Finite Group Databasebeta

← #381415

Subgroups — #381415

Order 512.

734 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1841C211yesmaximal, characteristic
2#20764C211yesmaximal, characteristic
3#20445C211yesmaximal, characteristic
4#11340C212yesmaximal
5#11333C212yesmaximal
6#11283C212yesmaximal
7#11262C212yesmaximal
8#10904C212yesmaximal
9#10889C212yesmaximal
10C2×#430C2211yescharacteristic
11#2911C2211yescharacteristic
12C2×#428C2212yes—
13#2873C2212yes—
14C2×C4×M16C2211yescharacteristic
15#2317C2212yes—
16#2285C2211yescharacteristic
17C2×#384C2211yescharacteristic
18#1841C418yescomplemented
19#1836—22no—
20#1834—22no—
21#1773—22no—
22#1772—22no—
23C2×#344C2212yes—
24C2×#344C2212yes—
25C2×#343C2212yes—
26#1740C2212yes—
27#1736C2212yes—
28#1736C2212yes—
29#1736C2212yes—
30#1733C2212yes—
31C2×#339C2212yes—
32C2×#339C2212yes—
33C2×#338C2211yescharacteristic
34C2×H(3,ℤ/4)C2212yes—
35C2×H(3,ℤ/4)C2211yescharacteristic
36C22×M16C2312yes—
37C22×M16C2312yes—
38C22×M16C2311yescharacteristic
39C22×M16C2311yescharacteristic
40C2×#117C2312yes—
41C2×#117C2312yes—
42C2×#117C2312yes—
43C22×C4⋊C4—41no—
44C22×C4⋊C4C2312yes—
45C22×C42C2311yescharacteristic
46#430—24no—
47C2×#108—22no—
48#428—22no—
49#428—22no—
50#428—42no—
51#428—42no—
52#427—22no—
53C2×#107—22no—
54C2×#106—22no—
55#405—24no—
56#405—42no—
57C4×M16—42no—
58C4×M16C2×C414yes—
59C4×M16Q814yes—
60C2×#97D412yes—
61C2×C4×C8D412yes—
62#389—42no—
63#385—42no—
64#384—22no—
65#384—21no—
66#384—21no—
67#384—21no—
68#384—41no—
69#382—22no—
70#382—21no—
71#380—41no—
72C4×C4⋊C4—41no—
73#344—24no—
74#344C2×C418yes—
75#343—42no—
76#342—42no—
77#340—24no—
78#340—24no—
79#340—42no—
80#339—22no—
81#339—22no—
82#339—42no—
83#339C2×C418yes—
84#338C2×C414yes—
85H(3,ℤ/4)—24no—
86H(3,ℤ/4)C2×C414yes—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4C2411yescharacteristic
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16C22×C414yes—
105C2×M16C22×C412yes—
106C2×M16C22×C412yes—
107C2×M16C2×Q812yes—
108C2×M16C2×Q812yes—
109C2×M16#4114yes—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114#117—22no—
115#117—22no—
116#117—22no—
117#117—22no—
118#117—22no—
119#117—22no—
120#117—22no—
121#117—22no—
122#117—22no—
123#117—22no—
124#117—22no—
125#117C22×C414yes—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—84no—
140C2×C4⋊C4#4112yes—
141C2×C4⋊C4#4112yes—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42C22×C412yes—
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×Q812yes—
155#108—28no—
156#107—22no—
157#107—22no—
158#107—44no—
159#106—24no—
160#106—44no—
161#97—44no—
162#97C4⋊C418yescomplemented
163C4×C8—44no—
164C4×C8C4⋊C418yescomplemented
165#95—42no—
166#95—44no—
167C24H(5,𝔽2)11yescharacteristic
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—84no—
182C22×C4C23×C411yescharacteristic
183C22×C4C22×D411yescharacteristic
184C22×C4C22×Q811yescharacteristic
185C22×C4C2×#4112yes—
186C22×C4C2×#4112yes—
187C22×C4H(5,𝔽2)12yes—
188C22×C4H(5,𝔽2)11yescharacteristic
189C22×C42−1+411yescharacteristic
190C22×C42−1+412yes—
191C22×C42−1+411yescharacteristic
192M16—24no—
193M16—24no—
194M16—24no—
195M16—24no—
196M16—24no—
197M16—24no—
198M16—24no—
199M16—24no—
200M16—42no—
201M16—42no—
202M16—42no—
203M16—42no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8C2×C4⋊C414yes—
215C2×C8C2×C4⋊C414yes—
216C4⋊C4—24no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—84no—
231C4⋊C4—84no—
232C4⋊C4—164no—
233C22⋊C4—24no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C42—22no—
240C42—21no—
241C42—21no—
242C42—22no—
243C42—24no—
244C42—42no—
245C42—42no—
246C42—42no—
247C42—42no—
248C42—42no—
249C42C2×C4⋊C412yes—
250C42C2×C4⋊C412yes—
251C23—41no—
252C23—41no—
253C23#52011yescharacteristic
254C23#53511yescharacteristic
255C23#53812yes—
256C23#54212yes—
257C23#54411yescharacteristic
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—84no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—84no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—162no—
280C2×C4C22×C4⋊C411yescharacteristic
281C2×C4C4×#4112yes—
282C2×C4#52012yes—
283C2×C4#52011yescharacteristic
284C2×C4#52511yescharacteristic
285C2×C4#52511yescharacteristic
286C2×C4#53312yes—
287C2×C4#53511yescharacteristic
288C2×C4#53611yescharacteristic
289C2×C4#53611yescharacteristic
290C2×C4#53812yes—
291C2×C4#53912yes—
292C2×C4#53912yes—
293C2×C4#54112yes—
294C2×C4#54112yes—
295C2×C4#54212yes—
296C2×C4#54412yes—
297C2×C4#54411yescharacteristic
298C8—24no—
299C8—24no—
300C8—44no—
301C8—44no—
302C22—21no—
303C22—22no—
304C22—21no—
305C22—22no—
306C22—41no—
307C22—41no—
308C22—42no—
309C22#231711yescharacteristic
310C22#233412yes—
311C22#263111yescharacteristic
312C22#263211yescharacteristic
313C22#266612yes—
314C4—24no—
315C4—42no—
316C4—84no—
317C4—82no—
318C4—82no—
319C4—162no—
320C4—162no—
321C4#229611yescharacteristic
322C4#231711yescharacteristic
323C4#233412yes—
324C4#263311yescharacteristic
325C4#263411yescharacteristic
326C4#266612yes—
327C2—21no—
328C2—22no—
329C2—21no—
330C2—22no—
331C2#1429811yescharacteristic
332C2#3005711yescharacteristic
333C2#3005811yescharacteristic
subgroupG/H
center, socleC22#2631
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1841C4semidirect
162. #97C4⋊C4semidirect
164. C4×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18412
#207642
#204452
#113402
#113332
#112832
#112622
#109042
#108892