Finite Group Databasebeta

← #508414

Subgroups — #508414

Order 512.

557 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 146 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×#2228C211yesmaximal, characteristic
2C2×#2208C211yesmaximal, characteristic
3#33249C211yesmaximal, characteristic, complemented
4#21282C211yesmaximal, characteristic
5#13477C212yesmaximal, complemented
6#13475C212yesmaximal, complemented
7#13449C212yesmaximal
8#13449C212yesmaximal
9#13277C211yesmaximal, characteristic, complemented
10#13253C211yesmaximal, characteristic, complemented
11#12899C211yesmaximal, characteristic, complemented
12C22×SD32C2212yes—
13C2×#499C2211yescharacteristic
14C2×#494C2211yescharacteristic
15C2×#491—21no—
16C2×C4×D8C2211yescharacteristic
17#2642—21no—
18#2319—21no—
19#2228—24no—
20#2208—24no—
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2212yes—
26C2×C4⋊C16C2212yes—
27C2×#358C2212yes—
28C2×#358C2212yes—
29C2×#358C2212yes—
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32#1971—22no—
33#1970C2212yes—
34#1969C2212yes—
35#1969C2212yes—
36#1942C2212yes—
37#1936—22no—
38C4×#107C2211yescharacteristic
39#1762—21no—
40#1726C2211yescharacteristic
41#1380C2212yes—
42#1380C2212yes—
43C22×Q16—21no—
44C22×Q16C2312yes—
45C22×D8C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×#125—21no—
48C2×C4×D4—21no—
49C2×SD32—24no—
50C2×SD32—42no—
51C2×SD32—42no—
52C22×C16C2312yes—
53C22×C16C2311yescharacteristic
54C22×C16C2311yescharacteristic
55#499—22no—
56#499—22no—
57#494—22no—
58#494—22no—
59#491—44no—
60C4×D8—22no—
61C4×D8D414yes—
62C2×#107C2311yescharacteristic
63C2×#107C2311yescharacteristic
64C2×#107C2312yes—
65C2×#107C2311yescharacteristic
66C2×#106—21no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#102—21no—
70C2×#102—22no—
71C2×#97D411yescharacteristic
72C2×#97D412yes—
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75#400—41no—
76#400—41no—
77#400—42no—
78#391—22no—
79#391—22no—
80#390—22no—
81#389—22no—
82#388—42no—
83#381—41no—
84#380—41no—
85C4×C4⋊C4—21no—
86#376D411yescharacteristic
87#367—22no—
88#366—22no—
89C4⋊C16—22no—
90C4⋊C16—22no—
91C4⋊C16—42no—
92C4⋊C16—42no—
93#358—24no—
94#358—42no—
95#358—42no—
96#357—22no—
97#357—22no—
98#340—21no—
99#340—22no—
100C22×Q8—22no—
101C22×D4—21no—
102C23×C4—41no—
103C2×Q16—22no—
104C2×Q16—22no—
105C2×Q16—22no—
106C2×Q16—22no—
107C2×Q16—22no—
108C2×Q16—22no—
109C2×Q16—44no—
110C2×Q16—42no—
111C2×D8—21no—
112C2×D8—22no—
113C2×D8—22no—
114C2×D8C2×D412yes—
115C22×C8C2×D412yes—
116C22×C8C2×D411yescharacteristic
117C22×C8C2411yescharacteristic
118C4⋊Q8—42no—
119C4⋊Q8—42no—
120#125—42no—
121#125—42no—
122C4×D4—24no—
123C4×D4—42no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C42—41no—
137C2×C42C2×D412yes—
138C2×C42C2×D411yescharacteristic
139SD32—48no—
140C2×C16—21no—
141C2×C16—21no—
142C2×C16—22no—
143C2×C16—22no—
144C2×C16—22no—
145C2×C16—22no—
146C2×C16—22no—
147#107—21no—
148#107—21no—
149#107—21no—
150#107—21no—
151#107—21no—
152#107—22no—
153#107—22no—
154#107C2×D412yes—
155#106—42no—
156#103—44no—
157#103—44no—
158#102—24no—
159#102—44no—
160#97—22no—
161#97—42no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8C2×D412yes—
165C4×C8#4112yes—
166#95—41no—
167#95—42no—
168#95—42no—
169#95—42no—
170#95—42no—
171#95—42no—
172#95D811yescharacteristic
173#95D811yescharacteristic
174#95D812yes—
175C24—41no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×D4—22no—
183C2×D4—42no—
184C2×D4—41no—
185C2×D4—42no—
186C22×C4—41no—
187C22×C4—44no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—162no—
193C22×C4C2×D812yes—
194C22×C4C2×D812yes—
195C22×C4C2×D811yescharacteristic
196C22×C4C2×D811yescharacteristic
197C22×C4C22×D411yescharacteristic
198Q16—42no—
199Q16—42no—
200Q16—42no—
201Q16—42no—
202Q16—84no—
203D8—22no—
204D8—22no—
205D8—42no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8#12112yes—
211C2×C8C22×D411yescharacteristic
212C2×C8C2×#4111yescharacteristic
213C2×C8H(5,𝔽2)12yes—
214C2×C82−1+412yes—
215C4⋊C4—22no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—162no—
228C22⋊C4—44no—
229C22⋊C4—162no—
230C22⋊C4—162no—
231C42—22no—
232C42—22no—
233C42—161no—
234C42#12112yes—
235C42AGL(1,ℤ/8)12yes—
236C16—41no—
237C16—41no—
238C16—42no—
239C23—42no—
240C23—81no—
241C23—82no—
242C23—81no—
243C23—82no—
244C23C22×D811yescharacteristic
245Q8—82no—
246Q8—82no—
247Q8—82no—
248Q8—82no—
249D4—42no—
250D4—42no—
251D4—82no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—164no—
275C2×C4#45912yes—
276C2×C4#45912yes—
277C2×C4C2×#12111yescharacteristic
278C2×C4#53712yes—
279C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
280C2×C4#57512yes—
281C8—22no—
282C8—42no—
283C8#53511yescharacteristic
284C8#53611yescharacteristic
285C8#54212yes—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—162no—
290C22—162no—
291C22C2×#45911yescharacteristic
292C22#324512yes—
293C22#341612yes—
294C22#341611yescharacteristic
295C22#341711yescharacteristic
296C4—22no—
297C4—22no—
298C4—42no—
299C4—42no—
300C4—82no—
301C4—162no—
302C4—161no—
303C4—161no—
304C4—162no—
305C4#262011yescharacteristic
306C4#304011yescharacteristic
307C4#325112yes—
308C2—162no—
309C2#2982411yescharacteristic
310C2#3358611yescharacteristic
311C2#3359011yescharacteristic
312C2#3382412yes—
313C2#3382612yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33249C2semidirect
5. #13477C2semidirect
6. #13475C2semidirect
9. #13277C2semidirect
10. #13253C2semidirect
11. #12899C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×#22082
#332492
#212822
#134772
#134752
#134492
#132772
#132532
#128992