Finite Group Databasebeta

← #511700

Subgroups — #511700

Order 512.

441 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 128 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#33261C211yesmaximal, characteristic
2#33249C211yesmaximal, characteristic
3#13531C211yesmaximal, characteristic
4#13530C211yesmaximal, characteristic
5#13483C211yesmaximal, characteristic
6#13483C211yesmaximal, characteristic
7#13477C212yesmaximal
8#13475C212yesmaximal
9#13453C212yesmaximal
10#13409C212yesmaximal
11#12880C211yesmaximal, characteristic
12C2×#499C2211yescharacteristic
13C2×#494C2211yescharacteristic
14C2×#491—21no—
15#2647—21no—
16#2642—21no—
17C2×#367C2212yes—
18C2×#366C2212yes—
19C2×C4⋊C16C2212yes—
20C2×#358C2212yes—
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24#1971—21no—
25#1971—21no—
26#1969C2211yescharacteristic
27#1969C2211yescharacteristic
28#1969C2211yescharacteristic
29#1969C2211yescharacteristic
30#1944—21no—
31#1943—21no—
32#1942C2212yes—
33#1942C2212yes—
34#1846C2211yescharacteristic
35#1845C2211yescharacteristic
36#1836—21no—
37#1726C2211yescharacteristic
38#1380C2212yes—
39#1380C2212yes—
40#1374C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×Q16—21no—
44C22×Q16C2311yescharacteristic
45C22×Q16C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×#125—21no—
48C22×C16C2312yes—
49C22×C16C2312yes—
50#499—42no—
51#494—42no—
52#491—44no—
53C2×#107—21no—
54C2×#107C2312yes—
55C2×#107C2312yes—
56C2×#107C2311yescharacteristic
57C2×#107C2311yescharacteristic
58C2×#106—21no—
59C2×#106—21no—
60C2×#103—21no—
61C2×#103—21no—
62C2×#103—21no—
63C2×#103—21no—
64C2×#97D411yescharacteristic
65C2×#97D411yescharacteristic
66C2×#97D411yescharacteristic
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#400—42no—
71#400—42no—
72#400—42no—
73#398—42no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#389—22no—
79#389—22no—
80#384—21no—
81#382—21no—
82#376D411yescharacteristic
83#367—42no—
84#366—42no—
85C4⋊C16—42no—
86#358—42no—
87#358—42no—
88#358—42no—
89#358—42no—
90#340—22no—
91#340—22no—
92C22×Q8—21no—
93C22×Q8—21no—
94C2×Q16—21no—
95C2×Q16—21no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—21no—
101C2×Q16—21no—
102C2×Q16—42no—
103C2×Q16—42no—
104C2×Q16—42no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2411yescharacteristic
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111#125—42no—
112#125—42no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C16—22no—
128C2×C16—22no—
129C2×C16—22no—
130C2×C16—22no—
131C2×C16—22no—
132C2×C16—22no—
133#107—22no—
134#107—22no—
135#107—22no—
136#107—22no—
137#107—22no—
138#107—22no—
139#107—42no—
140#106—42no—
141#106—42no—
142#103—42no—
143#103—42no—
144#103—42no—
145#103—42no—
146#97—22no—
147#97—22no—
148#97—41no—
149C4×C8—21no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—21no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—42no—
157#95—42no—
158#95—42no—
159#95—42no—
160#95D812yes—
161#95D812yes—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4C2×D812yes—
181C22×C4C2×D812yes—
182C22×C4C2×D811yescharacteristic
183C22×C4C2×D811yescharacteristic
184C22×C4C22×D411yescharacteristic
185Q16—42no—
186Q16—42no—
187Q16—42no—
188Q16—42no—
189Q16—84no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8AGL(1,ℤ/8)12yes—
198C2×C8#13712yes—
199C2×C8H(5,𝔽2)11yescharacteristic
200C2×C8H(5,𝔽2)12yes—
201C2×C82−1+412yes—
202C2×C82−1+411yescharacteristic
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C4⋊C4—162no—
216C4⋊C4—162no—
217C4⋊C4—162no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C16—42no—
225C16—42no—
226C23C22×D811yescharacteristic
227Q8—81no—
228Q8—81no—
229Q8—81no—
230Q8—81no—
231Q8—81no—
232Q8—81no—
233Q8—81no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4#53511yescharacteristic
262C2×C4#53611yescharacteristic
263C2×C4#53711yescharacteristic
264C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
265C2×C4C2×#13711yescharacteristic
266C2×C4#57511yescharacteristic
267C8—22no—
268C8—22no—
269C8—41no—
270C8#56212yes—
271C8#56312yes—
272C22#320511yescharacteristic
273C22#322611yescharacteristic
274C22#324511yescharacteristic
275C22#341612yes—
276C22#341712yes—
277C4—42no—
278C4—42no—
279C4—41no—
280C4—41no—
281C4—162no—
282C4—161no—
283C4—161no—
284C4—162no—
285C4—161no—
286C4—161no—
287C4#282011yescharacteristic
288C4#321511yescharacteristic
289C4#323411yescharacteristic
290C4#325411yescharacteristic
291C2#3281811yescharacteristic
292C2#3378711yescharacteristic
293C2#3378911yescharacteristic
294C2#3380812yes—
295C2#3382412yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332612
#332492
#135312
#135302
#134832
#134772
#134752
#134532
#134092
#128802