D4—24no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C22⋊C4C2×C412yes—
49C22×C22⋊C4C2×C412yes—
50C22×C22⋊C4C2×C412yes—
51C22×C42—24no—
52C22×C42—24no—
53C22×C42—48no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C412yes—
58C22×C42C2×C412yes—
59C22×C42C2×C411yescharacteristic
60C22×C42C2312yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C22⋊C8—216no—
65C2×C22⋊C8—216no—
66C2×C22⋊C8—48no—
67C2×#97—28no—
68C2×#97—24no—
69C4×C4⋊C4—24no—
70C4×C4⋊C4—24no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C22⋊C4—28no—
74C4×C22⋊C4—28no—
75C4×C22⋊C4—24no—
76C4×C22⋊C4—24no—
77C2×#95C2×C412yes—
78C2×#95C2×C412yes—
79C2×#95C2×C412yes—
80C43D414yes—
81C43D414yes—
82#336—24no—
83#336—24no—
84#336—24no—
85#336—24no—
86#336—24no—
87#336—48no—
88C25C22×C412yes—
89C22×D4—24no—
90C22×D4—24no—
91C22×D4—22no—
92C22×D4—22no—
93C22×D4M1614yes—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—28no—
103C23×C4—28no—
104C23×C4—28no—
105C23×C4—44no—
106C23×C4—44no—
107C23×C4—44no—
108C23×C4M1618yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C412yes—
113C23×C4C22×C412yes—
114C23×C4C22×C412yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4#4118yes—
119C23×C4C2411yescharacteristic
120C22×C8—24no—
121C22×C8—24no—
122C22×C8—28no—
123C22×C8—28no—
124C22×C8—24no—
125C22×C8—28no—
126C22×C8—28no—
127C22×C8—24no—
128C22×C8—24no—
129C22×C8#4118yes—
130C4×D4—216no—
131C4×D4—216no—
132C4×D4—232no—
133C4×D4—232no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4M1614yes—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C42—24no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—28no—
162C2×C42—416no—
163C2×C42—416no—
164C2×C42—44no—
165C2×C42—44no—
166C2×C42—42no—
167C2×C42C22⋊C414yes—
168C2×C42C22⋊C414yes—
169C2×C42C22⋊C414yes—
170C2×C42C22⋊C414yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C22⋊C8—432no—
178#97—216no—
179#95—24no—
180#95—24no—
181#95—24no—
182C24—22no—
183C24—22no—
184C24—24no—
185C24—24no—
186C24—22no—
187C24—22no—
188C24—24no—
189C24—24no—
190C24#11714yes—
191C24C2×M1614yes—
192C24C2×M1614yes—
193C24C23×C411yescharacteristic
194C2×D4—28no—
195C2×D4—28no—
196C2×D4—216no—
197C2×D4—216no—
198C2×D4—28no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—216no—
207C22×C4—22no—
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—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—216no—
232C22×C4—216no—
233C22×C4—216no—
234C22×C4—416no—
235C22×C4—416no—
236C22×C4—416no—
237C22×C4—48no—
238C22×C4—48no—
239C22×C4—48no—
240C22×C4—416no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—48no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4#11714yes—
255C22×C4C2×M1614yes—
256C22×C4C2×M1614yes—
257C22×C4C2×M1612yes—
258C22×C4C2×M1612yes—
259C22×C4C2×M1614yes—
260C22×C4#13114yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C2×#4114yes—
264C22×C4C2×#4114yes—
265C22×C4H(5,𝔽2)14yes—
266C2×C8—28no—
267C2×C8—216no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—48no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C2×C8—48no—
275C2×C8—48no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C22⋊C4—216no—
280C22⋊C4—216no—
281C22⋊C4—216no—
282C42—24no—
283C42—24no—
284C42—24no—
285C42—24no—
286C42—24no—
287C42—24no—
288C42—28no—
289C42—28no—
290C42—28no—
291C42—44no—
292C42—48no—
293C42—48no—
294C42—44no—
295C42—44no—
296C23—28no—
297C23—24no—
298C23—24no—
299C23—24no—
300C23—24no—
301C23—24no—
302C23—28no—
303C23—28no—
304C23—28no—
305C23—28no—
306C23—28no—
307C23—24no—
308C23—42no—
309C23—44no—
310C23—42no—
311C23—42no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23—42no—
318C23#43218yes—
319C23C22×C22⋊C411yescharacteristic
320C23C2×#11712yes—
321C23C4×#4112yes—
322C23#51812yes—
323C23C22×M1611yescharacteristic
324C23C22×M1612yes—
325C23C22×M1611yescharacteristic
326C23C2×#13112yes—
327C23#56812yes—
328D4—232no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—28no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—28no—
350C2×C4—28no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—432no—
356C2×C4—432no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—416no—
371C2×C4—416no—
372C2×C4#40714yes—
373C2×C4#40714yes—
374C2×C4#40814yes—
375C2×C4#52512yes—
376C2×C4#52512yes—
377C2×C4#52512yes—
378C2×C4#53212yes—
379C2×C4#53512yes—
380C2×C4#53512yes—
381C2×C4#55118yes—
382C2×C4#55318yes—
383C8—48no—
384C8—84no—
385C22—216no—
386C22—28no—
387C22—48no—
388C22—48no—
389C22—48no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—44no—
395C22—44no—
396C22#228011yescharacteristic
397C22#229711yescharacteristic
398C22#230911yescharacteristic
399C22#235314yes—
400C22C2×#40711yescharacteristic
401C22C2×#40711yescharacteristic
402C22C2×#40811yescharacteristic
403C22#287911yescharacteristic
404C22C2×#43212yes—
405C22#292012yes—
406C22#292514yes—
407C22#296514yes—
408C22#297012yes—
409C22#297112yes—
410C22#297314yes—
411C22#297412yes—
412C22#297412yes—
413C4—24no—
414C4—24no—
415C4—24no—
416C4—44no—
417C4—44no—
418C4—42no—
419C4—42no—
420C4—44no—
421C4—42no—
422C4#259814yes—
423C4#260314yes—
424C2—48no—
425C2#1395911yescharacteristic
426C2#2071511yescharacteristic
427C2#2072111yescharacteristic
428C2#2137411yescharacteristic
429C2#2138011yescharacteristic
430C2#2178811yescharacteristic
431C2#2179011yescharacteristic
432C2#2278714yes—
433C2#2278814yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1839C2semidirect
3. C2×#1745C2semidirect
4. #11346C2semidirect
6. #10922C2semidirect
7. #10712C2semidirect

Maximal subgroups

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

subgroupindex
C2×C42×D42
C2×#18392
C2×#17452
#113462
#109642
#109222
#107122