Finite Group Databasebeta

← #160944

Subgroups — #160944

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34026C211yesmaximal, characteristic
2#20777C212yesmaximal, complemented
3#7644C214yesmaximal, complemented
4#7642C214yesmaximal, complemented
5#7564C212yesmaximal, complemented
6#7563C212yesmaximal
7C2×C4×#41C411yescharacteristic
8#2298—22no—
9#2297C411yescharacteristic
10#2297C2211yescharacteristic
11#2296C411yescharacteristic
12#2296C2211yescharacteristic
13#2295C412yes—
14#2295C2212yes—
15#2293C412yes—
16#2293C2212yes—
17#2290—22no—
18#2290—22no—
19C4×#117—22no—
20C4×#117C411yescharacteristic
21C2×C43C2211yescharacteristic
22#1839C2214yes—
23#1831C2214yes—
24#1817—22no—
25#1797—22no—
26C4×C4⋊C8—22no—
27C4×C22⋊C8C2214yes—
28#1744—22no—
29#1272C2214yes—
30#1271C2218yes—
31#1270C2214yes—
32C22×#41C2×C411yescharacteristic
33C4×#41—24no—
34C4×#41—24no—
35C2×C4×Q8C2×C412yes—
36C2×C4×D4—22no—
37C2×C4×D4C2×C412yes—
38C2×#117—21no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117C2×C411yescharacteristic
42C2×#117C2×C411yescharacteristic
43C2×#117C2×C411yescharacteristic
44C22×C42—21no—
45C22×C42C2×C411yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D412yes—
48C22×C42D412yes—
49C22×C42C2311yescharacteristic
50C2×C4⋊C8—24no—
51C2×C22⋊C8—24no—
52C2×C4×C8—24no—
53#391C2×C411yescharacteristic
54#391C2×C412yes—
55#391C2311yescharacteristic
56#390C2×C411yescharacteristic
57#390C2×C412yes—
58#390C2311yescharacteristic
59#389C2×C412yes—
60#389C2×C412yes—
61#389C2×C412yes—
62#389C2312yes—
63#388—22no—
64#388—22no—
65#388—22no—
66#388—22no—
67#387—22no—
68#387—22no—
69#387—22no—
70#387—22no—
71#386—21no—
72#386—21no—
73#386—22no—
74#385—21no—
75#385—21no—
76#385—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4C2×C412yes—
82C4×C4⋊C4C2×C412yes—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—22no—
88#376—22no—
89#376—21no—
90#376—22no—
91C43—21no—
92C43C2×C412yes—
93C43D412yes—
94C43D412yes—
95C43C2312yes—
96#336—24no—
97#336—24no—
98#336—24no—
99#336C2318yes—
100C2×#41—22no—
101C2×#41—42no—
102C2×#41—42no—
103C2×#41M1614yes—
104C22×Q8C22×C411yescharacteristic
105C22×D4—21no—
106C22×D4C22×C411yescharacteristic
107C23×C4—22no—
108C23×C4—21no—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D412yes—
114C22×C8—28no—
115C4×Q8—24no—
116C4×Q8—24no—
117C4×D4—216no—
118C4×D4—24no—
119C4×D4—24no—
120#117—28no—
121#117—22no—
122#117—48no—
123#117—48no—
124#117—42no—
125#117—42no—
126#117M1614yes—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4C22×C412yes—
133C2×C4⋊C4C22×C411yescharacteristic
134C2×C4⋊C4C22×C411yescharacteristic
135C2×C4⋊C4C22×C412yes—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C42—28no—
145C2×C42—22no—
146C2×C42—21no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—28no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—42no—
155C2×C42—44no—
156C2×C42—42no—
157C2×C42C22⋊C412yes—
158C2×C42C22⋊C412yes—
159C2×C42C22⋊C412yes—
160C2×C42C22⋊C412yes—
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C412yes—
164C2×C42C22×C412yes—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2411yescharacteristic
173C4⋊C8—48no—
174C22⋊C8—48no—
175C4×C8—48no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186C24—21no—
187C24C2×C22⋊C411yescharacteristic
188#41—48no—
189#41—48no—
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8C2×M1612yes—
194C2×D4—24no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4C2×M1612yes—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—28no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×M1612yes—
231C22×C4C2×M1612yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D412yes—
234C2×C8—48no—
235C2×C8—48no—
236C2×C8—48no—
237C4⋊C4—28no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4C2×M1614yes—
247C22⋊C4—28no—
248C22⋊C4—24no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C42—28no—
256C42—22no—
257C42—22no—
258C42—24no—
259C42—22no—
260C42—24no—
261C42—22no—
262C42—24no—
263C42—24no—
264C42—24no—
265C42—24no—
266C42—24no—
267C42—24no—
268C42—41no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42—41no—
273C42—44no—
274C42—42no—
275C42—42no—
276C42—44no—
277C42#13518yes—
278C23—21no—
279C23—21no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23#40712yes—
288C23C22×C22⋊C411yescharacteristic
289Q8—42no—
290Q8—42no—
291D4—42no—
292D4—42no—
293D4—48no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—48no—
317C2×C4—416no—
318C2×C4—44no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—48no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4#40712yes—
332C2×C4#41012yes—
333C2×C4#41312yes—
334C2×C4#41614yes—
335C2×C4#41614yes—
336C2×C4#42114yes—
337C2×C4C22×M1611yescharacteristic
338C2×C4C22×M1611yescharacteristic
339C2×C4C2×#13514yes—
340C8—88no—
341C22—24no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—42no—
346C22—44no—
347C22—42no—
348C22—81no—
349C22—81no—
350C22—82no—
351C22C2×#40711yescharacteristic
352C22C2×#41011yescharacteristic
353C22C2×#41311yescharacteristic
354C22C2×#41612yes—
355C22C2×#42112yes—
356C4—24no—
357C4—24no—
358C4—24no—
359C4—24no—
360C4—21no—
361C4—21no—
362C4—42no—
363C4—42no—
364C4—44no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—82no—
369C4#146414yes—
370C2—42no—
371C2—82no—
372C2C2×#146411yescharacteristic
373C2#1744312yes—
374C2#1757312yes—
375C2#1766511yescharacteristic
376C2#1770011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20777C2semidirect
3. #7644C2semidirect
4. #7642C2semidirect
5. #7564C2semidirect

Maximal subgroups

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

subgroupindex
#340262
#207772
#76442
#76422
#75642
#75632