Finite Group Databasebeta

← #6434097

Subgroups — #6434097

Order 512.

1973 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#29915C218yesmaximal, complemented
4#29914C218yesmaximal, complemented
5#23745C212yesmaximal, complemented
6#23734C212yesmaximal, complemented
7#23659C212yesmaximal, complemented
8#23631C212yesmaximal, complemented
9#20910C212yesmaximal
10#20845C212yesmaximal, complemented
11#20682C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13C22×#117C2212yes—
14C23×C42C2211yescharacteristic
15C2×#428C2214yes—
16C2×#417C2214yes—
17C2×#407C2212yes—
18#2855C2214yes—
19#2854C2214yes—
20#2853—22no—
21#2849—22no—
22C4×#123—24no—
23#2297C2211yescharacteristic
24#2297C2211yescharacteristic
25#2296C2211yescharacteristic
26#2296C2211yescharacteristic
27#2285—24no—
28#2285C2218yes—
29C2×#384C2212yes—
30C2×#382C2212yes—
31#2280C2211yescharacteristic
32C2×#380—22no—
33C2×C4×C4⋊C4—22no—
34#2100C2218yescomplemented
35#2099C2218yescomplemented
36#2096C2214yescomplemented
37#2095C2214yescomplemented
38#2094C2214yescomplemented
39#2093C2214yescomplemented
40#2085C22116yescomplemented
41#2074C2218yescomplemented
42#2073C2218yescomplemented
43#1875C2218yes—
44#1873C2218yes—
45#1854C22116yes—
46#1817C2218yes—
47#1736C2214yes—
48#1736C2214yes—
49#1728C2212yes—
50#1728C2212yes—
51C22×#41C2311yescharacteristic
52C24×C4D412yes—
53C24×C4C2311yescharacteristic
54C22×M16C2314yes—
55#525—28no—
56C2×#123—24no—
57C2×#123—21no—
58C2×C22⋊Q8C2312yes—
59C2×#121C2312yes—
60C2×#120—21no—
61C2×C4×D4—22no—
62C2×#117—24no—
63C2×#117—48no—
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2318yes—
68C22×C4⋊C4—24no—
69C22×C4⋊C4C2314yes—
70C22×C22⋊C4—22no—
71C22×C42—28no—
72C22×C42—22no—
73C22×C42D412yes—
74C22×C42D414yes—
75C22×C42C2314yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#428—48no—
79C2×#107C2318yes—
80C2×#106C2318yes—
81#417—48no—
82C2×#103C2318yes—
83C2×#102C2318yes—
84#407—28no—
85C2×C22⋊C8C2318yes—
86#391C2312yes—
87#391C2312yes—
88#390C2312yes—
89#390C2312yes—
90#389C2314yes—
91#389C2314yes—
92#388—24no—
93#388—24no—
94#387—24no—
95#387—24no—
96#386—22no—
97#386—22no—
98#386—24no—
99#385—22no—
100#385—22no—
101#385—24no—
102#384—24no—
103#384C2314yes—
104#384C2314yes—
105#382—24no—
106#382C2318yes—
107#381—24no—
108#381—21no—
109#380—28no—
110#380—28no—
111#380C2314yes—
112#379—21no—
113C4×C4⋊C4—28no—
114C4×C4⋊C4—28no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C2×#95—24no—
121#340C23116yes—
122#340C23116yes—
123#336C2318yes—
124#336C2314yes—
125#336C2314yes—
126C25C2×D411yescharacteristic
127C2×#41—24no—
128C2×#41—24no—
129C2×#41—24no—
130C22×Q8C2411yescharacteristic
131C22×D4—22no—
132C22×D4—21no—
133C22×D4C2411yescharacteristic
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—28no—
137C23×C4—24no—
138C23×C4—21no—
139C23×C4—48no—
140C23×C4C2×D412yes—
141C23×C4C2×D414yes—
142C23×C4C2×D414yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4#4118yes—
147C23×C4C2412yes—
148C23×C4C2411yescharacteristic
149C2×M16—28no—
150C2×M16—28no—
151C2×M16—28no—
152C22×C8C2418yes—
153#123—416no—
154#123—44no—
155C22⋊Q8—28no—
156#121—28no—
157#120—44no—
158C4×D4—48no—
159#117—24no—
160#117—24no—
161#117—24no—
162#117—416no—
163#117—416no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—416no—
173C2×C4⋊C4C2418yes—
174C2×C4⋊C4C2412yes—
175C2×C4⋊C4C2412yes—
176C2×C4⋊C4C2412yes—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—48no—
187C2×C42—28no—
188C2×C42—216no—
189C2×C42—24no—
190C2×C42—216no—
191C2×C42—24no—
192C2×C42—28no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—48no—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D414yes—
