Finite Group Databasebeta

← #6487668

Subgroups — #6487668

Order 512.

2062 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
1C2×#3104C212yesmaximal
2#35173C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic
4#32906C214yesmaximal
5#29757C214yesmaximal
6#24817C218yesmaximal, complemented
7#24732C218yesmaximal, complemented
8#21951C211yesmaximal, characteristic
9#21860C212yesmaximal
10C2×#533C2212yes—
11C22×C4×Q8C2211yescharacteristic
12C22×C4×Q8C2212yes—
13#3104—216no—
14C2×#484C2214yes—
15C2×#477C2214yes—
16C2×#462C2214yes—
17C2×#462C2218yes—
18C2×#451C2214yes—
19C2×C4×Q16C2218yes—
20C2×#431C2212yes—
21#2857C2214yes—
22#2659—22no—
23#2584—24no—
24#2567C2214yes—
25#2546C2214yes—
26Q8×C4⋊C4—24no—
27Q8×C4⋊C4C2218yes—
28#2316—21no—
29#2316—21no—
30C2×#389—22no—
31C2×#389C2212yes—
32C2×#384—21no—
33C2×#384C2211yescharacteristic
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#2083C2218yes—
37#2068C22116yes—
38#2043C2218yes—
39#1990C2214yes—
40#1956C2214yes—
41#1939C2218yes—
42#1878C22116yes—
43#1872C2218yes—
44#1803C2218yes—
45#1803C2218yes—
46#1797C2214yes—
47C23×Q8C2311yescharacteristic
48C22×Q16C2318yes—
49#533—28no—
50#533—28no—
51C2×C4⋊Q8C2312yes—
52C2×C4⋊Q8C2314yes—
53C2×C4⋊Q8C2312yes—
54C2×#125—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8C2314yes—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—48no—
62C2×C4×Q8D418yes—
63C2×C4×Q8C2314yes—
64C2×C4×Q8C2318yes—
65C2×C4×Q8C2314yes—
66C2×#117—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C42—21no—
77C22×C42—22no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#484—216no—
81#477—216no—
82#462—216no—
83#462—416no—
84#451—216no—
85C4×Q16—232no—
86#431—28no—
87C2×#107C2318yes—
88C2×C4⋊C8C2314yes—
89C2×C4⋊C8C2318yes—
90C2×#103C2318yes—
91C2×#103C2318yes—
92C2×#103C23116yes—
93C2×C22⋊C8C2314yes—
94C2×C4×C8C2314yes—
95#398—28no—
96#395—28no—
97#391C2318yes—
98#389—24no—
99#389—24no—
100#389—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389—24no—
105#389—216no—
106#389C2318yes—
107#384—28no—
108#384—22no—
109#384C2314yes—
110#384C2314yes—
111#380—24no—
112#380—24no—
113C4×C4⋊C4—28no—
114C4×C4⋊C4D414yes—
115C4×C4⋊C4C2314yes—
116C2×#95—22no—
117C2×#95D412yes—
118#340C23116yes—
119#336C2318yes—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8C2×D414yes—
126C22×Q8C2414yes—
127C22×Q8C2412yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C2×Q16—216no—
137C2×Q16—216no—
138C2×Q16—216no—
139C22×C8C2418yes—
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—48no—
145#125—48no—
146C22⋊Q8—28no—
147C22⋊Q8—28no—
148C22⋊Q8—48no—
149C4×Q8—216no—
150C4×Q8—28no—
151C4×Q8—216no—
152C4×Q8—28no—
153C4×Q8—28no—
154C4×Q8—48no—
155C4×Q8—416no—
156C4×Q8—48no—
157C4×Q8C2×D4116yes—
158#117—48no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—84no—
183C2×C4⋊C4Q1618yescomplemented
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4C2414yes—
188C2×C4⋊C4C2414yes—
189C2×C4⋊C4C2412yes—
190C2×C22⋊C4—24no—
191C2×C42—28no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—28no—
195C2×C42—44no—
196C2×C42—48no—
197C2×C42C2×D414yes—
198C2×C42C2×D412yes—
