Finite Group Databasebeta

← #136374

Subgroups — #136374

Order 512.

353 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 98 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#22688C211yesmaximal, characteristic
2#12903C211yesmaximal, characteristic, complemented
3#12816C211yesmaximal, characteristic
4#11598C211yesmaximal, characteristic
5#9612C211yesmaximal, characteristic, complemented
6#7280C211yesmaximal, characteristic, complemented
7#7169C211yesmaximal, characteristic, complemented
8C2×#443—21no—
9C4×#135C2211yescharacteristic
10#2906—21no—
11C2×#365—22no—
12#2150C2211yescharacteristic
13C2×#360—21no—
14C4×M32C2211yescharacteristic
15#1948—21no—
16#1945—21no—
17#1866—22no—
18#1848C2211yescharacteristic
19#1844C2211yescharacteristic
20#1806—21no—
21C8×C4⋊C4C2211yescharacteristic
22C2×#334C2211yescharacteristic
23#1386—21no—
24#1334C412yescomplemented
25#1333C412yescomplemented
26#1296—21no—
27C2×#135D411yescharacteristic
28C2×#131—22no—
29C4×#41—21no—
30C2×M32—21no—
31C2×M32—21no—
32#443—42no—
33#443—42no—
34C4×Q16C2×C411yescharacteristic
35C4×Q16Q811yescharacteristic
36C4×SD16—21no—
37C4×SD16—21no—
38C4×D8C2×C411yescharacteristic
39C4×D8Q811yescharacteristic
40C2×#108—21no—
41C2×#108D412yes—
42#427D411yescharacteristic
43C2×#106—21no—
44C2×#106—21no—
45C2×#106—41no—
46#424—22no—
47#424—42no—
48#423—21no—
49#423—41no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—41no—
52C2×C4⋊C8D411yescharacteristic
53C2×#104—22no—
54#416—21no—
55C2×C4×C8—21no—
56C2×C4×C8C2311yescharacteristic
57#384—21no—
58C4×C4⋊C4D411yescharacteristic
59#365—44no—
60C4⋊C16—21no—
61C4⋊C16—21no—
62#360—42no—
63#346C2×C412yes—
64C4×C16—21no—
65#340—41no—
66#339—22no—
67#336—21no—
68#334—21no—
69#334C2×C412yes—
70C2×#41—21no—
71#135—21no—
72#135—21no—
73#135SD1612yes—
74#135SD1612yes—
75C2×Q16C4⋊C411yescharacteristic
76C2×SD16—21no—
77C2×D8C4⋊C411yescharacteristic
78#131—44no—
79#131—82no—
80C2×M16—22no—
81C2×M16—42no—
82C22×C8—21no—
83C22×C8—42no—
84C22×C8—41no—
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C4×Q8—21no—
88C4×Q8—21no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—41no—
92C4×D4—41no—
93#117—21no—
94#117—41no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C42—21no—
100C2×C42—41no—
101C2×C42C2×D411yescharacteristic
102M32—22no—
103M32—22no—
104M32—41no—
105M32—41no—
106C2×C16—22no—
107C2×C16—41no—
108#108—42no—
109#108—41no—
110#108—41no—
111#107C4⋊C411yescharacteristic
112#107C4⋊C411yescharacteristic
113#106—21no—
114#106—42no—
115#106—41no—
116#106—41no—
117#106—82no—
118C4⋊C8—22no—
119C4⋊C8—21no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8SD1611yescharacteristic
126C4⋊C8SD1611yescharacteristic
127#104—44no—
128#103—21no—
129#103—21no—
130#102—21no—
131#102—21no—
132C4×C8—41no—
133C4×C8—41no—
134C4×C8C22×C411yescharacteristic
135C4×C8C2×Q811yescharacteristic
136C4×C8#4111yescharacteristic
137C4×C8#4111yescharacteristic
138#95—21no—
139#41—22no—
140#41—22no—
141#41—41no—
142#41—41no—
143C2×Q8—21no—
144C2×D4—21no—
145C2×D4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—81no—
151C22×C4#12011yescharacteristic
152C22×C4#13511yescharacteristic
153C22×C4AGL(1,ℤ/8)11yescharacteristic
154Q16—21no—
155Q16#10612yescomplemented
156SD16—22no—
157SD16—22no—
158D8—21no—
159D8#10612yescomplemented
160M16—42no—
161M16—42no—
162M16—84no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—84no—
172C2×C8—81no—
173C2×C8—81no—
174C2×C8C2×C4⋊C411yescharacteristic
175C2×C8C4×D411yescharacteristic
176C2×C8C22⋊Q811yescharacteristic
177C2×C8#12311yescharacteristic
178C2×C8C2×SD1611yescharacteristic
179C2×C8C2×SD1611yescharacteristic
180C2×C8AGL(1,ℤ/8)11yescharacteristic
181C2×C8#13711yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4—161no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C42—41no—
197C42—41no—
198C42—41no—
199C42—41no—
200C42—41no—
201C42C4×D411yescharacteristic
202C42C22⋊Q811yescharacteristic
203C42C2×SD1611yescharacteristic
204C42AGL(1,ℤ/8)11yescharacteristic
205C16—24no—
206C16—81no—
207C23—41no—
208C23#44911yescharacteristic
209Q8—22no—
210Q8—41no—
211D4—22no—
212D4—42no—
213D4—42no—
214D4—41no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—161no—
229C2×C4—161no—
230C2×C4—161no—
231C2×C4—161no—
232C2×C4#38511yescharacteristic
233C2×C4C4×SD1611yescharacteristic
234C2×C4#44211yescharacteristic
235C2×C4#44811yescharacteristic
236C2×C4#45011yescharacteristic
237C2×C4#45011yescharacteristic
238C2×C4#45311yescharacteristic
239C2×C4#45611yescharacteristic
240C2×C4#47611yescharacteristic
241C2×C4#47811yescharacteristic
242C8—21no—
243C8—21no—
244C8—41no—
245C8—41no—
246C8—82no—
247C8—82no—
248C8C2×#10611yescharacteristic
249C8#42811yescharacteristic
250C8#48111yescharacteristic
251C8#48311yescharacteristic
252C22—21no—
253C22—21no—
254C22—42no—
255C22—81no—
256C22—81no—
257C22#162211yescharacteristic
258C22#165411yescharacteristic
259C22#186411yescharacteristic
260C4—21no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—82no—
265C4—81no—
266C4—81no—
267C4—321no—
268C4#162911yescharacteristic
269C4#165111yescharacteristic
270C4#185311yescharacteristic
271C4#185911yescharacteristic
272C4#223011yescharacteristic
273C4#223111yescharacteristic
274C2—21no—
275C2—21no—
276C2—82no—
277C2#888611yescharacteristic
278C2#1295011yescharacteristic
279C2#1295211yescharacteristic
subgroupG/H
center, socleC22#1864
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#135D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12903C2semidirect
5. #9612C2semidirect
6. #7280C2semidirect
7. #7169C2semidirect
24. #1334C4semidirect
25. #1333C4semidirect
155. Q16#106semidirect
159. D8#106semidirect

Maximal subgroups

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

subgroupindex
#226882
#129032
#128162
#115982
#96122
#72802
#71692