Finite Group Databasebeta

← #143409

Subgroups — #143409

Order 512.

458 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#2241C211yesmaximal, characteristic
2C2×#2225C211yesmaximal, characteristic
3#28462C211yesmaximal, characteristic
4C2×#1362C211yesmaximal, characteristic
5C2×#1332C211yesmaximal, characteristic
6#9731C211yesmaximal, characteristic
7#7167C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#494—21no—
10C2×#494C2211yescharacteristic
11C2×#462—21no—
12C2×C4×Q16C2211yescharacteristic
13#2546—21no—
14#2241—22no—
15#2241—22no—
16#2225—22no—
17#2225—22no—
18C2×#367—21no—
19C2×C4⋊C16C2211yescharacteristic
20C2×#358—21no—
21C2×#358—22no—
22C2×C4×C16C2211yescharacteristic
23#2016—22no—
24#1981—21no—
25#1969—21no—
26C8×C4⋊C4C2211yescharacteristic
27C2×#335C2211yescharacteristic
28C2×#332—21no—
29#1362—22no—
30#1332—22no—
31C22×Q16—22no—
32C22×Q16—21no—
33C22×Q16D411yescharacteristic
34C22×Q16D412yes—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4×Q8—21no—
39C22×C16—21no—
40C22×C16—21no—
41#500—21no—
42#500—21no—
43#500—21no—
44#500D412yes—
45#494—21no—
46#494—21no—
47#494—21no—
48#494—21no—
49#494—42no—
50#494—42no—
51#462—42no—
52#462—42no—
53C4×Q16—21no—
54C4×Q16—21no—
55C4×Q16—21no—
56C4×Q16—21no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107D411yescharacteristic
60C2×C4⋊C8—21no—
61C2×C4⋊C8D411yescharacteristic
62C2×#103—22no—
63C2×#103—21no—
64C2×#103—42no—
65C2×C4×C8—21no—
66C2×C4×C8C2311yescharacteristic
67#398—42no—
68#391—22no—
69#389—42no—
70C4×C4⋊C4D411yescharacteristic
71#367—24no—
72C4⋊C16—21no—
73C4⋊C16D412yes—
74#358—24no—
75#358—28no—
76C4×C16—21no—
77C4×C16D412yes—
78#336—21no—
79#335—21no—
80#335—21no—
81#332—42no—
82C22×Q8—21no—
83C22×Q8—22no—
84C22×Q8—41no—
85C2×Q16—21no—
86C2×Q16—22no—
87C2×Q16—21no—
88C2×Q16—22no—
89C2×Q16—21no—
90C2×Q16—21no—
91C2×Q16—21no—
92C2×Q16—42no—
93C2×Q16—42no—
94C2×Q16—44no—
95C2×Q16—44no—
96C2×Q16—41no—
97C2×Q16—41no—
98C2×Q16—41no—
99C2×Q16—41no—
100C2×Q16—41no—
101C2×Q16—41no—
102C2×Q16—82no—
103C2×Q16SD1612yes—
104C22×C8—21no—
105C22×C8—41no—
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C4⋊Q8—22no—
109C4⋊Q8—44no—
110C4⋊Q8—41no—
111C4⋊Q8—41no—
112C4⋊Q8—41no—
113C4⋊Q8—41no—
114C4⋊Q8—41no—
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117C4⋊Q8—82no—
118C4×Q8—41no—
119C4×Q8—41no—
120C4×Q8—41no—
121C4×Q8—41no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4D812yes—
130C2×C42—21no—
131C2×C42—41no—
132C2×C42C2×D411yescharacteristic
133C2×C16—22no—
134C2×C16—22no—
135C2×C16—41no—
136C2×C16—41no—
137C2×C16—41no—
138C2×C16—41no—
139#107—22no—
140#107—22no—
141#107—21no—
142#107—21no—
143#107—41no—
144#107—41no—
145C4⋊C8—21no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8D812yes—
149#103—44no—
150#103—41no—
151#103—41no—
152#103—84no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8C2×D411yescharacteristic
156C4×C8C2×D411yescharacteristic
157C4×C8C2×D411yescharacteristic
158C4×C8#4111yescharacteristic
159#95—21no—
160#95—44no—
161C2×Q8—22no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—42no—
165C2×Q8—41no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—41no—
169C2×Q8—42no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—81no—
175C2×Q8—82no—
176C2×Q8—82no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—81no—
182C22×C4#12011yescharacteristic
183C22×C4C2×D811yescharacteristic
184C22×C4AGL(1,ℤ/8)11yescharacteristic
185Q16—22no—
186Q16—22no—
187Q16—44no—
188Q16—41no—
189Q16—41no—
190Q16—82no—
191Q16—88no—
192Q16—81no—
193Q16—81no—
194Q16—81no—
195Q16—81no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—81no—
202C2×C8—82no—
203C2×C8#12011yescharacteristic
204C2×C8#12011yescharacteristic
205C2×C8#12111yescharacteristic
206C2×C8#12111yescharacteristic
207C2×C8C2×SD1611yescharacteristic
208C2×C8#13511yescharacteristic
209C2×C8AGL(1,ℤ/8)11yescharacteristic
210C2×C8#13711yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—88no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4SD3214yes—
226C42—41no—
227C42—41no—
228C42—81no—
229C42—81no—
230C42#12111yescharacteristic
231C42#12711yescharacteristic
232C42C2×D811yescharacteristic
233C42AGL(1,ℤ/8)11yescharacteristic
234C16—42no—
235C16—42no—
236C23#44711yescharacteristic
237Q8—42no—
238Q8—42no—
239Q8—81no—
240Q8—81no—
241Q8—82no—
242Q8—82no—
243Q8—84no—
244Q8—82no—
245Q8—82no—
246Q8—161no—
247Q8—161no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—162no—
262C2×C4#39211yescharacteristic
263C2×C4#44711yescharacteristic
264C2×C4#45011yescharacteristic
265C2×C4#45211yescharacteristic
266C2×C4#45511yescharacteristic
267C2×C4#45511yescharacteristic
268C2×C4#45911yescharacteristic
269C2×C4#45911yescharacteristic
270C2×C4#49311yescharacteristic
271C2×C4#49611yescharacteristic
272C2×C4C2×SD3212yes—
273C2×C4#51012yes—
274C8—21no—
275C8—21no—
276C8—42no—
277C8—161no—
278C8#44811yescharacteristic
279C8#44911yescharacteristic
280C8#46511yescharacteristic
281C8#46811yescharacteristic
282C22#163911yescharacteristic
283C22#164111yescharacteristic
284C22#201111yescharacteristic
285C22#218512yes—
286C22#219512yes—
287C4—21no—
288C4—21no—
289C4—44no—
290C4—82no—
291C4—162no—
292C4—161no—
293C4—162no—
294C4#163911yescharacteristic
295C4#164111yescharacteristic
296C4#200011yescharacteristic
297C4#200411yescharacteristic
298C4#220812yes—
299C4#224811yescharacteristic
300C4#225111yescharacteristic
301C2#1007011yescharacteristic
302C2#1237012yes—
303C2#1238212yes—
304C2#1332711yescharacteristic
305C2#1333911yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#22412
C2×#22252
#284622
C2×#13622
C2×#13322
#97312
#71672