Finite Group Databasebeta

← #141135

Subgroups — #141135

Order 512.

594 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 112 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#15151C211yesmaximal, characteristic
2C2×#1346C214yesmaximal
3C4×#343C211yesmaximal, characteristic
4#8558C211yesmaximal, characteristic
5C2×C4×M16—21no—
6C2×#398—21no—
7C2×#398C2212yes—
8C2×#389—21no—
9C2×#384C411yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797—21no—
12C2×#343—21no—
13C2×#343C2212yes—
14C2×#324C2212yes—
15#1445—21no—
16#1445C412yes—
17C4⋊#95—21no—
18C4⋊#95C411yescharacteristic
19#1346—28no—
20#1280—21no—
21C22×M16—22no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4C2×C412yes—
27C22×C4⋊C4D411yescharacteristic
28C22×C4⋊C4C2311yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D411yescharacteristic
31C22×C42D411yescharacteristic
32C2×C4⋊C8—41no—
33C2×C22⋊C8—22no—
34C4×M16—24no—
35C4×M16—42no—
36C2×#97—41no—
37C2×C4×C8—41no—
38#398—24no—
39#398—24no—
40#398—44no—
41#389—42no—
42#389—42no—
43#384—41no—
44#384—41no—
45C4×C4⋊C4—41no—
46C4×C4⋊C4—41no—
47C2×#95—22no—
48C2×#95—21no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95C2×C412yes—
52C2×#95C2×C411yescharacteristic
53C2×#95D411yescharacteristic
54#343—24no—
55#343—22no—
56#343D414yes—
57#336—41no—
58#324—24no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22×C411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C2×M16—24no—
69C2×M16—24no—
70C2×M16—24no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—42no—
74C22×C8—42no—
75C22×C8—42no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4C2×D412yes—
99C2×C4⋊C4#4112yes—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—22no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42D812yes—
110C2×C42SD1612yes—
111C4⋊C8—82no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114#97—44no—
115C4×C8—44no—
116#95—24no—
117#95—41no—
118#95—41no—
119#95—42no—
120#95—44no—
121#95—44no—
122#95—42no—
123#95—42no—
124C24C2×C22⋊C411yescharacteristic
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4#9911yescharacteristic
167C22×C4#9911yescharacteristic
168C22×C4#10212yes—
169C22×C4#10212yes—
170C22×C4#10412yes—
171C22×C4#10412yes—
172C22×C4C4×D412yes—
173C22×C4#12112yes—
174C22×C4#12411yescharacteristic
175C22×C4#12711yescharacteristic
176C22×C4C2×D811yescharacteristic
177C22×C4C2×SD1611yescharacteristic
178C22×C4AGL(1,ℤ/8)12yes—
179M16—28no—
180M16—44no—
181M16—44no—
182M16—44no—
183C2×C8—44no—
184C2×C8—44no—
185C2×C8—44no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—82no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—44no—
192C4⋊C4—48no—
193C4⋊C4—84no—
194C4⋊C4—84no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4—84no—
198C4⋊C4—84no—
199C42—22no—
200C42—22no—
201C42—22no—
202C42—22no—
203C42—42no—
204C42—42no—
205C42—42no—
206C42—81no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23#39011yescharacteristic
212C23C2×#9911yescharacteristic
213C23#41011yescharacteristic
214C23C2×#10211yescharacteristic
215C23#41811yescharacteristic
216C23C2×#10411yescharacteristic
217C23C2×#10411yescharacteristic
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—81no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4#32712yes—
272C2×C4#32712yes—
273C2×C4#35212yes—
274C2×C4#35612yes—
275C2×C4#45912yes—
276C2×C4#46012yes—
277C8—48no—
278C8—162no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22C2×#32711yescharacteristic
292C22#150211yescharacteristic
293C22#150511yescharacteristic
294C22#150711yescharacteristic
295C22#196211yescharacteristic
296C22#196411yescharacteristic
297C22#198211yescharacteristic
298C22C2×#35211yescharacteristic
299C22#212311yescharacteristic
300C22C2×#35611yescharacteristic
301C22#213511yescharacteristic
302C4—22no—
303C4—22no—
304C4—22no—
305C4—42no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—42no—
310C4—84no—
311C4—81no—
312C4—162no—
313C4#134614yescomplemented
314C2—21no—
315C2—21no—
316C2—21no—
317C2—21no—
318C2#971211yescharacteristic
319C2C2×#134611yescharacteristic
320C2#1170011yescharacteristic
321C2#1178011yescharacteristic
322C2#1178211yescharacteristic
323C2#1321711yescharacteristic
324C2#1322711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
313. C4#1346semidirect

Maximal subgroups

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

subgroupindex
#151512
C2×#13462
C4×#3432
#85582