Finite Group Databasebeta

← #6434799

Subgroups — #6434799

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34638C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic
3#30102C214yesmaximal, complemented
4#30101C214yesmaximal, complemented
5#30100C218yesmaximal, complemented
6#23747C212yesmaximal
7#23746C212yesmaximal
8#23657C212yesmaximal
9#23552C212yesmaximal
10#21033C214yesmaximal
11#20773C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#428C2214yes—
14#2647C2214yes—
15#2647C2212yes—
16#2647C2212yes—
17#2646—22no—
18#2646—22no—
19#2402—24no—
20#2331—24no—
21#2331C2214yes—
22#2331C2214yes—
23#2296C2212yes—
24#2296C2212yes—
25C2×#391—22no—
26C2×#389C2212yes—
27C2×#387—22no—
28#2285C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32C2×#376C2211yescharacteristic
33#2101C2218yes—
34#2101C2218yes—
35#2101C2218yes—
36#2100C2214yes—
37#2099C2214yes—
38#2085C2218yes—
39#2084C2214yes—
40#2083C2214yes—
41#2058C2218yes—
42#2057C2218yes—
43#1921C2218yes—
44#1920C2218yes—
45#1919C2218yes—
46#1918C2218yes—
47#1840C2214yes—
48#1840C2214yes—
49#1736C2214yes—
50#1736C2214yes—
51#1736C2214yes—
52#1728C2212yes—
53#1728C2212yes—
54#1728C2212yes—
55C23×Q8—21no—
56C22×M16C2314yes—
57C2×C4⋊Q8—28no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2314yes—
60C2×#117C2312yes—
61C2×#117C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C42—22no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#428—216no—
71C2×#107C2318yes—
72C2×#106C2318yes—
73#400—24no—
74#400—24no—
75#400—24no—
76#400—24no—
77#398—24no—
78#398—24no—
79#398—24no—
80#398—24no—
81#391—28no—
82#391—28no—
83#389—22no—
84#389—22no—
85#389C2314yes—
86#389C2314yes—
87#389C2314yes—
88#389C2314yes—
89#389C2314yes—
90#389C2314yes—
91#387—28no—
92#387—28no—
93#385—22no—
94#385—22no—
95#385—24no—
96#384C2311yescharacteristic
97#384C2311yescharacteristic
98#384C2312yes—
99#382—22no—
100#382C2314yes—
101#382C2312yes—
102#382C2312yes—
103#380—22no—
104#380—22no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2318yes—
112#376D414yes—
113#376C2312yes—
114#376C2312yes—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95D412yes—
120C2×#95D412yes—
121#340C2318yes—
122#340C2318yes—
123#340C2318yes—
124#340C2318yes—
125#340C2318yes—
126#340C2318yes—
127#336C2318yes—
128#336C2314yes—
129#336C2314yes—
130#336C2314yes—
131#336C2314yes—
132C22×Q8—28no—
133C22×Q8—24no—
134C22×Q8—84no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C2×M16—28no—
143C2×M16—28no—
144C2×M16—28no—
145C22×C8C2418yes—
146C4⋊Q8—216no—
147C4⋊Q8—216no—
148#117—28no—
149#117—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4#4118yes—
168C2×C4⋊C4C2414yes—
169C2×C4⋊C4C2414yes—
170C2×C4⋊C4C2412yes—
171C2×C4⋊C4C2412yes—
172C2×C4⋊C4C2414yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—28no—
181C2×C42—84no—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2412yes—
186C2×C42C2412yes—
187C2×C42C2411yescharacteristic
188C2×C42C2411yescharacteristic
189#107—216no—
190#106—216no—
191#95—28no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95C2×D414yes—
203#95C2×D414yes—
204#95C2×D414yes—
205#95C2×D414yes—
206C24C22×D411yescharacteristic
207C2×Q8—216no—
208C2×Q8—88no—
209C2×Q8—816no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—28no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4C22×D414yes—
232C22×C4C22×D412yes—
233C22×C4C22×D412yes—
234C22×C4C22×D412yes—
235C22×C4C22×D412yes—
236C22×C4C22×D411yescharacteristic
237C22×C4C2×#4114yes—
238C22×C42−1+414yes—
239C22×C42−1+414yes—
240C22×C4C2511yescharacteristic
241M16—416no—
242C2×C8—28no—
243C2×C8—28no—
244C2×C8—28no—
245C4⋊C4—216no—
246C4⋊C4—28no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—28no—
251C4⋊C4—28no—
252C4⋊C4—44no—
253C4⋊C4—84no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—84no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—84no—
260C4⋊C4—84no—
261C4⋊C4—84no—
262C4⋊C4—84no—
263C4⋊C4C2×#4118yes—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C42—24no—
267C42—24no—
268C42—24no—
269C42—24no—
270C42—24no—
271C42—22no—
272C42—22no—
273C42—44no—
274C42—44no—
275C42—88no—
276C23—22no—
277C23—22no—
278C23#52512yes—
279C23#53612yes—
280C23#53612yes—
281C23C23×D411yescharacteristic
282Q8—816no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—28no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—416no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—84no—
314C2×C4—88no—
315C2×C4#52512yes—
316C2×C4#53612yes—
317C2×C4#53612yes—
318C2×C4#541116yes—
319C2×C4C22×#4112yes—
320C2×C4C2×2−1+412yes—
321C2×C4C2×2−1+412yes—
322C8—48no—
323C22—22no—
324C22—22no—
325C22—24no—
326C22—24no—
327C22#263414yes—
328C22C2×#52511yescharacteristic
329C22C2×#53611yescharacteristic
330C22C2×#53611yescharacteristic
331C22#358512yes—
332C22#358612yes—
333C4—42no—
334C4—48no—
335C4—42no—
336C4—44no—
337C4—42no—
338C4—42no—
339C4—82no—
340C4—82no—
341C4—84no—
342C4—84no—
343C4#263414yes—
344C4C2×#54114yes—
345C2—24no—
346C2C2×#263411yescharacteristic
347C2#6056411yescharacteristic
348C2#6056611yescharacteristic
349C2#6086712yes—
350C2#6087012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30102C2semidirect
4. #30101C2semidirect
5. #30100C2semidirect

Maximal subgroups

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

subgroupindex
#346382
C2×#23312
#301022
#301012
#301002
#237472
#237462
#236572
#235522
#210332
#207732