Finite Group Databasebeta

← #6468754

Subgroups — #6468754

Order 512.

1881 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C212yesmaximal
2#29425C214yesmaximal, complemented
3#29251C214yesmaximal, complemented
4#28132C214yesmaximal, complemented
5#27609C211yesmaximal, characteristic
6#24016C212yesmaximal
7#23928C211yesmaximal, characteristic
8#23709C214yesmaximal, complemented
9#21583C214yesmaximal, complemented
10#21455C214yesmaximal, complemented
11#20719C211yesmaximal, characteristic
12C2×#536C2212yes—
13C2×C4×#41C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C2×#485C2214yes—
16C2×#451C2214yes—
17C2×#441C2214yes—
18C2×#432C2212yes—
19#2327—22no—
20#2326—22no—
21#2323C2214yes—
22#2323C2214yes—
23#2322—22no—
24#2322—22no—
25#2317C2212yes—
26#2311C2212yes—
27#2310C2212yes—
28C4×C4⋊Q8—22no—
29C4×#124—22no—
30C4×C22⋊Q8C2214yes—
31C4×C22⋊Q8C2214yes—
32C4×#123—22no—
33C4×#123—22no—
34C4×#117C2212yes—
35#2088C2214yes—
36#2077C2214yes—
37#2068C2214yes—
38#2066C2214yes—
39#2056C2214yes—
40#2036C2214yes—
41#2035C2214yes—
42#2001C2214yes—
43#1928C2214yes—
44#1927C2214yes—
45#1921C2214yes—
46#1920C2214yes—
47#1878C2214yes—
48#1878C2214yes—
49#1873C2214yes—
50#1872C2214yes—
51#1865C2214yes—
52#1865C2214yes—
53#1854C2214yes—
54#1853C2214yes—
55#1825C2214yes—
56#1818C2212yes—
57#1798C2214yes—
58#1787C2214yes—
59#1780C2211yescharacteristic
60#1780C2211yescharacteristic
61#1776C2214yes—
62#1763C2214yes—
63#1763C2214yes—
64#1749C2214yes—
65C22×#41C2311yescharacteristic
66C23×Q8C2312yes—
67C22×Q16C2314yes—
68#536—216no—
69C2×C4⋊Q8—22no—
70C2×#124—22no—
71C2×#123—22no—
72C2×#123—22no—
73C2×C22⋊Q8C2314yes—
74C2×C22⋊Q8C2314yes—
75C4×#41—44no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—44no—
79C2×C4×Q8—44no—
80C2×C4×Q8D412yes—
81C2×C4×Q8C2314yes—
82C2×C4×Q8C2314yes—
83C2×C4×D4D412yes—
84C2×C4×D4D412yes—
85C2×C4×D4D412yes—
86C2×#117C2312yes—
87C2×#117C2312yes—
88C2×#117C2312yes—
89C2×#117C2311yescharacteristic
90C22×C4⋊C4—22no—
91C22×C4⋊C4C2312yes—
92C22×C42—22no—
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#485—216no—
96#451—216no—
97#441—48no—
98#432—44no—
99C2×#107C2314yes—
100C2×#106C2314yes—
101C2×#106C2314yes—
102C2×C4⋊C8C2314yes—
103C2×#103C2314yes—
104C2×#103C2314yes—
105C2×#103C2314yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×C22⋊C8C2314yes—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2312yes—
112C2×C22⋊C8C2312yes—
113C2×#97C2314yes—
114#391—22no—
115#391—22no—
116#391—22no—
117#391C2314yes—
118#391C2314yes—
119#390—22no—
120#389—24no—
121#389—24no—
122#389C2314yes—
123#389C2314yes—
124#388—24no—
125#388—24no—
126#388—24no—
127#388—24no—
128#387—24no—
129#387—24no—
130#387—24no—
131#387—24no—
132#386—22no—
133#386—22no—
134#385—24no—
135#385—24no—
136#385—22no—
137#385—22no—
138#384—22no—
139#382—22no—
140#381—22no—
141#381—22no—
142#380C2314yes—
143#380C2314yes—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4C2314yes—
147C4×C4⋊C4C2314yes—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—24no—
152#376—22no—
153C43—22no—
154#340C2314yes—
155#340C2314yes—
156#340C2314yes—
157#340C2314yes—
158#340C2314yes—
159#340C2314yes—
160#336C2314yes—
161#336C2314yes—
162C2×#41—28no—
163C2×#41—42no—
164C22×Q8—28no—
165C22×Q8—22no—
166C22×Q8—28no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8C2×D411yescharacteristic
170C22×Q8C2414yes—
171C22×D4C2×D411yescharacteristic
172C22×D4C2×D412yes—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C2×Q16—28no—
181C2×Q16—28no—
182C2×Q16—44no—
