Finite Group Databasebeta

← #244157

Subgroups — #244157

Order 512.

595 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 595 conjugacy classes of subgroups — too many to draw. Only its 142 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#21735C211yesmaximal, characteristic, complemented
2#21638C211yesmaximal, characteristic
3#17934C212yesmaximal, complemented
4#17905C212yesmaximal
5#10669C212yesmaximal, complemented
6#10662C211yesmaximal, characteristic, complemented
7#10661C211yesmaximal, characteristic
8#10037C211yesmaximal, characteristic, complemented
9#10036C211yesmaximal, characteristic
10#10016C212yesmaximal
11#8140C211yesmaximal, characteristic, complemented
12#3162C2211yescharacteristic
13#3104C2212yes—
14#3101C2212yes—
15#2960C2212yes—
16C2×#446C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18#2398C2211yescharacteristic
19#1995C2212yes—
20#1985C2211yescharacteristic
21#1955C2211yescharacteristic
22#1946—22no—
23#1923—22no—
24#1849C2211yescharacteristic
25#1832C2211yescharacteristic
26#1772—22no—
27#1752—22no—
28#1726C2211yescharacteristic
29#1593—24no—
30#1587—24no—
31#1585—24no—
32SD16⋊C8—24no—
33#1571C2212yes—
34#1570C2212yes—
35#1487C2212yes—
36#1486C2212yes—
37#1481C2212yes—
38#1481C2212yes—
39#1452C2212yes—
40#1275C2211yescharacteristic
41#1275C2211yescharacteristic
42#1274C2212yes—
43C23×C8—21no—
44#533C2312yes—
45C2×C4×Q8C2311yescharacteristic
46C2×C4×D4C2311yescharacteristic
47#501—22no—
48#501—22no—
49#484—22no—
50#483—22no—
51#477—22no—
52#476—22no—
53#462—22no—
54#461—22no—
55#451—22no—
56#450—22no—
57#446—22no—
58#446—22no—
59#446—22no—
60#446—22no—
61C8×Q8—22no—
62C8×Q8—22no—
63C4×Q16—24no—
64C4×SD16—24no—
65C8×D4—22no—
66C8×D4—22no—
67#431C2312yes—
68#431C2312yes—
69C2×#108—22no—
70#427—22no—
71#424—22no—
72#423C2312yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C22⋊C8—21no—
77C2×C22⋊C8—22no—
78#405—22no—
79C2×#97—21no—
80C2×#97D412yes—
81C2×#97C2311yescharacteristic
82C2×C4×C8D412yes—
83C2×C4×C8C2311yescharacteristic
84#400—21no—
85#397—21no—
86#394—21no—
87#393—21no—
88C4×C22⋊C4—22no—
89#376C2311yescharacteristic
90#339—24no—
91#336—22no—
92#336—22no—
93#335—24no—
94#334—24no—
95Q8⋊C8—24no—
96Q8⋊C8—24no—
97Q8⋊C8—24no—
98D4⋊C8—24no—
99#322D414yes—
100#322D414yes—
101C22×Q8#4111yescharacteristic
102C22×D4#4111yescharacteristic
103C23×C4—21no—
104C2×Q16—24no—
105C2×SD16—24no—
106C2×M16—22no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—44no—
110C22×C8—42no—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D412yes—
114C4⋊Q8#4112yes—
115C4⋊Q8#4112yes—
116#125—22no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×D4—22no—
124C4×D4—22no—
125#117—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4#4111yescharacteristic
128C2×C4⋊C4#4111yescharacteristic
129C2×C22⋊C4—22no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—22no—
132C2×C42—21no—
133C2×C42C2×D412yes—
134C2×C42C2411yescharacteristic
135#108—44no—
136#107—22no—
137#107—22no—
138#107—24no—
139#106—24no—
140#106—24no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—42no—
150C4⋊C8—42no—
151#103—24no—
152#103—24no—
153#103—24no—
154#102—24no—
155C22⋊C8—22no—
156C22⋊C8—22no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—44no—
160#97—22no—
161#97—42no—
162#97—42no—
163#97—42no—
164C4×C8—24no—
165C4×C8—42no—
166C4×C8C2×D414yes—
167C4×C8C2×D414yes—
168C4×C8C2×D412yes—
169C4×C8C2×D412yes—
170#95—22no—
171#95—22no—
172#95—21no—
173#95—21no—
174C24—21no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—81no—
179C2×D4—22no—
180C2×D4—42no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4C22×D411yescharacteristic
190C22×C4C2×#4111yescharacteristic
191C22×C4C2×#4111yescharacteristic
192Q16—84no—
193SD16—48no—
194M16—44no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—84no—
207C2×C8#12711yescharacteristic
208C2×C8#12711yescharacteristic
209C2×C8#12712yes—
210C2×C8#13714yes—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C22⋊C4—44no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—82no—
224C42—22no—
225C42—22no—
226C42—41no—
227C42—42no—
228C42—41no—
229C42C22×D412yes—
230C42C2×#4112yes—
231C23—22no—
232C23—42no—
233C23—41no—
234C23—42no—
235C23—41no—
236C23#52511yescharacteristic
237Q8—44no—
238Q8—42no—
239Q8—44no—
240Q8—42no—
241D4—24no—
242D4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4#52512yes—
261C2×C4#52512yes—
262C2×C4#52511yescharacteristic
263C2×C4C2×#12711yescharacteristic
264C2×C4C2×#13712yes—
265C2×C4#57512yes—
266C8—24no—
267C8—21no—
268C8—21no—
269C8—22no—
270C8—44no—
271C8—44no—
272C8—81no—
273C8—81no—
274C8—82no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22—42no—
279C22—82no—
280C22—81no—
281C22—81no—
282C22#239711yescharacteristic
283C22#311612yes—
284C4—22no—
285C4—22no—
286C4—82no—
287C4—81no—
288C4—82no—
289C4—81no—
290C4—82no—
291C4—81no—
292C4#311712yes—
293C4#315211yescharacteristic
294C4#315311yescharacteristic
295C2—22no—
296C2—82no—
297C2#2379411yescharacteristic
298C2#2379511yescharacteristic
299C2#2380711yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutator, Frattini, cosocleC2×C42C24
socleC22#2397
prime radicalC22×D4#41
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21735C2semidirect
3. #17934C2semidirect
5. #10669C2semidirect
6. #10662C2semidirect
8. #10037C2semidirect
11. #8140C2semidirect

Maximal subgroups

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

subgroupindex
#217352
#216382
#179342
#179052
#106692
#106622
#106612
#100372
#100362
#100162
#81402