Finite Group Databasebeta

← #145040

Subgroups — #145040

Order 512.

408 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 98 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#28462C211yesmaximal, characteristic
2#13203C211yesmaximal, characteristic
3#12812C211yesmaximal, characteristic
4C2×#1362C211yesmaximal, characteristic
5C2×#1332C211yesmaximal, characteristic
6#9615C211yesmaximal, characteristic
7#7167C211yesmaximal, characteristic
8C2×#494—21no—
9C2×#494C2211yescharacteristic
10C2×#462—21no—
11C2×C4×Q16C2211yescharacteristic
12#2546—21no—
13C2×C4⋊C16C2211yescharacteristic
14C2×#358—21no—
15C2×#358—22no—
16C2×C4×C16C2211yescharacteristic
17#2016—22no—
18#1969—21no—
19#1948—21no—
20#1845C2211yescharacteristic
21C8×C4⋊C4C2211yescharacteristic
22C2×#335C2211yescharacteristic
23#1380—21no—
24#1362—22no—
25#1332—22no—
26#1296—21no—
27C22×Q16—22no—
28C22×Q16—21no—
29C22×Q16D411yescharacteristic
30C22×Q16D412yes—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—21no—
33C2×C4×Q8—21no—
34C22×C16—21no—
35C22×C16—21no—
36#494—21no—
37#494—21no—
38#494—21no—
39#494—21no—
40#494—42no—
41#494—42no—
42#462—42no—
43#462—42no—
44C4×Q16—21no—
45C4×Q16—21no—
46C4×Q16—21no—
47C4×Q16—21no—
48C2×#107—21no—
49C2×#107—21no—
50C2×#107D411yescharacteristic
51C2×#106—41no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8D411yescharacteristic
54C2×#103—22no—
55C2×#103—21no—
56C2×C4×C8—21no—
57C2×C4×C8C2311yescharacteristic
58#398—42no—
59#391—22no—
60#389—41no—
61#389—41no—
62#384—21no—
63C4×C4⋊C4D411yescharacteristic
64C4⋊C16—21no—
65C4⋊C16—21no—
66#358—44no—
67#358—42no—
68C4×C16—21no—
69C4×C16—21no—
70#340—41no—
71#336—21no—
72#335—21no—
73#335D412yes—
74C22×Q8—21no—
75C22×Q8—22no—
76C2×Q16—21no—
77C2×Q16—22no—
78C2×Q16—21no—
79C2×Q16—22no—
80C2×Q16—21no—
81C2×Q16—21no—
82C2×Q16—21no—
83C2×Q16—42no—
84C2×Q16—42no—
85C2×Q16—44no—
86C2×Q16—44no—
87C2×Q16—41no—
88C2×Q16—41no—
89C2×Q16—41no—
90C2×Q16—41no—
91C2×Q16—41no—
92C2×Q16—41no—
93C2×Q16—82no—
94C2×Q16SD1612yes—
95C22×C8—21no—
96C22×C8—41no—
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C4⋊Q8—44no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104C4⋊Q8—82no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4D811yescharacteristic
119C2×C4⋊C4D811yescharacteristic
120C2×C42—21no—
121C2×C42—41no—
122C2×C42C2×D411yescharacteristic
123C2×C16—22no—
124C2×C16—41no—
125C2×C16—41no—
126C2×C16—41no—
127C2×C16—41no—
128C2×C16—41no—
129#107—21no—
130#107—21no—
131#107—42no—
132#107—41no—
133#107—41no—
134#106—82no—
135C4⋊C8—22no—
136C4⋊C8—21no—
137C4⋊C8—41no—
138C4⋊C8SD1612yes—
139#103—44no—
140#103—41no—
141#103—41no—
142C4×C8—41no—
143C4×C8—41no—
144C4×C8C2×D411yescharacteristic
145C4×C8#4111yescharacteristic
146C4×C8#4111yescharacteristic
147C4×C8#4111yescharacteristic
148#95—21no—
149#95—42no—
150#95—42no—
151C2×Q8—22no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—42no—
155C2×Q8—41no—
156C2×Q8—42no—
157C2×Q8—41no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—41no—
161C2×Q8—42no—
162C2×Q8—42no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—81no—
169C22×C4#12011yescharacteristic
170C22×C4C2×D811yescharacteristic
171C22×C4AGL(1,ℤ/8)11yescharacteristic
172Q16—22no—
173Q16—22no—
174Q16—44no—
175Q16—41no—
176Q16—41no—
177Q16—82no—
178Q16—88no—
179Q16—81no—
180Q16—81no—
181Q16—81no—
182Q16—81no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—22no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—81no—
190C2×C8—81no—
191C2×C8#12111yescharacteristic
192C2×C8#12111yescharacteristic
193C2×C8#12311yescharacteristic
194C2×C8#12311yescharacteristic
195C2×C8C2×SD1611yescharacteristic
196C2×C8#13511yescharacteristic
197C2×C8AGL(1,ℤ/8)11yescharacteristic
198C2×C8AGL(1,ℤ/8)11yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—161no—
216C4⋊C4SD3212yes—
217C4⋊C4Q3212yes—
218C42—41no—
219C42—41no—
220C42—81no—
221C42—81no—
222C42#12111yescharacteristic
223C42#12411yescharacteristic
224C42C2×SD1611yescharacteristic
225C42AGL(1,ℤ/8)11yescharacteristic
226C16—42no—
227C16—81no—
228C23#44711yescharacteristic
229Q8—42no—
230Q8—42no—
231Q8—81no—
232Q8—81no—
233Q8—82no—
234Q8—82no—
235Q8—82no—
236Q8—82no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—161no—
255C2×C4—161no—
256C2×C4—161no—
257C2×C4#39411yescharacteristic
258C2×C4#44911yescharacteristic
259C2×C4#45011yescharacteristic
260C2×C4#45011yescharacteristic
261C2×C4#45311yescharacteristic
262C2×C4#45511yescharacteristic
263C2×C4#45911yescharacteristic
264C2×C4#46011yescharacteristic
265C2×C4#48611yescharacteristic
266C2×C4#49011yescharacteristic
267C2×C4C2×SD3211yescharacteristic
268C2×C4C2×Q3211yescharacteristic
269C2×C4#51011yescharacteristic
270C2×C4#51011yescharacteristic
271C8—21no—
272C8—21no—
273C8—42no—
274C8—82no—
275C8#46511yescharacteristic
276C8#46911yescharacteristic
277C8#48111yescharacteristic
278C8#48211yescharacteristic
279C22#162111yescharacteristic
280C22#163911yescharacteristic
281C22#204111yescharacteristic
282C22#218511yescharacteristic
283C22#218811yescharacteristic
284C22#219311yescharacteristic
285C22#219511yescharacteristic
286C4—21no—
287C4—21no—
288C4—42no—
289C4—42no—
290C4—82no—
291C4—161no—
292C4—162no—
293C4—321no—
294C4#162911yescharacteristic
295C4#164211yescharacteristic
296C4#203411yescharacteristic
297C4#203711yescharacteristic
298C4#220811yescharacteristic
299C4#220911yescharacteristic
300C4#224111yescharacteristic
301C4#224511yescharacteristic
302C2#1040911yescharacteristic
303C2#1233011yescharacteristic
304C2#1233411yescharacteristic
305C2#1237011yescharacteristic
306C2#1237411yescharacteristic
307C2#1340911yescharacteristic
308C2#1341211yescharacteristic
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
#284622
#132032
#128122
C2×#13622
C2×#13322
#96152
#71672