Finite Group Databasebeta

← #6434796

Subgroups — #6434796

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2C2×C4×#125C211yesmaximal, characteristic
3#30099C214yesmaximal, complemented
4#30098C214yesmaximal, complemented
5#30097C218yesmaximal, complemented
6#23747C212yesmaximal
7#23746C212yesmaximal
8#23657C212yesmaximal
9#23553C212yesmaximal
10#21033C214yesmaximal
11#20772C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C2×#428C2214yes—
14#2864C2214yes—
15#2859C2212yes—
16#2859C2212yes—
17#2851—22no—
18#2851—22no—
19C4×#125—22no—
20C4×#125—22no—
21C4×#125—22no—
22C4×#125—22no—
23C4×#125C2214yes—
24C4×#125C2214yes—
25#2296C2212yes—
26#2296C2212yes—
27C2×#389C2212yes—
28C2×#387—22no—
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4—22no—
33C2×C4×C4⋊C4C2212yes—
34C2×C43C2211yescharacteristic
35#2101C2218yes—
36#2101C2218yes—
37#2101C2218yes—
38#2100C2214yes—
39#2099C2214yes—
40#2085C2218yes—
41#2084C2214yes—
42#2083C2214yes—
43#2059C2218yes—
44#2059C2218yes—
45#1921C2218yes—
46#1920C2218yes—
47#1919C2218yes—
48#1918C2218yes—
49#1839C2214yes—
50#1839C2214yes—
51#1736C2214yes—
52#1736C2214yes—
53#1736C2214yes—
54#1728C2212yes—
55#1728C2212yes—
56#1728C2212yes—
57C22×M16C2314yes—
58C2×#125—28no—
59C2×#125—21no—
60C2×#125—21no—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125C2314yes—
64C2×#117C2312yes—
65C2×#117C2312yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C42—22no—
71C22×C42D412yes—
72C22×C42D412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#428—216no—
77C2×#107C2318yes—
78C2×#106C2318yes—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389C2314yes—
86#389C2314yes—
87#389C2314yes—
88#389C2314yes—
89#389C2314yes—
90#389C2314yes—
91#387—24no—
92#387—24no—
93#387—24no—
94#387—24no—
95#387—24no—
96#387—24no—
97#387—24no—
98#387—24no—
99#385—22no—
100#385—24no—
101#385—22no—
102#384C2312yes—
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#382—21no—
106#382—21no—
107#382—22no—
108#382—22no—
109#382C2314yes—
110#382C2312yes—
111#382C2312yes—
112#380—22no—
113#380—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4C2314yes—
126C4×C4⋊C4C2314yes—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C43D412yes—
131C43D412yes—
132C43C2312yes—
133C43C2312yes—
134#340C2318yes—
135#340C2318yes—
136#340C2318yes—
137#340C2318yes—
138#340C2318yes—
139#340C2318yes—
140#336C2318yes—
141#336C2314yes—
142#336C2314yes—
143#336C2318yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C2×M16—28no—
152C2×M16—28no—
153C2×M16—28no—
154C22×C8C2418yes—
155#125—28no—
156#125—28no—
157#125—216no—
158#125—48no—
159#125—48no—
160#117—28no—
161#117—28no—
162C2×C4⋊C4—216no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4#4118yes—
181C2×C4⋊C4C2414yes—
182C2×C4⋊C4C2412yes—
183C2×C4⋊C4C2412yes—
184C2×C4⋊C4C2414yes—
185C2×C4⋊C4C2414yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—44no—
196C2×C42—44no—
197C2×C42—48no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2412yes—
211C2×C42C2412yes—
212C2×C42C2411yescharacteristic
213C2×C42C2411yescharacteristic
214#107—216no—
215#106—216no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228C24C22×D411yescharacteristic
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—28no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4C22×D412yes—
251C22×C4C22×D412yes—
252C22×C4C22×D412yes—
253C22×C4C22×D412yes—
254C22×C4C22×D412yes—
255C22×C4C22×D412yes—
256C22×C4C22×D411yescharacteristic
257C22×C4C2×#4114yes—
258C22×C42−1+414yes—
259C22×C42−1+414yes—
260C22×C4C2511yescharacteristic
261M16—416no—
262C2×C8—28no—
263C2×C8—28no—
264C2×C8—28no—
265C4⋊C4—216no—
266C4⋊C4—216no—
267C4⋊C4—28no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—48no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—48no—
278C4⋊C4—48no—
279C4⋊C4—48no—
280C4⋊C4—44no—
281C4⋊C4—48no—
282C4⋊C4—48no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4C2×#4118yes—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C42—22no—
289C42—22no—
290C42—24no—
291C42—24no—
292C42—24no—
293C42—24no—
294C42—24no—
295C42—24no—
296C42—24no—
297C42—22no—
298C42—24no—
299C42—22no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—48no—
305C42—44no—
306C42—44no—
307C42—48no—
308C23—22no—
309C23—22no—
310C23#52512yes—
311C23#53612yes—
312C23#53612yes—
313C23C23×D411yescharacteristic
314C2×C4—28no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—28no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—416no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—88no—
349C2×C4#52512yes—
350C2×C4#53612yes—
351C2×C4#53612yes—
352C2×C4#541116yes—
353C2×C4#57514yes—
354C2×C4#57514yes—
355C2×C4C22×#4112yes—
356C2×C4C2×2−1+412yes—
357C2×C4C2×2−1+412yes—
358C8—48no—
359C22—24no—
360C22—22no—
361C22—22no—
362C22—24no—
363C22#263414yes—
364C22C2×#52511yescharacteristic
365C22C2×#53611yescharacteristic
366C22C2×#53611yescharacteristic
367C22C2×#57512yes—
368C22C2×#57512yes—
369C4—24no—
370C4—24no—
371C4—42no—
372C4—42no—
373C4—48no—
374C4—42no—
375C4—42no—
376C4—84no—
377C4—82no—
378C4—82no—
379C4—84no—
380C4#263414yes—
381C4C2×#54114yes—
382C2—24no—
383C2C2×#263411yescharacteristic
384C2#6055912yes—
385C2#6086111yescharacteristic
386C2#6086211yescharacteristic
387C2#6086612yes—
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. #30099C2semidirect
4. #30098C2semidirect
5. #30097C2semidirect

Maximal subgroups

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

subgroupindex
#366402
C2×C4×#1252
#300992
#300982
#300972
#237472
#237462
#236572
#235532
#210332
#207722