Finite Group Databasebeta

← #139195

Subgroups — #139195

Order 512.

312 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 84 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#33505C211yesmaximal, characteristic
2#19273C211yesmaximal, characteristic
3#13278C211yesmaximal, characteristic, complemented
4#11968C211yesmaximal, characteristic, complemented
5#8666C211yesmaximal, characteristic
6#7299C211yesmaximal, characteristic, complemented
7#7244C211yesmaximal, characteristic, complemented
8#3160C2211yescharacteristic
9#3116—21no—
10C8×#41C2211yescharacteristic
11Q8×C16—21no—
12#2169—22no—
13#2169—21no—
14#2162—21no—
15#2162C2211yescharacteristic
16#2150—21no—
17#2150C2211yescharacteristic
18#1806—21no—
19C4×#106C2211yescharacteristic
20#1676C411yescharacteristic, complemented
21#1675C411yescharacteristic, complemented
22#1674C411yescharacteristic, complemented
23#1673C411yescharacteristic, complemented
24#1672—21no—
25#1671—21no—
26#1568C2211yescharacteristic
27#1566C2211yescharacteristic
28#1554—21no—
29#1386—22no—
30#1363—21no—
31#1363—21no—
32#1303—21no—
33#533—21no—
34C4×#41D411yescharacteristic
35C2×M32—22no—
36C2×M32D412yes—
37#500C2×C411yescharacteristic
38#500C2×C411yescharacteristic
39#499—21no—
40#499—21no—
41#498—21no—
42#485—41no—
43#477—21no—
44#476—21no—
45#462—21no—
46#461—21no—
47#452—41no—
48C8×Q8C2×C411yescharacteristic
49C8×Q8D411yescharacteristic
50#441—41no—
51#440—41no—
52C8×D4—21no—
53C8×D4—21no—
54C8×D4C2×C411yescharacteristic
55C8×D4D411yescharacteristic
56#432—21no—
57#431—21no—
58#431D411yescharacteristic
59#427—21no—
60#427—21no—
61C2×#106—21no—
62#423—21no—
63C2×#104—41no—
64C2×C4×C8—21no—
65C2×C4×C8C2311yescharacteristic
66C4×C4⋊C4—21no—
67C4⋊C16—22no—
68C4⋊C16—21no—
69C4⋊C16—21no—
70C4⋊C16D411yescharacteristic
71C4⋊C16D411yescharacteristic
72C22⋊C16—22no—
73C22⋊C16—21no—
74C4×C16—22no—
75C4×C16—21no—
76#340—21no—
77#339—41no—
78#335C2×C411yescharacteristic
79#335C2×C411yescharacteristic
80#334—21no—
81#334—21no—
82#334—21no—
83#333—21no—
84#333—21no—
85#332—21no—
86#332—21no—
87C2×#41D811yescharacteristic
88C2×Q16—41no—
89C2×SD16—41no—
90C2×M16—41no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8C2×D411yescharacteristic
94C22×C8#4111yescharacteristic
95C4⋊Q8—21no—
96C4⋊Q8—21no—
97#125—41no—
98C22⋊Q8—41no—
99C22⋊Q8—41no—
100C4×Q8—41no—
101C4×Q8—41no—
102C4×Q8C22⋊C411yescharacteristic
103C4×Q8SD1611yescharacteristic
104C4×D4—21no—
105C4×D4—21no—
106C4×D4C22⋊C411yescharacteristic
107C4×D4SD1611yescharacteristic
108#117—21no—
109#117—41no—
110#117D811yescharacteristic
111C2×C4⋊C4—41no—
112C2×C42—21no—
113C2×C42—41no—
114C2×C42C2×D411yescharacteristic
115M32—42no—
116M32—82no—
117C2×C16—21no—
118C2×C16—22no—
119C2×C16—21no—
120#107—21no—
121#107—21no—
122#107—21no—
123#107—21no—
124#107—41no—
125#106—22no—
126#106—41no—
127#106—41no—
128#106—41no—
129#106—41no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8C22⋊C411yescharacteristic
137C4⋊C8C22⋊C411yescharacteristic
138#104—82no—
139#103—41no—
140#103—41no—
141#103—41no—
142#102—41no—
143C22⋊C8—21no—
144C22⋊C8—21no—
145C22⋊C8—21no—
146C22⋊C8—41no—
147#97—41no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8C22×C411yescharacteristic
152C4×C8C2×D411yescharacteristic
153C4×C8C2×D411yescharacteristic
154C4×C8#4111yescharacteristic
155#95—41no—
156#41—22no—
157#41—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8#10211yescharacteristic
161C2×D4—21no—
162C2×D4#10211yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—81no—
167C22×C4#12111yescharacteristic
168C22×C4C2×D811yescharacteristic
169C22×C4#13511yescharacteristic
170Q16—82no—
171SD16—82no—
172M16—81no—
173M16—81no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—81no—
184C2×C8—81no—
185C2×C8C4×D411yescharacteristic
186C2×C8C4×D411yescharacteristic
187C2×C8#12011yescharacteristic
188C2×C8#12311yescharacteristic
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—42no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4#10211yescharacteristic
200C4⋊C4#10211yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—81no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—81no—
209C42—81no—
210C42—81no—
211C42C2×C22⋊C411yescharacteristic
212C42#12111yescharacteristic
213C42C2×SD1611yescharacteristic
214C42#13511yescharacteristic
215C16—42no—
216C16—41no—
217C16—41no—
218C23—21no—
219C23#46611yescharacteristic
220Q8—21no—
221Q8—81no—
222Q8—81no—
223Q8—161no—
224Q8#36012yes—
225D4—21no—
226D4—21no—
227D4—42no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—82no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—161no—
244C2×C4—161no—
245C2×C4#38611yescharacteristic
246C2×C4C2×#10211yescharacteristic
247C2×C4#41611yescharacteristic
248C2×C4#46511yescharacteristic
249C2×C4#46511yescharacteristic
250C2×C4#46611yescharacteristic
251C8—21no—
252C8—21no—
253C8—21no—
254C8—41no—
255C8—41no—
256C8—81no—
257C8—81no—
258C8—161no—
259C8#45511yescharacteristic
260C8#45611yescharacteristic
261C22—21no—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22#189311yescharacteristic
268C4—21no—
269C4—22no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—162no—
276C4—161no—
277C4—161no—
278C4#188211yescharacteristic
279C4#190211yescharacteristic
280C4C2×#36011yescharacteristic
281C4#214511yescharacteristic
282C2—41no—
283C2—81no—
284C2#931611yescharacteristic
285C2#1306511yescharacteristic
286C2#1307111yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#41D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13278C2semidirect
4. #11968C2semidirect
6. #7299C2semidirect
7. #7244C2semidirect
20. #1676C4semidirect
21. #1675C4semidirect
22. #1674C4semidirect
23. #1673C4semidirect

Maximal subgroups

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

subgroupindex
#335052
#192732
#132782
#119682
#86662
#72992
#72442