Finite Group Databasebeta

← #137646

Subgroups — #137646

Order 512.

406 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 102 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#33419C211yesmaximal, characteristic, complemented
2#21494C211yesmaximal, characteristic
3#12786C211yesmaximal, characteristic, complemented
4#11587C212yesmaximal, complemented
5#7169C212yesmaximal, complemented
6C2×#436—21no—
7C2×#436C2211yescharacteristic
8C2×#405C2211yescharacteristic
9#2392C411yescharacteristic
10#2164—24no—
11#2150—21no—
12#2150C2212yes—
13C2×#349—21no—
14#2112—21no—
15C4×M32C2212yes—
16#1955C411yescharacteristic
17#1940C411yescharacteristic
18#1923—21no—
19#1869—21no—
20#1798C411yescharacteristic
21C8×C4⋊C4C2211yescharacteristic
22#1753—21no—
23#1377—21no—
24#1330—24no—
25C22×M16—21no—
26C22×M16D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4C2×C411yescharacteristic
30C2×#117C2×C411yescharacteristic
31C2×M32—22no—
32C2×M32—22no—
33#436—22no—
34#436—22no—
35#436—44no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43#405—22no—
44#405—22no—
45C2×#97C2×C411yescharacteristic
46C2×#97D411yescharacteristic
47C2×C4×C8—21no—
48C2×C4×C8D411yescharacteristic
49C2×C4×C8C2311yescharacteristic
50#398C811yescharacteristic
51#397C811yescharacteristic
52#396—21no—
53#394—21no—
54#394C811yescharacteristic
55#392C811yescharacteristic
56C4×C4⋊C4C2×C411yescharacteristic
57C4×C22⋊C4—21no—
58C4⋊C16—24no—
59C4⋊C16—22no—
60C4⋊C16—22no—
61#349—42no—
62C22⋊C16—24no—
63#346D414yescomplemented
64C4×C16—22no—
65#336—21no—
66#336—21no—
67#336—21no—
68C22×D4—21no—
69C22×D4C2×C811yescharacteristic
70C23×C4—21no—
71C23×C4C22⋊C411yescharacteristic
72C2×M16—22no—
73C2×M16—21no—
74C2×M16—22no—
75C2×M16—21no—
76C2×M16—42no—
77C2×M16—41no—
78C2×M16—42no—
79C2×M16—41no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—42no—
93C4×D4—42no—
94#117—22no—
95#117—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4C2×C811yescharacteristic
99C2×C4⋊C4C2×C811yescharacteristic
100C2×C4⋊C4M1611yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4C2×C811yescharacteristic
107C2×C22⋊C4M1611yescharacteristic
108C2×C42—21no—
109C2×C42C22⋊C411yescharacteristic
110C2×C42C22×C411yescharacteristic
111M32—42no—
112M32—42no—
113M32—42no—
114M32—42no—
115C2×C16—24no—
116C2×C16—42no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125#97—22no—
126#97—22no—
127C4×C8—21no—
128C4×C8—21no—
129C4×C8—42no—
130C4×C8C2×D412yes—
131C4×C8#4112yes—
132#95—21no—
133#95—21no—
134#95C2×C811yescharacteristic
135#95C2×C811yescharacteristic
136C24—21no—
137C24C22⋊C811yescharacteristic
138C2×D4—22no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—81no—
142C2×D4—81no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4C22⋊C811yescharacteristic
159C22×C4C22⋊C811yescharacteristic
160C22×C4C22⋊C811yescharacteristic
161C22×C4C2×C22⋊C411yescharacteristic
162C22×C4C22×C811yescharacteristic
163C22×C4C2×M1611yescharacteristic
164M16—42no—
165M16—42no—
166M16—82no—
167M16—82no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8#12011yescharacteristic
185C2×C8#12112yes—
186C2×C8#12311yescharacteristic
187C2×C8AGL(1,ℤ/8)12yes—
188C2×C8AGL(1,ℤ/8)11yescharacteristic
189C2×C8#13711yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—81no—
202C42—21no—
203C42—21no—
204C42—41no—
205C42—41no—
206C42C4×D412yes—
207C42#13112yes—
208C16—44no—
209C16—82no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23C2×C22⋊C811yescharacteristic
223D4—82no—
224D4—84no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
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—41no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4#38611yescharacteristic
252C2×C4#40811yescharacteristic
253C2×C4#41711yescharacteristic
254C2×C4#41811yescharacteristic
255C2×C4#42111yescharacteristic
256C2×C4#42111yescharacteristic
257C2×C4C8×D412yes—
258C2×C4#43512yes—
259C8—21no—
260C8—21no—
261C8—41no—
262C8—41no—
263C8—81no—
264C8—81no—
265C8#46812yes—
266C8#46912yes—
267C22—21no—
268C22—21no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22#147711yescharacteristic
282C22#147811yescharacteristic
283C22#186911yescharacteristic
284C4—22no—
285C4—22no—
286C4—42no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—81no—
292C4#149511yescharacteristic
293C4#149611yescharacteristic
294C4#189511yescharacteristic
295C4#189711yescharacteristic
296C2—21no—
297C2—21no—
298C2—81no—
299C2—161no—
300C2#907411yescharacteristic
301C2#1298211yescharacteristic
302C2#1298911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radical#392C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33419C2semidirect
3. #12786C2semidirect
4. #11587C2semidirect
5. #7169C2semidirect
63. #346D4semidirect

Maximal subgroups

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

subgroupindex
#334192
#214942
#127862
#115872
#71692