Finite Group Databasebeta

← #124439

Subgroups — #124439

Order 512.

389 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 94 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#21494C211yesmaximal, characteristic
2#12786C211yesmaximal, characteristic, complemented
3#12551C211yesmaximal, characteristic, complemented
4#7124C212yesmaximal, complemented
5#7105C212yesmaximal, complemented
6C22×M32—21no—
7C2×#436—21no—
8C2×#436C411yescharacteristic
9C2×#405C2211yescharacteristic
10#2392C411yescharacteristic
11#2150—21no—
12C2×#349—21no—
13#1955C2211yescharacteristic
14#1940C2211yescharacteristic
15#1923—21no—
16#1869—21no—
17#1798C411yescharacteristic
18C8×C4⋊C4C411yescharacteristic
19#1753—21no—
20#1377—21no—
21#1377C2212yes—
22#1377C2212yes—
23C22×M16—21no—
24C22×M16D411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C4×D4—21no—
27C2×C4×D4C2×C411yescharacteristic
28C2×#117C2×C411yescharacteristic
29C2×M32—24no—
30C2×M32—22no—
31C2×M32—22no—
32C2×M32—41no—
33C2×M32—42no—
34C22×C16—41no—
35#436—41no—
36#436—41no—
37#436—42no—
38#436—42no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46#405—41no—
47#405—41no—
48C2×#97D411yescharacteristic
49C2×#97C2311yescharacteristic
50C2×C4×C8—21no—
51C2×C4×C8C2×C411yescharacteristic
52C2×C4×C8D411yescharacteristic
53#398C811yescharacteristic
54#397C811yescharacteristic
55#396—21no—
56#394—21no—
57#394C811yescharacteristic
58#392C811yescharacteristic
59C4×C4⋊C4C2×C411yescharacteristic
60C4×C22⋊C4—21no—
61C4⋊C16—41no—
62C4⋊C16—41no—
63#349—42no—
64#336—21no—
65#336—21no—
66#336—21no—
67C22×D4—21no—
68C22×D4C2×C811yescharacteristic
69C23×C4—21no—
70C23×C4C22⋊C411yescharacteristic
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16D812yes—
77C2×M16SD1612yes—
78C22×C8—22no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22⋊C411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C4×D4—41no—
90C4×D4—41no—
91C4×D4—42no—
92C4×D4—42no—
93#117—41no—
94#117—41no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4C2×C811yescharacteristic
98C2×C4⋊C4C2×C811yescharacteristic
99C2×C4⋊C4M1611yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4C2×C811yescharacteristic
106C2×C22⋊C4M1611yescharacteristic
107C2×C42—21no—
108C2×C42C22⋊C411yescharacteristic
109C2×C42C22×C411yescharacteristic
110M32—24no—
111M32—44no—
112M32—42no—
113M32—42no—
114M32—42no—
115C2×C16—44no—
116C2×C16—42no—
117C2×C16—42no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C22⋊C8—42no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129#97—41no—
130#97—41no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—42no—
136#95—21no—
137#95—21no—
138#95C2×C811yescharacteristic
139#95C2×C811yescharacteristic
140C24—21no—
141C24C22⋊C811yescharacteristic
142C2×D4—41no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—81no—
146C2×D4—81no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4C22⋊C811yescharacteristic
161C22×C4C22⋊C811yescharacteristic
162C22×C4C22⋊C811yescharacteristic
163C22×C4#10212yes—
164C22×C4#10412yes—
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4C22×C811yescharacteristic
167C22×C4C2×M1611yescharacteristic
168M16—42no—
169M16—42no—
170M16—84no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8#9911yescharacteristic
190C2×C8#9911yescharacteristic
191C2×C8#10212yes—
192C2×C8C2×D811yescharacteristic
193C2×C8C2×SD1611yescharacteristic
194C2×C8AGL(1,ℤ/8)11yescharacteristic
195C2×C8#13711yescharacteristic
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—41no—
213C42—41no—
214C16—44no—
215C16—82no—
216C23—21no—
217C23—21no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23D4⋊C812yes—
228C23C2×C22⋊C811yescharacteristic
229D4—84no—
230D4—82no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4D4⋊C812yes—
259C2×C4C2×#9911yescharacteristic
260C2×C4#41311yescharacteristic
261C2×C4C2×#10211yescharacteristic
262C2×C4#41711yescharacteristic
263C2×C4C2×#10411yescharacteristic
264C2×C4#42111yescharacteristic
265C8—21no—
266C8—21no—
267C8—21no—
268C8—21no—
269C8—21no—
270C8—22no—
271C8—21no—
272C8—81no—
273C8—81no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22#145911yescharacteristic
289C22C2×D4⋊C811yescharacteristic
290C22#147711yescharacteristic
291C4—22no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4#150411yescharacteristic
300C4#151911yescharacteristic
301C4#153211yescharacteristic
302C4#154711yescharacteristic
303C2—21no—
304C2—21no—
305C2—42no—
306C2—161no—
307C2#758611yescharacteristic
308C2#1148411yescharacteristic
309C2#1148511yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22#1459
prime radical#392C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12786C2semidirect
3. #12551C2semidirect
4. #7124C2semidirect
5. #7105C2semidirect

Maximal subgroups

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

subgroupindex
#214942
#127862
#125512
#71242
#71052