Finite Group Databasebeta

← #6494922

Subgroups — #6494922

Order 512.

1738 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic
3#33196C214yesmaximal, complemented
4#29996C214yesmaximal, complemented
5#29287C214yesmaximal, complemented
6#28283C214yesmaximal, complemented
7#24838C218yesmaximal
8#21959C212yesmaximal
9#21885C212yesmaximal
10C2×#1842C211yesmaximal, characteristic
11C2×#533—21no—
12C2×#533C2211yescharacteristic
13C2×#475C2218yes—
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2214yes—
16#2868C2212yes—
17#2862—21no—
18#2859—21no—
19#2859C2212yes—
20#2857—22no—
21#2857C2214yes—
22C4×C4⋊Q8—22no—
23C4×C4⋊Q8C2214yes—
24C4×C4⋊Q8C2212yes—
25C4×C4⋊Q8C2212yes—
26C4×#125—22no—
27C4×C22⋊Q8—22no—
28C4×C22⋊Q8C2214yes—
29#2296—21no—
30#2295—21no—
31#2295C2212yes—
32#2285—21no—
33#2285C2212yes—
34#2285C2211yescharacteristic
35C42×Q8—21no—
36C42×Q8C2212yes—
37C4×#117—21no—
38C2×C43C2211yescharacteristic
39#2088C22116yes—
40#2066C22116yes—
41#1988C2218yes—
42#1987C2214yes—
43#1981C2214yes—
44#1948C2218yes—
45#1948C2218yes—
46#1948C2218yes—
47#1892C2218yes—
48#1842C2212yes—
49#1842C2214yes—
50#1842C2212yes—
51#1807C2218yes—
52#1803C2218yes—
53#1803C2218yes—
54C23×C8C2314yes—
55#533—24no—
56#533—48no—
57#533Q818yescomplemented
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2314yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125—21no—
64C2×#125—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8C2312yes—
68C2×C4×Q8—21no—
69C2×C4×Q8—24no—
70C2×C4×Q8—22no—
71C2×C4×Q8C2314yes—
72C2×C4×Q8C2312yes—
73C2×#117—21no—
74C2×#117—22no—
75C2×#117—21no—
76C22×C42D411yescharacteristic
77C22×C42D412yes—
78C22×C42D411yescharacteristic
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#475—416no—
82C2×#106C23116yes—
83C2×C4⋊C8—28no—
84C2×C4⋊C8—48no—
85C2×C4⋊C8C2318yes—
86C2×C4⋊C8C2314yes—
87C2×C4⋊C8C2314yes—
88C2×C4⋊C8C2318yes—
89C2×#103C23116yes—
90#391—24no—
91#391—24no—
92#391—22no—
93#391C2314yes—
94#389—22no—
95#389—22no—
96#389—24no—
97#389—24no—
98#389—24no—
99#389—22no—
100#389C2318yes—
101#389C2314yes—
102#387—24no—
103#387—24no—
104#385—24no—
105#384—22no—
106#384—22no—
107#384C2312yes—
108#384C2314yes—
109#384C2314yes—
110#382—21no—
111#382—22no—
112#382—22no—
113#380—22no—
114#380—24no—
115#380—21no—
116#380C2312yes—
117#380C2314yes—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2314yes—
129C4×C4⋊C4C2314yes—
130C4×C4⋊C4C2312yes—
131C4×C22⋊C4—24no—
132#376—22no—
133C43—22no—
134C43D412yes—
135C43D411yescharacteristic
136C43D411yescharacteristic
137C43C2312yes—
138C43C2311yescharacteristic
139C43C2311yescharacteristic
140#340C23116yes—
141#340C23116yes—
142C22×Q8—21no—
143C22×Q8—24no—
144C22×Q8C2412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—28no—
151C22×C8—48no—
152C22×C8C2418yes—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—28no—
156C4⋊Q8—28no—
157C4⋊Q8—48no—
158C4⋊Q8—48no—
159C4⋊Q8C2×Q814yes—
160C4⋊Q8C2×Q814yes—
161#125—28no—
162#125—48no—
163C22⋊Q8—28no—
164C22⋊Q8—24no—
165C22⋊Q8—416no—
166C22⋊Q8C2×Q818yes—
167C4×Q8—28no—
168C4×Q8—24no—
169C4×Q8—28no—
170C4×Q8—28no—
171C4×Q8—48no—
172C4×Q8—416no—
173C4×Q8C2×Q818yes—
174#117—28no—
175#117—48no—
176#117—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2418yes—
194C2×C4⋊C4C2412yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—44no—
207C2×C42SD1618yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×Q814yes—
223C2×C42#4114yes—
224C2×C42C2412yes—
225C2×C42C2411yescharacteristic
226C2×C42C2411yescharacteristic
