Finite Group Databasebeta

← #6408584

Subgroups — #6408584

Order 512.

1490 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2399C211yesmaximal, characteristic
2#23368C214yesmaximal, complemented
3#23252C214yesmaximal, complemented
4#22603C214yesmaximal
5#22601C214yesmaximal
6#22316C214yesmaximal, complemented
7#22312C214yesmaximal, complemented
8#21822C212yesmaximal
9#21820C212yesmaximal
10#20779C212yesmaximal
11C22×C4×Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2212yes—
14C2×#432C2214yes—
15C2×#431C2212yes—
16#2399C412yes—
17#2399C414yes—
18#2399C412yes—
19#2399C2212yes—
20#2399C2212yes—
21#2399C2212yes—
22#2399C2212yes—
23C2×#398C412yes—
24C2×#398C412yes—
25C2×#395C412yes—
26C2×#395C412yes—
27C2×C4×C4⋊C4C2214yes—
28C2×#376C2211yescharacteristic
29#1985C2214yes—
30#1985C2218yes—
31#1985C2214yes—
32#1985C2218yes—
33#1984C2214yes—
34#1984C2214yes—
35#1940C2218yes—
36#1939C2218yes—
37#1917C2218yes—
38#1917C2218yes—
39#1917C2218yes—
40#1916C2218yes—
41#1840C2214yes—
42#1832C2214yes—
43#1817C2214yes—
44#1817C2214yes—
45#1817C2212yes—
46#1817C2212yes—
47#1797C2214yes—
48#1771C2218yes—
49#1770C2218yes—
50#1768C2214yes—
51#1749C2212yes—
52C4×C22⋊C8C2212yes—
53#1746C2214yes—
54C23×Q8C2×C411yescharacteristic
55C23×Q8C2×C411yescharacteristic
56C2×C4×Q8—44no—
57C2×C4×Q8—44no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2311yescharacteristic
64C2×C4×Q8C2312yes—
65C2×C4×Q8C2311yescharacteristic
66C2×C4×Q8C2312yes—
67C2×C4×Q8C2312yes—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C414yes—
71C22×C4⋊C4C2×C414yes—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C22×C42C2312yes—
77C22×C42C2312yes—
78#432—48no—
79#432—44no—
80#431—44no—
81C2×C4⋊C8C2318yes—
82C2×C4⋊C8C2314yes—
83C2×C4⋊C8C2318yes—
84C2×C4⋊C8C2314yes—
85C2×C22⋊C8C2314yes—
86C2×C22⋊C8C2314yes—
87C2×#97C2314yes—
88C2×#97C2314yes—
89C2×#97C2318yes—
90C2×C4×C8C2314yes—
91C2×C4×C8C2314yes—
92#398C2×C414yes—
93#398C2×C412yes—
94#398C2×C412yes—
95#398C2×C414yes—
96#398C2×C414yes—
97#398C2×C414yes—
98#398C2×C412yes—
99#398C2×C414yes—
100#398C2×C412yes—
101#398C2×C414yes—
102#395C2×C414yes—
103#395C2×C414yes—
104#395C2×C414yes—
105#395C2×C414yes—
106#395C2×C414yes—
107#395C2×C414yes—
108#395C2×C412yes—
109#395C2×C412yes—
110#395C2×C412yes—
111#395C2×C412yes—
112C4×C4⋊C4C2×C414yes—
113C4×C4⋊C4C2×C414yes—
114C4×C4⋊C4C2×C414yes—
115C4×C4⋊C4C2×C414yes—
116C4×C4⋊C4C2314yes—
117C4×C4⋊C4C2314yes—
118C4×C4⋊C4C2314yes—
119C4×C4⋊C4C2314yes—
120#376C2×C412yes—
121#376C2×C412yes—
122#376C2312yes—
123#376C2311yescharacteristic
124#376C2311yescharacteristic
125C2×#95C2×C414yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C414yes—
129#336C2318yes—
130#336C2318yes—
131#336C2318yes—
132#336C2318yes—
133#336C2318yes—
134#336C2318yes—
135#336C2318yes—
136#336C2314yes—
137#336C2314yes—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—44no—
141C22×Q8—44no—
142C22×Q8C22×C411yescharacteristic
143C22×Q8C22×C412yes—
144C22×Q8C22×C412yes—
145C22×Q8C22×C412yes—
146C22×Q8C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C22×C412yes—
151C23×C4C22×C414yes—
152C23×C4C22×C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8C2418yes—
156C22×C8C2418yes—
157C4×Q8—42no—
158C4×Q8—44no—
159C4×Q8—42no—
160C4×Q8—44no—
161C4×Q8—44no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×Q8—44no—
167C4×Q8—42no—
