Finite Group Databasebeta

← #136428

Subgroups — #136428

Order 512.

357 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 96 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
1#22666C211yesmaximal, characteristic
2#13185C211yesmaximal, characteristic, complemented
3#12887C211yesmaximal, characteristic, complemented
4#11587C211yesmaximal, characteristic
5#8662C211yesmaximal, characteristic
6#7279C211yesmaximal, characteristic, complemented
7#7175C211yesmaximal, characteristic, complemented
8C2×#436C2211yescharacteristic
9C2×#428—21no—
10#2352—21no—
11C2×#365—21no—
12#2150C2211yescharacteristic
13C4×M32C2211yescharacteristic
14#1949—21no—
15#1854—21no—
16#1853—21no—
17#1848C2211yescharacteristic
18#1846C2211yescharacteristic
19#1809—21no—
20C4×#106C2211yescharacteristic
21#1564C2211yescharacteristic
22#1486—21no—
23#1383—21no—
24#1383—21no—
25#1383—21no—
26#1330C414yescomplemented
27#1301—21no—
28C22×M16—21no—
29C2×C4×D4D411yescharacteristic
30C2×#117—21no—
31C22×C4⋊C4—21no—
32C2×M32—21no—
33C2×M32—21no—
34#436—21no—
35#436—21no—
36#436C2×C412yes—
37#436Q812yes—
38#431—21no—
39C2×#108—21no—
40C2×#108—21no—
41#428—42no—
42#428—42no—
43C2×#107—21no—
44C2×#107—21no—
45C2×#106—21no—
46C2×#106D411yescharacteristic
47C2×#106D411yescharacteristic
48C2×#106D411yescharacteristic
49C2×#106D411yescharacteristic
50#423—21no—
51C2×C4⋊C8D411yescharacteristic
52C2×C22⋊C8—21no—
53C2×C4×C8C2311yescharacteristic
54#388—41no—
55#387—41no—
56#386—41no—
57#382—21no—
58#380—21no—
59#380—21no—
60C4×C4⋊C4—21no—
61C4×C22⋊C4—41no—
62#365—42no—
63C4⋊C16—21no—
64C4⋊C16C2×C412yescomplemented
65#346C2×C412yescomplemented
66C4×C16—21no—
67#340—21no—
68#340—21no—
69#340—21no—
70#335—21no—
71#334—21no—
72D4⋊C8—42no—
73C22×D4SD1611yescharacteristic
74C23×C4—21no—
75C23×C4—41no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C22×C8—21no—
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C4×D4—21no—
87C4×D4—21no—
88C4×D4C4⋊C412yes—
89C4×D4SD1612yes—
90#117—42no—
91#117—81no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—81no—
99C2×C4⋊C4SD1611yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42C2×D411yescharacteristic
107M32—41no—
108M32—41no—
109M32—41no—
110M32—41no—
111C2×C16—22no—
112C2×C16—22no—
113#108—42no—
114#108—41no—
115#108—41no—
116#107—42no—
117#107—41no—
118#107—41no—
119#106—21no—
120#106—21no—
121#106—21no—
122#106—42no—
123#106—41no—
124#106—41no—
125#106D812yes—
126C4⋊C8—21no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8C4⋊C412yes—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—41no—
134C4×C8—41no—
135C4×C8C22×C411yescharacteristic
136C4×C8C2×Q811yescharacteristic
137C4×C8#4111yescharacteristic
138C4×C8#4111yescharacteristic
139#95—41no—
140#95—41no—
141#95—41no—
142#95—41no—
143#95—41no—
144C24—21no—
145C2×D4—21no—
146C2×D4—81no—
147C2×D4#10612yes—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—82no—
160C22×C4—82no—
161C22×C4—161no—
162C22×C4#12011yescharacteristic
163C22×C4C2×SD1611yescharacteristic
164C22×C4AGL(1,ℤ/8)11yescharacteristic
165M16—42no—
166M16—42no—
167M16—42no—
168M16—42no—
169M16—81no—
170M16—81no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—22no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—81no—
177C2×C8—81no—
178C2×C8C4×D411yescharacteristic
179C2×C8C4×D411yescharacteristic
180C2×C8C22⋊Q811yescharacteristic
181C2×C8C22⋊Q811yescharacteristic
182C2×C8C2×D811yescharacteristic
183C2×C8#13511yescharacteristic
184C2×C8AGL(1,ℤ/8)11yescharacteristic
185C2×C8#13711yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—82no—
196C4⋊C4#10612yes—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C22⋊C4—161no—
202C42—82no—
203C42—161no—
204C42C2×C4⋊C411yescharacteristic
205C42#12311yescharacteristic
206C42C2×SD1611yescharacteristic
207C42#13711yescharacteristic
208C16—42no—
209C16—42no—
210C23—22no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23#45011yescharacteristic
218D4—42no—
219D4—81no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—164no—
242C2×C4#38511yescharacteristic
243C2×C4C2×#10611yescharacteristic
244C2×C4#42811yescharacteristic
245C2×C4#44711yescharacteristic
246C2×C4#44811yescharacteristic
247C2×C4#45211yescharacteristic
248C2×C4#45411yescharacteristic
249C2×C4#45511yescharacteristic
250C2×C4#48111yescharacteristic
251C2×C4#48311yescharacteristic
252C8—21no—
253C8—21no—
254C8—42no—
255C8—161no—
256C8—161no—
257C8C4×D811yescharacteristic
258C8#44011yescharacteristic
259C8#47411yescharacteristic
260C8#47811yescharacteristic
261C22—21no—
262C22—21no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22—81no—
270C22#163511yescharacteristic
271C22#163911yescharacteristic
272C22#185311yescharacteristic
273C4—21no—
274C4—21no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—82no—
279C4—161no—
280C4—161no—
281C4—161no—
282C4#162711yescharacteristic
283C4#164311yescharacteristic
284C4#186011yescharacteristic
285C4#186211yescharacteristic
286C2—21no—
287C2—21no—
288C2—82no—
289C2#889411yescharacteristic
290C2#1293211yescharacteristic
291C2#1293411yescharacteristic
subgroupG/H
center, socleC22#1853
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#436 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13185C2semidirect
3. #12887C2semidirect
6. #7279C2semidirect
7. #7175C2semidirect
26. #1330C4semidirect
64. C4⋊C16C2×C4semidirect
65. #346C2×C4semidirect

Maximal subgroups

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

subgroupindex
#226662
#131852
#128872
#115872
#86622
#72792
#71752