Finite Group Databasebeta

← #136913

Subgroups — #136913

Order 512.

417 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#107C211yesmaximal, characteristic
2C2×#1670C211yesmaximal, characteristic
3#13253C211yesmaximal, characteristic
4#13203C211yesmaximal, characteristic
5C2×Q8⋊C16C211yesmaximal, characteristic
6#7274C211yesmaximal, characteristic
7#7171C211yesmaximal, characteristic
8C2×#494C2211yescharacteristic
9C2×#475—21no—
10C2×C8×Q8C2211yescharacteristic
11Q8×C4⋊C4—21no—
12C2×C4⋊C16C2211yescharacteristic
13C2×#358—22no—
14C2×#358—21no—
15C2×#358—21no—
16C2×C4×C16C2211yescharacteristic
17#1947—22no—
18#1947—21no—
19#1845—21no—
20#1845C2211yescharacteristic
21C4×#107—21no—
22C4×#107C2211yescharacteristic
23#1670—22no—
24#1670C414yescomplemented
25C2×#335C2211yescharacteristic
26C2×#332—21no—
27Q8⋊C16—22no—
28C22×Q16—21no—
29C22×Q16—21no—
30C2×C4⋊Q8—21no—
31C2×C4×Q8—41no—
32C2×C4×Q8D411yescharacteristic
33C22×C16—21no—
34C22×C16—21no—
35#494—21no—
36#494—21no—
37#494—21no—
38#494C2×C412yes—
39#475—24no—
40#475—42no—
41C8×Q8—21no—
42C8×Q8—21no—
43C8×Q8C2×C412yes—
44C8×Q8D412yes—
45C2×#107—21no—
46C2×#107—22no—
47C2×#107D411yescharacteristic
48C2×#107D411yescharacteristic
49C2×#107D412yes—
50C2×#106—41no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8D411yescharacteristic
54C2×#103—41no—
55C2×C4×C8—21no—
56C2×C4×C8C2311yescharacteristic
57#389—42no—
58#389—41no—
59#384—22no—
60#384—21no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—21no—
63C4⋊C16—21no—
64C4⋊C16—21no—
65#358—42no—
66#358—44no—
67#358—42no—
68C4×C16—21no—
69C4×C16—21no—
70#340—22no—
71#340—21no—
72#335—21no—
73#335C2×C412yes—
74#332—24no—
75C22×Q8—41no—
76C22×Q8D811yescharacteristic
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—42no—
80C2×Q16—42no—
81C2×Q16—41no—
82C2×Q16—41no—
83C2×Q16—41no—
84C2×Q16—41no—
85C2×Q16—41no—
86C2×Q16—41no—
87C22×C8—21no—
88C22×C8—41no—
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C4⋊Q8—22no—
92C4⋊Q8—41no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—82no—
98C4×Q8—82no—
99C4×Q8C22⋊C412yes—
100C4×Q8SD1612yes—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4D811yescharacteristic
112C2×C42—21no—
113C2×C42—42no—
114C2×C42—41no—
115C2×C42C2×D411yescharacteristic
116C2×C16—22no—
117C2×C16—41no—
118C2×C16—41no—
119C2×C16—41no—
120C2×C16—41no—
121C2×C16—41no—
122#107—21no—
123#107—21no—
124#107—21no—
125#107—41no—
126#107—41no—
127#107—42no—
128#107—82no—
129#107SD1612yes—
130#106—44no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—21no—
135C4⋊C8—41no—
136C4⋊C8C22⋊C412yes—
137#103—44no—
138C4×C8—22no—
139C4×C8—22no—
140C4×C8C22×C411yescharacteristic
141C4×C8C2×D411yescharacteristic
142C4×C8#4111yescharacteristic
143C4×C8#4111yescharacteristic
144#95—42no—
145#95—41no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—42no—
150C2×Q8—81no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×Q8—81no—
154C2×Q8—81no—
155C2×Q8#10212yes—
156C2×Q8SD3212yes—
157C2×Q8Q3212yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—81no—
164C22×C4#12011yescharacteristic
165C22×C4C2×D811yescharacteristic
166C22×C4AGL(1,ℤ/8)11yescharacteristic
167Q16—44no—
168Q16—42no—
169Q16—42no—
170Q16—82no—
171Q16—81no—
172Q16—81no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—22no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—42no—
179C2×C8—81no—
180C2×C8—81no—
181C2×C8C4×D411yescharacteristic
182C2×C8C4×D411yescharacteristic
183C2×C8#12111yescharacteristic
184C2×C8#12111yescharacteristic
185C2×C8C2×SD1611yescharacteristic
186C2×C8#13511yescharacteristic
187C2×C8AGL(1,ℤ/8)11yescharacteristic
188C2×C8#13711yescharacteristic
189C4⋊C4—22no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—41no—
200C4⋊C4—84no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4—161no—
206C4⋊C4—162no—
207C4⋊C4#10212yes—
208C42—22no—
209C42—22no—
210C42—84no—
211C42—81no—
212C42—81no—
213C42C2×C22⋊C411yescharacteristic
214C42#12311yescharacteristic
215C42C2×SD1611yescharacteristic
216C42#13711yescharacteristic
217C16—42no—
218C16—81no—
219C23#44711yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—82no—
224Q8—82no—
225Q8—161no—
226Q8—161no—
227Q8#35814yescomplemented
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—161no—
247C2×C4—161no—
248C2×C4#38611yescharacteristic
249C2×C4C2×#10211yescharacteristic
250C2×C4#41711yescharacteristic
251C2×C4#44811yescharacteristic
252C2×C4#44811yescharacteristic
253C2×C4#44911yescharacteristic
254C2×C4#45311yescharacteristic
255C2×C4#45411yescharacteristic
256C2×C4#48011yescharacteristic
257C2×C4#48111yescharacteristic
258C2×C4C2×SD3211yescharacteristic
259C2×C4C2×Q3211yescharacteristic
260C2×C4#50911yescharacteristic
261C2×C4#51011yescharacteristic
262C8—21no—
263C8—21no—
264C8—42no—
265C8—82no—
266C8C4×SD1611yescharacteristic
267C8#44011yescharacteristic
268C8#46011yescharacteristic
269C8#46411yescharacteristic
270C22#162111yescharacteristic
271C22#162711yescharacteristic
272C22#187411yescharacteristic
273C22#218711yescharacteristic
274C22#218811yescharacteristic
275C22#219411yescharacteristic
276C22#219511yescharacteristic
277C4—24no—
278C4—21no—
279C4—21no—
280C4—42no—
281C4—82no—
282C4—162no—
283C4—161no—
284C4—162no—
285C4#162311yescharacteristic
286C4#162611yescharacteristic
287C4#188011yescharacteristic
288C4#188311yescharacteristic
289C4C2×#35811yescharacteristic
290C4#213911yescharacteristic
291C4#223511yescharacteristic
292C4#223611yescharacteristic
293C2#897911yescharacteristic
294C2#1233211yescharacteristic
295C2#1233411yescharacteristic
296C2#1235611yescharacteristic
297C2#1235811yescharacteristic
298C2#1299311yescharacteristic
299C2#1299711yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1670C4semidirect
227. Q8#358semidirect

Maximal subgroups

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

subgroupindex
Q8×#1072
C2×#16702
#132532
#132032
C2×Q8⋊C162
#72742
#71712