Finite Group Databasebeta

← #138020

Subgroups — #138020

Order 512.

499 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2162C211yesmaximal, characteristic, complemented
2#33129C211yesmaximal, characteristic
3C2×#1347C212yesmaximal
4#10820C211yesmaximal, characteristic, complemented
5#7258C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2854—21no—
9#2162—24no—
10C2×C4⋊C16—21no—
11C2×C22⋊C16C2212yes—
12C2×C4×C16—21no—
13#1970C411yescharacteristic, complemented
14#1970C411yescharacteristic, complemented
15#1942C411yescharacteristic, complemented
16#1942C411yescharacteristic, complemented
17#1934—21no—
18#1934—21no—
19#1893—21no—
20#1808C2211yescharacteristic
21C2×#340—21no—
22C2×#340C2212yes—
23C4⋊#95—21no—
24#1347—24no—
25C22×D8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×#123—41no—
29C2×#121—21no—
30C22×C4⋊C4—22no—
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—
55C2×#95—41no—
56C2×#95—41no—
57C4⋊C16—24no—
58C22⋊C16—22no—
59C22⋊C16D414yes—
60C4×C16—24no—
61#340—24no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—22no—
66#340—22no—
67C22×D4—21no—
68C22×D4—41no—
69C23×C4—42no—
70C23×C4D811yescharacteristic
71C23×C4D811yescharacteristic
72C23×C4C2×D411yescharacteristic
73C2×D8—22no—
74C2×D8—41no—
75C2×D8—41no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D412yes—
86C22×C8#4112yes—
87#123—82no—
88#123—82no—
89#121—42no—
90#121—42no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42C22⋊C411yescharacteristic
111C2×C42C22⋊C411yescharacteristic
112C2×C42SD1611yescharacteristic
113C2×C42SD1611yescharacteristic
114C2×C16—24no—
115C2×C16—42no—
116C2×C16—42no—
117#107—22no—
118#107—41no—
119#107—41no—
120#102—42no—
121#102—42no—
122#102—41no—
123#102—41no—
124C4×C8—22no—
125C4×C8—22no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—21no—
129C4×C8—21no—
130#95—41no—
131#95—41no—
132#95—41no—
133#95—82no—
134#95—82no—
135C24—41no—
136C24C2×D811yescharacteristic
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—41no—
143C2×D4—82no—
144C2×D4—81no—
145C2×D4—81no—
146C2×D4—82no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—41no—
156C22×C4—44no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—82no—
161C22×C4—82no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4#10211yescharacteristic
167C22×C4#10211yescharacteristic
168C22×C4#10211yescharacteristic
169C22×C4#10211yescharacteristic
170C22×C4D1612yes—
171C22×C4SD3212yes—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4#12112yes—
174C22×C4C2×SD1611yescharacteristic
175C22×C4#13512yes—
176D8—82no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—44no—
189C2×C8#10412yes—
190C2×C8C4×D412yes—
191C2×C8#12011yescharacteristic
192C2×C8#12311yescharacteristic
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C22⋊C4—82no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—22no—
217C42—22no—
218C16—44no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—81no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23—81no—
230C23C2×#10211yescharacteristic
231C23#46612yes—
232C23C2×D1611yescharacteristic
233C23C2×SD3211yescharacteristic
234C23#50812yes—
235D4—81no—
236D4—81no—
237D4—81no—
238D4—81no—
239D4—162no—
240D4—162no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—44no—
259C2×C4—84no—
260C2×C4—84no—
261C2×C4—84no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—162no—
272C2×C4—162no—
273C2×C4—162no—
274C2×C4—162no—
275C2×C4#35712yes—
276C2×C4#35712yes—
277C2×C4#36012yes—
278C2×C4#38611yescharacteristic
279C2×C4#41011yescharacteristic
280C2×C4#41011yescharacteristic
281C2×C4#41611yescharacteristic
282C2×C4C2×#10411yescharacteristic
283C2×C4#42111yescharacteristic
284C2×C4#46512yes—
285C8—22no—
286C8—21no—
287C8—22no—
288C8—21no—
289C8—41no—
290C22—22no—
291C22—22no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—22no—
296C22—21no—
297C22—21no—
298C22—161no—
299C22—161no—
300C22—161no—
301C22—161no—
302C22—161no—
303C22—161no—
304C22—161no—
305C22#150211yescharacteristic
306C22#151711yescharacteristic
307C22#189311yescharacteristic
308C22C2×#35711yescharacteristic
309C22#213811yescharacteristic
310C22C2×#36011yescharacteristic
311C22#214511yescharacteristic
312C22#221512yes—
313C22#221712yes—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—41no—
319C4—164no—
320C4—161no—
321C4—161no—
322C4—161no—
323C4#150011yescharacteristic
324C4#152111yescharacteristic
325C4#187111yescharacteristic
326C4#189911yescharacteristic
327C2—22no—
328C2—22no—
329C2—321no—
330C2#911411yescharacteristic
331C2#1174111yescharacteristic
332C2#1174211yescharacteristic
333C2#1185511yescharacteristic
334C2#1185611yescharacteristic
335C2#1305911yescharacteristic
336C2#1306511yescharacteristic
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×#2162C2semidirect
4. #10820C2semidirect
5. #7258C2semidirect
13. #1970C4semidirect
14. #1970C4semidirect
15. #1942C4semidirect
16. #1942C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21622
#331292
C2×#13472
#108202
#72582