183C22×C8C2414yes—
184C22×C8C2414yes—
185C4⋊Q8—216no—
186#124—216no—
187#123—216no—
188#123—216no—
189C22⋊Q8—28no—
190C22⋊Q8—28no—
191C22⋊Q8—28no—
192C22⋊Q8#41116yes—
193C4×Q8—28no—
194C4×Q8—28no—
195C4×Q8—44no—
196C4×Q8—44no—
197C4×Q8—44no—
198C4×Q8—48no—
199C4×Q8—48no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—44no—
203#117—28no—
204#117—28no—
205#117—42no—
206#117—42no—
207#117—42no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2414yes—
222C2×C4⋊C4C2414yes—
223C2×C4⋊C4C2414yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4C2×D412yes—
231C2×C22⋊C4C2×D412yes—
232C2×C22⋊C4C2×D411yescharacteristic
233C2×C22⋊C4C2×D411yescharacteristic
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—48no—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2412yes—
248C2×C42C2411yescharacteristic
249C2×C42C2411yescharacteristic
250#107—28no—
251#106—28no—
252#106—44no—
253C4⋊C8—44no—
254#103—28no—
255#103—28no—
256#103—28no—
257#103—28no—
258#103—44no—
259#103—44no—
260C22⋊C8—28no—
261C22⋊C8—28no—
262C22⋊C8—42no—
263C22⋊C8—42no—
264#97—28no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275C24C22×D411yescharacteristic
276C24C22×D412yes—
277#41—416no—
278C2×Q8—28no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—41no—
286C2×Q8—416no—
287C2×Q8—416no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8C2×#4118yes—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—42no—
295C2×D4—41no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—28no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—84no—
316C22×C4C22×D412yes—
317C22×C4C22×D412yes—
318C22×C4C22×D412yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D412yes—
321C22×C4C22×D412yes—
322C22×C4C2×#4118yes—
323C22×C4C2511yescharacteristic
324Q16—416no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C4⋊C4—216no—
332C4⋊C4—216no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C4⋊C4—48no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C4⋊C4C2×#4118yes—
347C22⋊C4—28no—
348C22⋊C4—216no—
349C22⋊C4—216no—
350C22⋊C4—41no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—42no—
354C22⋊C4—44no—
355C22⋊C4—84no—
356C22⋊C4—84no—
357C42—28no—
358C42—24no—
359C42—24no—
360C42—24no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—48no—
365C42—48no—
366C42—44no—
367C42—48no—
368C42—84no—
369C42H(5,𝔽2)14yes—
370C23—24no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—22no—
376C23—21no—
377C23#52514yes—
378C23C23×D411yescharacteristic
379Q8—28no—
380Q8—28no—
381Q8—44no—
382Q8—416no—
383Q8—44no—
384Q8—44no—
385D4—48no—
386D4—44no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—48no—
403C2×C4—416no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—88no—
418C2×C4#52514yes—
419C2×C4#53414yes—
420C2×C4C22×#4114yes—
421C2×C4C2×H(5,𝔽2)12yes—
422C8—44no—
423C8—44no—
424C22—24no—
425C22—28no—
426C22—42no—
427C22—42no—
428C22—44no—
429C22—42no—
430C22—41no—
431C22—44no—
432C22C2×#52512yes—
433C22C2×#53411yescharacteristic
434C22#358614yes—
435C4—24no—
436C4—24no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—41no—
441C4—48no—
442C4—48no—
443C4—84no—
444C4—84no—
445C4#348114yes—
446C2—44no—
447C2—81no—
448C2#3495611yescharacteristic
449C2#6057714yes—
450C2#6075412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29425C2semidirect
3. #29251C2semidirect
4. #28132C2semidirect
8. #23709C2semidirect
9. #21583C2semidirect
10. #21455C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
#294252
#292512
#281322
#276092
#240162
#239282
#237092
#215832
#214552
#207192