Finite Group Databasebeta

← #508513

Subgroups — #508513

Order 512.

496 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 136 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#33136C211yesmaximal, characteristic
2C2×#1849C211yesmaximal, characteristic
3#13541C212yesmaximal, complemented
4#13536C212yesmaximal, complemented
5#13510C212yesmaximal, complemented
6#13488C212yesmaximal, complemented
7#13003C212yesmaximal
8#12928C212yesmaximal
9#12846C211yesmaximal, characteristic
10C2×#467C2211yescharacteristic
11C22×#108—21no—
12C22×#108C2212yes—
13C2×#423—21no—
14C22×#97C2211yescharacteristic
15#2855—21no—
16C2×#365C2214yes—
17C2×#362C2214yes—
18#2112C2212yes—
19#1971C2211yescharacteristic
20#1971C2211yescharacteristic
21#1944C2211yescharacteristic
22#1944C2211yescharacteristic
23#1935—21no—
24#1935—21no—
25#1892—21no—
26#1849—21no—
27#1849—21no—
28#1849C2212yes—
29#1849C2212yes—
30#1807C2211yescharacteristic
31#1387C2214yes—
32#1383C2214yes—
33#1377C2212yes—
34#1377C2212yes—
35C22×Q16C2311yescharacteristic
36C22×M16—22no—
37C23×C8D411yescharacteristic
38C23×C8C2311yescharacteristic
39C2×C22⋊Q8—21no—
40C22×C42D411yescharacteristic
41C2×M32C2314yes—
42#467—22no—
43#467—22no—
44C2×#108—22no—
45C2×#108—22no—
46C2×#108—22no—
47C2×#108—44no—
48C2×#108—44no—
49C2×#108C2314yes—
50C2×#107C2311yescharacteristic
51C2×#106C2311yescharacteristic
52C2×#106C2311yescharacteristic
53#423—22no—
54#423—22no—
55#423—42no—
56C2×C4⋊C8—42no—
57C2×#103—21no—
58C2×#103—21no—
59C2×#103—21no—
60C2×#97—22no—
61C2×#97—22no—
62C2×#97D411yescharacteristic
63C2×#97D411yescharacteristic
64C2×#97C2311yescharacteristic
65C2×#97C2311yescharacteristic
66#391—21no—
67#391—21no—
68#389—21no—
69#389—21no—
70#387—41no—
71#387—41no—
72#385—41no—
73#381—41no—
74#380—21no—
75#365—44no—
76#362—44no—
77C22⋊C16—22no—
78C22⋊C16—22no—
79#340—21no—
80#340—21no—
81#340—21no—
82C22×Q8—21no—
83C23×C4SD1611yescharacteristic
84C23×C4SD1611yescharacteristic
85C23×C4C2×D411yescharacteristic
86C2×Q16—22no—
87C2×Q16—22no—
88C2×Q16—22no—
89C2×M16—24no—
90C2×M16—42no—
91C2×M16—44no—
92C2×M16—44no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—42no—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8#4112yes—
104C22×C8#4112yes—
105C22×C8C2411yescharacteristic
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42SD1611yescharacteristic
123C2×C42SD1611yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126M32—24no—
127M32—24no—
128C2×C16—24no—
129#108—24no—
130#108—24no—
131#108—44no—
132#108—44no—
133#108—44no—
134#107—22no—
135#106—22no—
136#106—22no—
137C4⋊C8—44no—
138C4⋊C8—44no—
139#103—42no—
140#103—42no—
141#103—42no—
142#97—22no—
143#97—22no—
144#97—22no—
145#97—22no—
146#95—41no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153C24C2×SD1611yescharacteristic
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—81no—
174C22×C4#12311yescharacteristic
175C22×C4#12311yescharacteristic
176C22×C4C2×SD1611yescharacteristic
177C22×C4C2×SD1611yescharacteristic
178C22×C4C2×SD1611yescharacteristic
179C22×C4C2×SD1611yescharacteristic
180C22×C4C2×SD1611yescharacteristic
181C22×C4AGL(1,ℤ/8)11yescharacteristic
182C22×C4AGL(1,ℤ/8)11yescharacteristic
183C22×C4C22×D411yescharacteristic
184Q16—44no—
185M16—44no—
186M16—44no—
187M16—84no—
188C2×C8—24no—
189C2×C8—24no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—44no—
199C2×C8—84no—
200C2×C8#12312yes—
201C2×C8AGL(1,ℤ/8)12yes—
202C2×C8#13712yes—
203C2×C8C2×#4112yes—
204C2×C8H(5,𝔽2)12yes—
205C2×C82−1+412yes—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C22⋊C4—82no—
217C22⋊C4—81no—
218C22⋊C4—81no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C16—44no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—41no—
232C23#48111yescharacteristic
233C23#48111yescharacteristic
234C23C22×SD1611yescharacteristic
235Q8—81no—
236Q8—81no—
237Q8—81no—
238Q8—81no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4#48111yescharacteristic
274C2×C4#48111yescharacteristic
275C2×C4C2×#12311yescharacteristic
276C2×C4#53511yescharacteristic
277C2×C4#53611yescharacteristic
278C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
279C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
280C2×C4C2×#13711yescharacteristic
281C8—22no—
282C8—22no—
283C8—22no—
284C8—84no—
285C8#53914yes—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—42no—
294C22—42no—
295C22#223912yes—
296C22C2×#48111yescharacteristic
297C22#320611yescharacteristic
298C22#322711yescharacteristic
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—161no—
306C4—161no—
307C4—161no—
308C4—161no—
309C4#262311yescharacteristic
310C4#309211yescharacteristic
311C4#321711yescharacteristic
312C4#323811yescharacteristic
313C2—21no—
314C2—21no—
315C2—22no—
316C2—41no—
317C2#2990711yescharacteristic
318C2C2×#223911yescharacteristic
319C2#3365611yescharacteristic
subgroupG/H
center, socleC22C2×#481
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13541C2semidirect
4. #13536C2semidirect
5. #13510C2semidirect
6. #13488C2semidirect

Maximal subgroups

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

subgroupindex
#331362
C2×#18492
#135412
#135362
#135102
#134882
#130032
#129282
#128462