227C2×C42C2412yes—
228#106—416no—
229C4⋊C8—28no—
230C4⋊C8—28no—
231C4⋊C8—216no—
232C4⋊C8—48no—
233#103—416no—
234#95—24no—
235#95—24no—
236#95—22no—
237#95—24no—
238#95—22no—
239#95—24no—
240#95—28no—
241#95—24no—
242#95—24no—
243C24C22×D411yescharacteristic
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—44no—
249C2×Q8—48no—
250C2×Q8—48no—
251C2×Q8—48no—
252C2×Q8C22×Q814yes—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—28no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C22⋊Q814yes—
266C22×C4C22⋊Q814yes—
267C22×C4C2×SD1618yes—
268C22×C4C2×SD1614yes—
269C22×C4#13714yes—
270C22×C4C22×D412yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D412yes—
275C22×C4C22×D412yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D412yes—
279C22×C4C22×Q812yes—
280C22×C4C2×#4112yes—
281C22×C4C2511yescharacteristic
282C2×C8—28no—
283C2×C8—216no—
284C2×C8—88no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—28no—
289C4⋊C4—28no—
290C4⋊C4—22no—
291C4⋊C4—28no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—28no—
295C4⋊C4—44no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—48no—
300C4⋊C4—48no—
301C4⋊C4—48no—
302C4⋊C4—416no—
303C4⋊C4—48no—
304C4⋊C4—44no—
305C4⋊C4C22×Q814yes—
306C22⋊C4—28no—
307C22⋊C4—48no—
308C22⋊C4—44no—
309C42—28no—
310C42—24no—
311C42—24no—
312C42—22no—
313C42—24no—
314C42—22no—
315C42—24no—
316C42—24no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—42no—
321C42—48no—
322C42—48no—
323C42—44no—
324C42—48no—
325C42C22⋊Q814yes—
326C42C22⋊Q814yes—
327C42C2×SD1618yes—
328C42C2×SD1614yes—
329C42C2×SD1614yes—
330C42C22×Q812yes—
331C42C22×Q812yes—
332C42C2×#4112yes—
333C42C2×#4112yes—
334C42H(5,𝔽2)14yes—
335C422−1+414yes—
336C23—22no—
337C23—22no—
338C23C2×C22⋊Q812yes—
339C23C22×SD1612yes—
340C23C2×#13712yes—
341C23C23×D411yescharacteristic
342Q8—28no—
343Q8—48no—
344Q8—42no—
345Q8—42no—
346Q8—816no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—216no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—44no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4#475116yescomplemented
374C2×C4C2×C22⋊Q812yes—
375C2×C4C2×C22⋊Q812yes—
376C2×C4C2×C22⋊Q812yes—
377C2×C4C2×C22⋊Q812yes—
378C2×C4C2×C22⋊Q812yes—
379C2×C4#53714yes—
380C2×C4#53714yes—
381C2×C4C22×SD1614yes—
382C2×C4C22×SD1612yes—
383C2×C4C22×SD1612yes—
384C2×C4C22×SD1614yes—
385C2×C4C2×#13712yes—
386C2×C4C2×#13712yes—
387C2×C4#57514yes—
388C2×C4C23×Q811yescharacteristic
389C2×C4C22×#4111yescharacteristic
390C2×C4C2×H(5,𝔽2)12yes—
391C2×C4C2×2−1+412yes—
392C8—48no—
393C22—22no—
394C22—22no—
395C22—24no—
396C22—24no—
397C22C2×#47514yes—
398C22C22×C22⋊Q811yescharacteristic
399C22C2×#53712yes—
400C22C23×SD1611yescharacteristic
401C22C22×#13711yescharacteristic
402C22C2×#57512yes—
403C4—22no—
404C4—22no—
405C4—22no—
406C4—24no—
407C4—24no—
408C4—44no—
409C4—44no—
410C4—42no—
411C4—88no—
412C4—84no—
413C4C2×#47514yes—
414C4C2×#47514yes—
415C4#324618yes—
416C4#350312yes—
417C4#350811yescharacteristic
418C4#350911yescharacteristic
419C2—24no—
420C2#3528011yescharacteristic
421C2C22×#47511yescharacteristic
422C2#6048411yescharacteristic
423C2C2×#324612yes—
424C2#6091812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33196C2semidirect
4. #29996C2semidirect
5. #29287C2semidirect
6. #28283C2semidirect
57. #533Q8semidirect
373. C2×C4#475semidirect

Maximal subgroups

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

subgroupindex
#366452
C4×#5332
#331962
#299962
#292872
#282832
#248382
#219592
#218852
C2×#18422