Finite Group Databasebeta

← #109921

Subgroups — #109921

Order 512.

974 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20524C211yesmaximal, characteristic
2#13844C211yesmaximal, characteristic
3#10733C212yesmaximal
4C4×#342C211yesmaximal, characteristic
5C2×#1297C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2212yes—
8C2×C4×C4⋊C4—21no—
9C2×C4×C22⋊C4C2212yes—
10C2×C4×C22⋊C4C2211yescharacteristic
11C2×C4×C22⋊C4C2211yescharacteristic
12#1753—24no—
13#1749—21no—
14C4×C22⋊C8—21no—
15C2×#342—22no—
16C2×#339—24no—
17#1444C412yescomplemented
18#1442C412yescomplemented
19#1439—22no—
20C4×#95—21no—
21#1297C418yescomplemented
22C24×C4C2311yescharacteristic
23C22×M16D414yes—
24C2×#117—22no—
25C2×#117—22no—
26C2×#117—44no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4C2×C412yes—
29C22×C22⋊C4—22no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4D412yes—
33C22×C42—21no—
34C22×C42—21no—
35C22×C42—22no—
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38#407—22no—
39#407—22no—
40#407C2×C414yes—
41#407Q814yes—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×#97—22no—
45C2×C4×C8—22no—
46C4×C4⋊C4—44no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—22no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4C2×C414yes—
54C4×C22⋊C4C2×C414yes—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95C2×C412yes—
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62#342—28no—
63#339—48no—
64#336—22no—
65#336—22no—
66C25C2×D411yescharacteristic
67C23×C4—21no—
68C23×C4—22no—
69C23×C4—21no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4C22⋊C412yes—
74C23×C4C22⋊C412yes—
75C23×C4C22×C412yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D412yes—
78C23×C4C2×D411yescharacteristic
79C23×C4C2×Q812yes—
80C23×C4#4112yes—
81C23×C4#4112yes—
82C23×C4#4112yes—
83C2×M16—24no—
84C2×M16—44no—
85C2×M16—44no—
86C2×M16C4⋊C418yescomplemented
87C22×C8—24no—
88#117—28no—
89#117—48no—
90#117—44no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—44no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—44no—
96C2×C22⋊C4—28no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4—44no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C42—24no—
108C2×C42—24no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—24no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—41no—
118C2×C42—41no—
119C2×C42—44no—
120C2×C42—44no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42—42no—
124C22⋊C8—24no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127C22⋊C8—24no—
128#97—82no—
129C4×C8—82no—
130#95—21no—
131#95—21no—
132#95—24no—
133#95—24no—
134#95—44no—
135#95—44no—
136#95—44no—
137#95—42no—
138#95C4212yes—
139#95C4212yes—
140C24—22no—
141C24—22no—
142C24—41no—
143C24—42no—
144C24—41no—
145C24C2×C22⋊C411yescharacteristic
146C24C4×D412yes—
147C24#12112yes—
148C24#12411yescharacteristic
149C24C4⋊Q811yescharacteristic
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—48no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—82no—
191C22×C4—82no—
192C22×C4—84no—
193C22×C4#10414yes—
194C22×C4C2×C4211yescharacteristic
195C22×C4C2×C4⋊C412yes—
196C22×C4#11712yes—
197C22×C4#11711yescharacteristic
198C22×C4C4×D412yes—
199C22×C4C4×D412yes—
200C22×C4C4×D412yes—
201C22×C4C4×Q812yes—
202C22×C4C4×Q812yes—
203C22×C4#12011yescharacteristic
204C22×C4C22⋊Q812yes—
205C22×C4C22⋊Q812yes—
206C22×C4#12312yes—
207C22×C4#12311yescharacteristic
208C22×C4#12512yes—
209C22×C4#12612yes—
210M16—48no—
211M16—84no—
212C2×C8—28no—
213C2×C8—44no—
214C2×C8—44no—
215C4⋊C4—28no—
216C4⋊C4—44no—
217C4⋊C4—416no—
218C22⋊C4—28no—
219C22⋊C4—48no—
220C22⋊C4—48no—
221C22⋊C4—48no—
222C22⋊C4—44no—
223C22⋊C4—48no—
224C42—48no—
225C42—44no—
226C42—48no—
227C42—42no—
228C42—44no—
229C42—42no—
230C42—81no—
231C42—81no—
232C42—84no—
233C23—22no—
234C23—21no—
235C23—21no—
236C23—24no—
237C23—24no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—82no—
251C23—82no—
252C23—82no—
253C23—82no—
254C23C4×C22⋊C411yescharacteristic
255C23#38611yescharacteristic
256C23#38812yes—
257C23#38912yes—
258C23#39111yescharacteristic
259C23C2×#10412yes—
260C23#42112yes—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—48no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—82no—
299C2×C4—84no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—84no—
305C2×C4—88no—
306C2×C4—88no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4C4×C4⋊C412yes—
311C2×C4#38512yes—
312C2×C4#38712yes—
313C2×C4#38712yes—
314C2×C4B(4,𝔽2)12yes—
315C2×C4#45812yes—
316C8—48no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—44no—
332C22—44no—
333C22—82no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22#143611yescharacteristic
338C22C4×#10411yescharacteristic
339C22#175911yescharacteristic
340C22#187112yes—
341C22#188412yes—
342C22#188911yescharacteristic
343C22#189011yescharacteristic
344C22#198612yes—
345C4—22no—
346C4—22no—
347C4—22no—
348C4—44no—
349C4—44no—
350C4—82no—
351C4—88no—
352C4—82no—
353C4—84no—
354C4#186714yes—
355C2—21no—
356C2—21no—
357C2—22no—
358C2—42no—
359C2—42no—
360C2#901012yes—
361C2#901112yes—
362C2#1088611yescharacteristic
363C2#1089311yescharacteristic
364C2#1091811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC23#386
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1444C4semidirect
18. #1442C4semidirect
21. #1297C4semidirect
86. C2×M16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#205242
#138442
#107332
C4×#3422
C2×#12972