168C4×Q8—42no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4C22×C414yes—
176C2×C4⋊C4C22×C414yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C414yes—
179C2×C4⋊C4C22×C414yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C414yes—
189C2×C4⋊C4C22×C414yes—
190C2×C4⋊C4C22×C414yes—
191C2×C4⋊C4C22×C414yes—
192C2×C4⋊C4C22×C414yes—
193C2×C4⋊C4C22×C414yes—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—28no—
198C2×C42—28no—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C412yes—
202C2×C42C22×C414yes—
203C2×C42C22×C412yes—
204C2×C42C22×C414yes—
205C2×C42C2411yescharacteristic
206C2×C42C2411yescharacteristic
207C2×C42C2412yes—
208C2×C42C2412yes—
209C2×C42C2414yes—
210C2×C42C2412yes—
211C2×C42C2412yes—
212C4⋊C8—216no—
213C4⋊C8—44no—
214C4⋊C8—48no—
215C4⋊C8—44no—
216C22⋊C8—28no—
217C22⋊C8—28no—
218#97—28no—
219#97—48no—
220#97—44no—
221C4×C8—28no—
222C4×C8—44no—
223#95C22×C412yes—
224#95C22×C412yes—
225#95C22×C414yes—
226#95C22×C414yes—
227#95C22×C412yes—
228#95C22×C412yes—
229#95C22×C414yes—
230#95C22×C414yes—
231#95C22×C414yes—
232#95C22×C414yes—
233#95C22×C414yes—
234#95C22×C414yes—
235#95C22×C414yes—
236#95C22×C414yes—
237C24C23×C411yescharacteristic
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—41no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—42no—
249C2×Q8—416no—
250C2×Q8—416no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4C23×C414yes—
270C22×C4C23×C414yes—
271C22×C4C23×C412yes—
272C22×C4C23×C412yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C23×C411yescharacteristic
275C22×C4C23×C411yescharacteristic
276C22×C4C23×C411yescharacteristic
277C22×C4C23×C414yes—
278C22×C4C23×C412yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C23×C414yes—
281C22×C4C23×C412yes—
282C22×C4H(5,𝔽2)14yes—
283C22×C4C2511yescharacteristic
284C2×C8—28no—
285C2×C8—28no—
286C2×C8—28no—
287C2×C8—28no—
288C2×C8—28no—
289C2×C8—28no—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C42—24no—
310C42—24no—
311C42—28no—
312C42—22no—
313C42—22no—
314C42—28no—
315C42—22no—
316C42—22no—
317C42—24no—
318C42—22no—
319C42—22no—
320C42—24no—
321C42—22no—
322C42—22no—
323C42—24no—
324C422−1+418yes—
325C23—22no—
326C23—22no—
327C23#51812yes—
328C23#56814yes—
329C23C24×C411yescharacteristic
330Q8—44no—
331Q8—48no—
332Q8—48no—
333Q8—48no—
334Q8—44no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—28no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—416no—
366C2×C4—48no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—48no—
371C2×C4—84no—
372C2×C4#51812yes—
373C2×C4#51918yes—
374C2×C4#56814yes—
375C2×C4C2×H(5,𝔽2)12yes—
376C2×C4C2×2−1+414yes—
377C8—48no—
378C8—48no—
379C22—22no—
380C22—22no—
381C22—28no—
382C22#297814yes—
383C22C2×#51811yescharacteristic
384C22C2×#51912yes—
385C22C2×#56812yes—
386C22C2×#56812yes—
387C4—22no—
388C4—24no—
389C4—28no—
390C4—42no—
391C4—42no—
392C4—41no—
393C4—41no—
394C4—44no—
395C4—41no—
396C4—41no—
397C4—42no—
398C4—44no—
399C4—44no—
400C4#354414yes—
401C2—24no—
402C2#3436511yescharacteristic
403C2C2×#297811yescharacteristic
404C2#6023111yescharacteristic
405C2#6025114yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23368C2semidirect
3. #23252C2semidirect
6. #22316C2semidirect
7. #22312C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23992
#233682
#232522
#226032
#226012
#223162
#223122
#218222
#218202
#207792