200C2×C42C2×D414yes—
201C2×C42C2412yes—
202C2×C42C2412yes—
203#107—48no—
204#106—48no—
205#103—48no—
206#102—48no—
207C22⋊C8—216no—
208#95—216no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214C24—22no—
215C24—22no—
216C24—21no—
217C24—44no—
218C24#12314yes—
219C24C22×D412yes—
220C24C22×D411yescharacteristic
221#41—416no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—48no—
229C2×D4—44no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—232no—
239C22×C4—28no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—28no—
243C22×C4—28no—
244C22×C4—24no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—28no—
251C22×C4—22no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—432no—
256C22×C4—416no—
257C22×C4—48no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4#12318yes—
262C22×C4#12314yes—
263C22×C4C22×D414yes—
264C22×C4C22×D412yes—
265C22×C4C22×D414yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4118yes—
268C22×C4C2×#4114yes—
269C22×C4H(5,𝔽2)14yes—
270C22×C42−1+414yes—
271C22×C4C2511yescharacteristic
272M16—416no—
273C2×C8—28no—
274C2×C8—28no—
275C2×C8—28no—
276C4⋊C4—24no—
277C4⋊C4—28no—
278C4⋊C4—24no—
279C4⋊C4—28no—
280C4⋊C4—24no—
281C4⋊C4—416no—
282C4⋊C4—48no—
283C4⋊C4—44no—
284C4⋊C4—48no—
285C4⋊C4—416no—
286C4⋊C4—48no—
287C22⋊C4—48no—
288C22⋊C4—416no—
289C22⋊C4—48no—
290C22⋊C4—42no—
291C22⋊C4—44no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—44no—
295C22⋊C4—48no—
296C42—24no—
297C42—28no—
298C42—28no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—416no—
303C42—42no—
304C42—42no—
305C42—42no—
306C23—24no—
307C23—28no—
308C23—22no—
309C23—22no—
310C23—24no—
311C23—24no—
312C23—48no—
313C23—48no—
314C23—48no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—44no—
319C23C2×#12314yes—
320C23C2×#12312yes—
321C23#53512yes—
322C23#53612yes—
323C23C23×D411yescharacteristic
324Q8—44no—
325D4—44no—
326D4—88no—
327D4—84no—
328C2×C4—216no—
329C2×C4—28no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—216no—
335C2×C4—216no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—416no—
340C2×C4—416no—
341C2×C4—48no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—48no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—416no—
352C2×C4—44no—
353C2×C4—816no—
354C2×C4—84no—
355C2×C4C2×#12314yes—
356C2×C4C2×#12312yes—
357C2×C4#53512yes—
358C2×C4#53612yes—
359C2×C4#539116yes—
360C2×C4#57518yes—
361C2×C4C22×#4112yes—
362C2×C4C2×H(5,𝔽2)12yes—
363C2×C4C2×2−1+412yes—
364C8—48no—
365C22—24no—
366C22—24no—
367C22—24no—
368C22—24no—
369C22—24no—
370C22—24no—
371C22—416no—
372C22—44no—
373C22—44no—
374C22—81no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22#262314yes—
379C22C22×#12311yescharacteristic
380C22C2×#53511yescharacteristic
381C22C2×#53611yescharacteristic
382C22C2×#57514yes—
383C4—24no—
384C4—28no—
385C4—44no—
386C4—81no—
387C4—82no—
388C4—82no—
389C4—88no—
390C4—82no—
391C4#262314yes—
392C4C2×#53914yes—
393C2—24no—
394C2—44no—
395C2—161no—
396C2C2×#262311yescharacteristic
397C2#6051112yes—
398C2#6080112yes—
399C2#6086312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
3. #29915C2semidirect
4. #29914C2semidirect
5. #23745C2semidirect
6. #23734C2semidirect
7. #23659C2semidirect
8. #23631C2semidirect
10. #20845C2semidirect
11. #20682C2semidirect
34. #2100C22semidirect
35. #2099C22semidirect
36. #2096C22semidirect
37. #2095C22semidirect
38. #2094C22semidirect
39. #2093C22semidirect
40. #2085C22semidirect
41. #2074C22semidirect
42. #2073C22semidirect

Maximal subgroups

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

subgroupindex
#366262
C2×#22852
#299152
#299142
#237452
#237342
#236592
#236312
#209102
#208452
#206822