Finite Group Databasebeta

← #138027

Subgroups — #138027

Order 512.

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

normal subgroups only

This group has 445 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
1C2×#2163C211yesmaximal, characteristic, complemented
2#33129C211yesmaximal, characteristic
3C2×#1349C212yesmaximal
4#10822C211yesmaximal, characteristic, complemented
5#7259C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2854—21no—
9#2163—24no—
10C2×C4⋊C16—21no—
11C2×C22⋊C16C2212yes—
12C2×#346—21no—
13#1970C411yescharacteristic
14#1970C411yescharacteristic
15#1942C411yescharacteristic
16#1942C411yescharacteristic
17#1934—21no—
18#1934—21no—
19#1893—21no—
20#1808C2211yescharacteristic
21C2×#341—21no—
22C2×#341C2212yes—
23#1728—21no—
24#1349—24no—
25C22×D8C2×C411yescharacteristic
26C22×M16—22no—
27C23×C8D411yescharacteristic
28C23×C8C2311yescharacteristic
29C2×#123—41no—
30C2×#121—21no—
31C22×C42D411yescharacteristic
32C22×C16—22no—
33#466—22no—
34#466—22no—
35C2×#107C2×C411yescharacteristic
36C2×#107C2×C411yescharacteristic
37C2×#107C2×C411yescharacteristic
38C2×#102—21no—
39C2×#102—21no—
40C2×#102—21no—
41C2×C4×C8—22no—
42C2×C4×C8—22no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#390—21no—
48#390—21no—
49#389—21no—
50#389—21no—
51#388—41no—
52#388—41no—
53#386—41no—
54#380—21no—
55C4⋊C16—24no—
56C22⋊C16—22no—
57C22⋊C16D414yes—
58#346—24no—
59#341—24no—
60#341—22no—
61#341—22no—
62#340—21no—
63#340—21no—
64#340—21no—
65#336—41no—
66#336—41no—
67C22×D4—21no—
68C22×D4—41no—
69C23×C4SD1611yescharacteristic
70C23×C4SD1611yescharacteristic
71C23×C4C2×D411yescharacteristic
72C2×D8—22no—
73C2×D8—41no—
74C2×D8—41no—
75C2×M16—24no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—42no—
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22×C411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D412yes—
92C22×C8#4112yes—
93#123—84no—
94#121—42no—
95#121—42no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42C22⋊C411yescharacteristic
110C2×C42C22⋊C411yescharacteristic
111C2×C42D811yescharacteristic
112C2×C42D811yescharacteristic
113C2×C16—24no—
114C2×C16—42no—
115C2×C16—42no—
116#107—22no—
117#107—41no—
118#107—41no—
119#102—42no—
120#102—42no—
121#102—41no—
122#102—41no—
123C4×C8—22no—
124C4×C8—22no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—21no—
129#95—41no—
130#95—41no—
131#95—41no—
132C24—41no—
133C24C2×SD1611yescharacteristic
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—82no—
141C2×D4—82no—
142C2×D4—82no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4#10211yescharacteristic
157C22×C4#10211yescharacteristic
158C22×C4#10211yescharacteristic
159C22×C4#10211yescharacteristic
160C22×C4C2×C22⋊C411yescharacteristic
161C22×C4#12112yes—
162C22×C4C2×D811yescharacteristic
163C22×C4#13512yes—
164D8—82no—
165M16—44no—
166M16—44no—
167M16—82no—
168M16—82no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—82no—
183C2×C8—82no—
184C2×C8#10412yes—
185C2×C8C4×D412yes—
186C2×C8#12011yescharacteristic
187C2×C8#12311yescharacteristic
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C22⋊C4—82no—
197C22⋊C4—82no—
198C22⋊C4—82no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—22no—
207C42—22no—
208C16—44no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—81no—
216C23—81no—
217C23—81no—
218C23—81no—
219C23—81no—
220C23—81no—
221C23—81no—
222C23C2×#10211yescharacteristic
223C23#46512yes—
224D4—81no—
225D4—81no—
226D4—81no—
227D4—81no—
228D4—164no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—44no—
247C2×C4—44no—
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#36112yes—
258C2×C4#38611yescharacteristic
259C2×C4#41011yescharacteristic
260C2×C4#41011yescharacteristic
261C2×C4#41611yescharacteristic
262C2×C4C2×#10411yescharacteristic
263C2×C4#42111yescharacteristic
264C2×C4#46612yes—
265C8—22no—
266C8—21no—
267C8—22no—
268C8—21no—
269C8—41no—
270C8—84no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—161no—
279C22—161no—
280C22—161no—
281C22—161no—
282C22—161no—
283C22—161no—
284C22—161no—
285C22#150211yescharacteristic
286C22#151711yescharacteristic
287C22#189311yescharacteristic
288C22#214311yescharacteristic
289C22#214311yescharacteristic
290C22C2×#36111yescharacteristic
291C22#214811yescharacteristic
292C4—22no—
293C4—22no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—161no—
298C4—161no—
299C4—161no—
300C4#150011yescharacteristic
301C4#152111yescharacteristic
302C4#187111yescharacteristic
303C4#189911yescharacteristic
304C2—42no—
305C2—321no—
306C2#911411yescharacteristic
307C2#1175011yescharacteristic
308C2#1175111yescharacteristic
309C2#1186411yescharacteristic
310C2#1186611yescharacteristic
311C2#1306411yescharacteristic
312C2#1306711yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2163C2semidirect
4. #10822C2semidirect
5. #7259C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21632
#331292
C2×#13492
#108222
#72592