Finite Group Databasebeta

← #141557

Subgroups — #141557

Order 512.

360 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 72 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#27658C211yesmaximal, characteristic
2#11790C211yesmaximal, characteristic
3#11780C211yesmaximal, characteristic
4#11579C211yesmaximal, characteristic
5#11562C211yesmaximal, characteristic
6#8611C211yesmaximal, characteristic
7C8×#100C211yesmaximal, characteristic
8C2×C8×D4C2211yescharacteristic
9C22×#106C411yescharacteristic
10#2487C2211yescharacteristic
11#2486C2211yescharacteristic
12#2088—21no—
13#2077—21no—
14#1941—21no—
15#1824C411yescharacteristic
16#1823C411yescharacteristic
17#1810—21no—
18#1807—21no—
19#1807C411yescharacteristic
20C4×#100C2211yescharacteristic
21#1465C2211yescharacteristic
22#1465C2211yescharacteristic
23#1460C2211yescharacteristic
24C8×M16—21no—
25#1353—22no—
26#1346—22no—
27#1326—22no—
28#1317—22no—
29#1293—21no—
30#1283—21no—
31C23×C8C2×C411yescharacteristic
32C23×C8D411yescharacteristic
33C2×#123—21no—
34C2×C4×D4C2311yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C8×D4—41no—
38C8×D4—41no—
39#431—21no—
40C2×#108—41no—
41C2×#106—21no—
42C2×#106—21no—
43C2×#106—21no—
44C2×#106—41no—
45C2×#106—41no—
46C2×C4⋊C8D411yescharacteristic
47C2×#100—21no—
48#407—21no—
49#407—21no—
50C2×C22⋊C8C2×C411yescharacteristic
51C2×C22⋊C8D411yescharacteristic
52C4×M16—21no—
53C2×C4×C8D411yescharacteristic
54#400—21no—
55#399—21no—
56#398—21no—
57#397—21no—
58#389—21no—
59#389—21no—
60#386—21no—
61#385—21no—
62#381—21no—
63#380—21no—
64#380—21no—
65#380C2×C411yescharacteristic
66#380C2×C411yescharacteristic
67#343D412yes—
68#341—41no—
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#340—21no—
74#324—21no—
75#324—21no—
76#323—21no—
77#323—21no—
78#323—21no—
79#323D412yes—
80#322—41no—
81C82—41no—
82C22×D4#4111yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C2×M16—22no—
88C2×M16—41no—
89C2×M16—41no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—42no—
95C22×C8—42no—
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98C22×C8C22⋊C411yescharacteristic
99C22×C8C22⋊C411yescharacteristic
100#123—82no—
101C4×D4—41no—
102C4×D4—41no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4C2×D411yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4C2×D411yescharacteristic
121C2×C22⋊C4#4111yescharacteristic
122C2×C42C2×D411yescharacteristic
123#108—82no—
124#106—42no—
125#106—42no—
126#106—41no—
127#106—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130#100—24no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138#97—41no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—41no—
142C4×C8—41no—
143#95—21no—
144#95—21no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—41no—
149C24C4×D411yescharacteristic
150C24#12411yescharacteristic
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—41no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—161no—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C4×D411yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12711yescharacteristic
176M16—24no—
177M16—42no—
178M16—81no—
179M16—81no—
180M16—81no—
181M16—81no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8#9911yescharacteristic
198C2×C8#9911yescharacteristic
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—81no—
209C4⋊C4—81no—
210C4⋊C4—82no—
211C4⋊C4—161no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—82no—
216C22⋊C4—81no—
217C22⋊C4—161no—
218C42—21no—
219C42—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23#39011yescharacteristic
230C23#44312yes—
231D4—44no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—162no—
261C2×C4—162no—
262C2×C4C2×#9911yescharacteristic
263C2×C4#41011yescharacteristic
264C2×C4#47112yes—
265C8—21no—
266C8—44no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—82no—
271C8—161no—
272C8—161no—
273C8#35212yes—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22#196411yescharacteristic
287C22#197411yescharacteristic
288C22#197711yescharacteristic
289C4—22no—
290C4—21no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—161no—
295C4—161no—
296C4—161no—
297C4—161no—
298C4C2×#35211yescharacteristic
299C4#212311yescharacteristic
300C2—21no—
301C2—21no—
302C2—42no—
303C2—81no—
304C2#981311yescharacteristic
305C2#1321711yescharacteristic
306C2#1323211yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4#41
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#276582
#117902
#117802
#115792
#115622
#86112
C8×#1002