Finite Group Databasebeta

← #137642

Subgroups — #137642

Order 512.

441 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C2×#2163C211yesmaximal, characteristic, complemented
2#21487C211yesmaximal, characteristic
3#12783C211yesmaximal, characteristic, complemented
4C2×#1330C212yesmaximal, complemented
5#7170C212yesmaximal, complemented
6C2×#436C2211yescharacteristic
7C2×#435—21no—
8C22×C4×C8C2211yescharacteristic
9C4×#121C411yescharacteristic
10#2163—24no—
11C2×C4⋊C16—21no—
12C2×C4⋊C16C2212yes—
13C2×C22⋊C16—21no—
14C2×C22⋊C16—21no—
15C2×#346C2212yes—
16#1954C411yescharacteristic
17#1939C411yescharacteristic
18#1923—21no—
19#1869—21no—
20#1797C411yescharacteristic
21#1770C2211yescharacteristic
22#1753—21no—
23#1374—21no—
24#1330—24no—
25C22×M16—21no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#121C811yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C16—22no—
33C22×C16—22no—
34#436—22no—
35#436—22no—
36#435—44no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×#97—21no—
45C2×C4×C8—22no—
46C2×C4×C8—22no—
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8C2311yescharacteristic
51#390C811yescharacteristic
52#389C811yescharacteristic
53#388—21no—
54#386—21no—
55#380C811yescharacteristic
56C4×C4⋊C4C2×C411yescharacteristic
57C4×C22⋊C4—21no—
58C4⋊C16—24no—
59C4⋊C16—22no—
60C4⋊C16—22no—
61C22⋊C16—24no—
62C22⋊C16—42no—
63#346—22no—
64#346D414yescomplemented
65#336—21no—
66#336—21no—
67#336—21no—
68C22×D4—21no—
69C22×D4C2×C811yescharacteristic
70C23×C4—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C2×C811yescharacteristic
73C23×C4M1611yescharacteristic
74C2×M16—42no—
75C2×M16—41no—
76C2×M16—42no—
77C2×M16—41no—
78C22×C8—21no—
79C22×C8—22no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—22no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D411yescharacteristic
92#121—42no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—42no—
96C4×D4—42no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4C2×C811yescharacteristic
99C2×C4⋊C4C2×C811yescharacteristic
100C2×C4⋊C4C2×C811yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—21no—
109C2×C42C22⋊C411yescharacteristic
110C2×C42C2×C811yescharacteristic
111C2×C42M1611yescharacteristic
112C2×C42C22×C411yescharacteristic
113C2×C16—24no—
114C2×C16—42no—
115C2×C16—42no—
116C2×C16—42no—
117C2×C16—42no—
118C2×C16—42no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127#97—42no—
128C4×C8—21no—
129C4×C8—22no—
130C4×C8—21no—
131C4×C8—22no—
132C4×C8C2×D412yes—
133C4×C8#4112yes—
134#95—21no—
135#95—21no—
136#95—21no—
137C24—21no—
138C24C22⋊C811yescharacteristic
139C2×D4—22no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4C22⋊C811yescharacteristic
167C22×C4C22⋊C811yescharacteristic
168C22×C4C22⋊C811yescharacteristic
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C22×C811yescharacteristic
171C22×C4C2×M1611yescharacteristic
172M16—82no—
173M16—82no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8#12011yescharacteristic
193C2×C8#12112yes—
194C2×C8#12311yescharacteristic
195C2×C8AGL(1,ℤ/8)12yes—
196C2×C8AGL(1,ℤ/8)11yescharacteristic
197C2×C8#13711yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C42—21no—
208C42—21no—
209C42—22no—
210C42—22no—
211C42—41no—
212C42—41no—
213C42C4×D412yes—
214C42#13112yes—
215C16—44no—
216C16—82no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23#34912yes—
231C23C2×C22⋊C811yescharacteristic
232D4—82no—
233D4—84no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4#35012yes—
266C2×C4#38611yescharacteristic
267C2×C4#40811yescharacteristic
268C2×C4#41711yescharacteristic
269C2×C4#41811yescharacteristic
270C2×C4#42111yescharacteristic
271C2×C4#42111yescharacteristic
272C2×C4C8×D412yes—
273C2×C4#43512yes—
274C8—21no—
275C8—21no—
276C8—41no—
277C8—41no—
278C8—81no—
279C8—81no—
280C8#46812yes—
281C8#46912yes—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22#147711yescharacteristic
297C22#147811yescharacteristic
298C22#186911yescharacteristic
299C22C2×#34911yescharacteristic
300C22#211511yescharacteristic
301C22C2×#35011yescharacteristic
302C22#211711yescharacteristic
303C4—22no—
304C4—22no—
305C4—22no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4#149511yescharacteristic
312C4#149611yescharacteristic
313C4#189511yescharacteristic
314C4#189711yescharacteristic
315C2—42no—
316C2—161no—
317C2#907411yescharacteristic
318C2#1162611yescharacteristic
319C2#1162711yescharacteristic
320C2#1163311yescharacteristic
321C2#1163611yescharacteristic
322C2#1297811yescharacteristic
323C2#1298511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×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. C2×#2163C2semidirect
3. #12783C2semidirect
4. C2×#1330C2semidirect
5. #7170C2semidirect
64. #346D4semidirect

Maximal subgroups

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

subgroupindex
C2×#21632
#214872
#127832
C2×#13302
#71702