199C2×C42C2×D414yes—
200C2×C42#4114yes—
201C2×C42C2414yes—
202C2×C42C2412yes—
203#107—216no—
204C4⋊C8—28no—
205C4⋊C8—48no—
206#103—216no—
207#103—216no—
208#103—416no—
209C22⋊C8—28no—
210C4×C8—28no—
211#95—28no—
212#95—28no—
213#95—28no—
214#95C2×D418yes—
215C24C22×D411yescharacteristic
216C2×Q8—24no—
217C2×Q8—216no—
218C2×Q8—24no—
219C2×Q8—216no—
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—28no—
227C2×Q8—28no—
228C2×Q8—44no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8C22×D414yes—
232C2×Q8H(5,𝔽2)14yes—
233C2×Q82−1+414yes—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4#12114yes—
258C22×C4#12114yes—
259C22×C4C2×Q1618yes—
260C22×C4C2×Q1614yes—
261C22×C4#13714yes—
262C22×C4C22×D414yes—
263C22×C4C22×D412yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D412yes—
266C22×C4C22×D414yes—
267C22×C4C22×D412yes—
268C22×C4C2×#4112yes—
269C22×C4C2511yescharacteristic
270Q16—432no—
271C2×C8—28no—
272C2×C8—28no—
273C2×C8—28no—
274C4⋊C4—216no—
275C4⋊C4—24no—
276C4⋊C4—28no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—28no—
281C4⋊C4—416no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—48no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C4⋊C4—84no—
290C4⋊C4—84no—
291C4⋊C4—84no—
292C4⋊C4—88no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4C2×Q16116yes—
296C4⋊C4C22×D414yes—
297C22⋊C4—48no—
298C42—216no—
299C42—28no—
300C42—44no—
301C42—416no—
302C42—48no—
303C42#12118yes—
304C42C22×D414yes—
305C42C2×#4114yes—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23C2×#12112yes—
311C23C22×Q1612yes—
312C23C2×#13712yes—
313C23C23×D411yescharacteristic
314Q8—216no—
315Q8—24no—
316Q8—24no—
317Q8—24no—
318Q8—48no—
319Q8—48no—
320Q8—88no—
321Q8#53718yes—
322C2×C4—216no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—44no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—84no—
353C2×C4—88no—
354C2×C4#462116yescomplemented
355C2×C4C2×#12112yes—
356C2×C4C2×#12114yes—
357C2×C4C2×#12114yes—
358C2×C4C22×Q1618yes—
359C2×C4C22×Q1614yes—
360C2×C4C2×#13714yes—
361C2×C4C23×D411yescharacteristic
362C2×C4C22×#4111yescharacteristic
363C2×C4C2×H(5,𝔽2)12yes—
364C2×C4C2×2−1+412yes—
365C8—48no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22C2×#46214yes—
373C22C22×#12111yescharacteristic
374C22#348411yescharacteristic
375C22#348611yescharacteristic
376C22C23×Q1611yescharacteristic
377C22C22×#13711yescharacteristic
378C22#358511yescharacteristic
379C22#358611yescharacteristic
380C4—24no—
381C4—28no—
382C4—28no—
383C4—22no—
384C4—44no—
385C4—44no—
386C4—84no—
387C4—84no—
388C4—84no—
389C4C2×#46218yes—
390C4C2×#53712yes—
391C4#349212yes—
392C2—24no—
393C2#3522311yescharacteristic
394C2C22×#46211yescharacteristic
395C2#6042511yescharacteristic
396C2#6156311yescharacteristic
397C2#6158811yescharacteristic
398C2#6182111yescharacteristic
399C2#6182911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #24817C2semidirect
7. #24732C2semidirect
183. C2×C4⋊C4Q16semidirect
354. C2×C4#462semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
#351732
C2×Q8×C4⋊C42
#329062
#297572
#248172
#247322
#219512
#218602