Finite Group Databasebeta

← #136646

Subgroups — #136646

Order 512.

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

normal subgroups only

This group has 468 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
1C4×#469C211yesmaximal, characteristic
2#17795C211yesmaximal, characteristic, complemented
3C4×#349C211yesmaximal, characteristic, complemented
4#11599C212yesmaximal, complemented
5#7174C212yesmaximal, complemented
6C2×#469C411yescharacteristic
7C2×C4×D8C2211yescharacteristic
8#2920—21no—
9C2×C4×M16C2211yescharacteristic
10C4×#121—21no—
11C2×#349—21no—
12C4×M32C2212yes—
13#1972C411yescharacteristic
14#1944C411yescharacteristic
15#1936—21no—
16#1895—21no—
17#1809C411yescharacteristic
18C4×#106C2211yescharacteristic
19C4×#102—21no—
20#1569—24no—
21#1564—21no—
22#1564C2212yes—
23#1377—21no—
24#1333—24no—
25C22×D8C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C22×M16D411yescharacteristic
28C2×#121—21no—
29C2×C4×D4—21no—
30C2×C4×D4—41no—
31C22×C42D411yescharacteristic
32C2×M32—22no—
33#469—22no—
34#469—22no—
35C4×D8—22no—
36C4×D8—22no—
37#432—22no—
38#432—41no—
39#431—22no—
40#431—41no—
41C2×#107C2×C411yescharacteristic
42C2×#106C2×C411yescharacteristic
43C2×#106C2×C411yescharacteristic
44#423—22no—
45C2×#102—21no—
46C2×#102—21no—
47C2×#102—21no—
48#407—41no—
49C4×M16—22no—
50C4×M16—22no—
51C4×M16—41no—
52C2×#97C2×C411yescharacteristic
53C2×#97D411yescharacteristic
54C2×C4×C8D411yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#390—21no—
57#389—21no—
58#388—41no—
59#386—41no—
60#380—21no—
61C4×C4⋊C4—21no—
62C4×C22⋊C4—41no—
63#349—24no—
64#346—22no—
65C4×C16—22no—
66#340—21no—
67#340—21no—
68#340—21no—
69#335—22no—
70#335—22no—
71#334—22no—
72#334D414yescomplemented
73C22×D4—21no—
74C22×D4—41no—
75C23×C4—41no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4D811yescharacteristic
78C23×C4SD1611yescharacteristic
79C2×D8—22no—
80C2×D8—21no—
81C2×D8—22no—
82C2×D8M1612yescomplemented
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—21no—
87C2×M16—21no—
88C2×M16—22no—
89C2×M16—41no—
90C2×M16—42no—
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22×C411yescharacteristic
94C22×C8C2×D411yescharacteristic
95#121—24no—
96#121—42no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—84no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—41no—
116C2×C42C22⋊C411yescharacteristic
117C2×C42D811yescharacteristic
118C2×C42SD1611yescharacteristic
119C2×C42C2×D411yescharacteristic
120M32—24no—
121M32—42no—
122C2×C16—42no—
123#107—22no—
124#106—21no—
125#106—22no—
126#106M1612yescomplemented
127C4⋊C8—24no—
128C4⋊C8—42no—
129#102—24no—
130#102—42no—
131#102—42no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134#97—22no—
135#97—22no—
136#97—42no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—42no—
140C4×C8C2×D412yes—
141C4×C8#4112yes—
142#95—41no—
143#95—41no—
144#95—41no—
145C24—41no—
146C24#10211yescharacteristic
147C2×D4—22no—
148C2×D4—44no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—82no—
155C2×D4—82no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—82no—
171C22×C4—82no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4#10211yescharacteristic
176C22×C4#10211yescharacteristic
177C22×C4#10211yescharacteristic
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C2×D811yescharacteristic
180C22×C4C2×SD1611yescharacteristic
181D8—24no—
182D8—42no—
183M16—44no—
184M16—42no—
185M16—42no—
186M16—84no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—44no—
195C2×C8#10412yes—
196C2×C8C4×D412yes—
197C2×C8#12112yes—
198C2×C8C2×M1611yescharacteristic
199C2×C8C2×M1611yescharacteristic
200C2×C8#13112yes—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—82no—
208C22⋊C4—44no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—82no—
221C42#12011yescharacteristic
222C42#12311yescharacteristic
223C42AGL(1,ℤ/8)11yescharacteristic
224C42#13711yescharacteristic
225C16—44no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23C2×#10211yescharacteristic
239D4—42no—
240D4—42no—
241D4—88no—
242D4—81no—
243D4—81no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—162no—
271C2×C4—162no—
272C2×C4#38611yescharacteristic
273C2×C4#40711yescharacteristic
274C2×C4#40811yescharacteristic
275C2×C4#41711yescharacteristic
276C2×C4C2×#10411yescharacteristic
277C2×C4#42111yescharacteristic
278C2×C4#44711yescharacteristic
279C2×C4#44811yescharacteristic
280C2×C4#48011yescharacteristic
281C2×C4#48111yescharacteristic
282C8—22no—
283C8—22no—
284C8—41no—
285C8—84no—
286C8#43512yes—
287C8#43612yes—
288C22—21no—
289C22—21no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22—161no—
299C22—161no—
300C22—161no—
301C22—161no—
302C22#148611yescharacteristic
303C22#149211yescharacteristic
304C22#187411yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—161no—
313C4—161no—
314C4#148911yescharacteristic
315C4#149511yescharacteristic
316C4#187011yescharacteristic
317C4#187111yescharacteristic
318C4#223812yes—
319C2—21no—
320C2—21no—
321C2—81no—
322C2—162no—
323C2#894411yescharacteristic
324C2#1300811yescharacteristic
325C2#1301011yescharacteristic
subgroupG/H
centerC2×C4#481
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1874
prime radicalC2×#469C4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17795C2semidirect
3. C4×#349C2semidirect
4. #11599C2semidirect
5. #7174C2semidirect
72. #334D4semidirect
82. C2×D8M16semidirect
126. #106M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4692
#177952
C4×#3492
#115992
#71742