Finite Group Databasebeta

← #326607

Subgroups — #326607

Order 512.

519 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 126 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#21736C211yesmaximal, characteristic, complemented
2#21265C211yesmaximal, characteristic
3#10688C211yesmaximal, characteristic, complemented
4#10661C211yesmaximal, characteristic
5#10548C211yesmaximal, characteristic, complemented
6#10036C211yesmaximal, characteristic
7#9579C212yesmaximal, complemented
8#9231C212yesmaximal
9#9050C212yesmaximal
10#8881C212yesmaximal, complemented
11#8800C211yesmaximal, characteristic, complemented
12#3162C2211yescharacteristic
13#3101C2212yes—
14#2961C2212yes—
15C2×#431—21no—
16C42×D4C2211yescharacteristic
17#1994C2212yes—
18#1955C2211yescharacteristic
19#1955C2211yescharacteristic
20#1946—21no—
21#1945—21no—
22#1940C2211yescharacteristic
23#1940C2211yescharacteristic
24#1924—21no—
25#1924—21no—
26#1849C2211yescharacteristic
27#1773—21no—
28#1772—21no—
29C4×C4⋊C8C2211yescharacteristic
30C4×C22⋊C8—21no—
31C4×#97C2211yescharacteristic
32#1570C2212yes—
33#1542C2212yes—
34#1503C2212yes—
35#1486C2212yes—
36#1481C2212yes—
37#1303C2212yes—
38#1301C2212yes—
39#1281C2212yes—
40#1276C2212yes—
41#1275C2211yescharacteristic
42#1275C2211yescharacteristic
43#533C2312yes—
44C2×C4×D4—21no—
45C2×C4×D4C2311yescharacteristic
46C22×C42—21no—
47#501—22no—
48#501—22no—
49#483—22no—
50#476—22no—
51#461—22no—
52#450—22no—
53#446—42no—
54#446—42no—
55C4×SD16—42no—
56#431—44no—
57#431C2312yes—
58#431C2312yes—
59C2×#108—21no—
60C2×#108—21no—
61#427—21no—
62#427—21no—
63#424—22no—
64#423C2312yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73#405—22no—
74C2×#97D411yescharacteristic
75C2×#97D411yescharacteristic
76C2×#97C2311yescharacteristic
77C2×#97C2311yescharacteristic
78C2×C4×C8—21no—
79C2×C4×C8D411yescharacteristic
80C2×C4×C8D411yescharacteristic
81C4×C4⋊C4—21no—
82C4×C4⋊C4C2311yescharacteristic
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86C43C2311yescharacteristic
87#339—22no—
88#339—22no—
89#336—21no—
90#336—21no—
91#336—21no—
92#334—42no—
93#333—22no—
94#332—22no—
95#328—22no—
96#328—22no—
97Q8⋊C8—42no—
98D4⋊C8—42no—
99C22×D4#4111yescharacteristic
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C2×SD16—42no—
104C2×M16—22no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C4⋊Q8#4112yes—
112C4⋊Q8#4112yes—
113#125—22no—
114C22⋊Q8—22no—
115C22⋊Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×D4—22no—
119C4×D4—22no—
120C4×D4—44no—
121#117—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4#4111yescharacteristic
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—42no—
133C2×C42—42no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2411yescharacteristic
137#108—42no—
138#108—42no—
139#107—21no—
140#107—21no—
141#107—21no—
142#107—21no—
143#106—22no—
144#106—22no—
145#106—42no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—44no—
153C4⋊C8—42no—
154C4⋊C8—42no—
155#103—42no—
156#102—42no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8—42no—
160C22⋊C8—42no—
161C22⋊C8—42no—
162C22⋊C8—42no—
163#97—22no—
164#97—22no—
165#97—22no—
166#97—41no—
167#97—42no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—41no—
171C4×C8—42no—
172C4×C8—42no—
173C4×C8SD1614yescomplemented
174#95—21no—
175#95—21no—
176#95#4111yescharacteristic
177#95#4111yescharacteristic
178C24—21no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×D4—22no—
182C2×D4—42no—
183C22×C4—21no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4C22×D411yescharacteristic
201C22×C4C2×#4111yescharacteristic
202C22×C4C2×#4111yescharacteristic
203SD16—84no—
204M16—42no—
205M16—42no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—41no—
216C2×C8—42no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8C2×SD1612yes—
225C2×C8C2×SD1612yes—
226C2×C8#13712yes—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—42no—
243C42—42no—
244C42—42no—
245C42—42no—
246C42—42no—
247C42—41no—
248C42—41no—
249C42C2×SD1612yes—
250C42C2×#4112yes—
251C42H(5,𝔽2)12yes—
252C23—22no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#52511yescharacteristic
260Q8—82no—
261Q8—82no—
262D4—24no—
263D4—42no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—48no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—84no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4#52511yescharacteristic
290C2×C4#53511yescharacteristic
291C2×C4#55112yes—
292C2×C4#55312yes—
293C2×C4C22×SD1611yescharacteristic
294C2×C4C2×#13711yescharacteristic
295C2×C4#57511yescharacteristic
296C2×C4#57511yescharacteristic
297C8—22no—
298C8—22no—
299C8—22no—
300C8—41no—
301C8—82no—
302C8—82no—
303C8—81no—
304C8—81no—
305C8—82no—
306C22—21no—
307C22—21no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22#259811yescharacteristic
316C22#310111yescharacteristic
317C22#311611yescharacteristic
318C4—24no—
319C4—22no—
320C4—22no—
321C4—42no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—82no—
326C4—82no—
327C4#310311yescharacteristic
328C4#311711yescharacteristic
329C4#320611yescharacteristic
330C4#321911yescharacteristic
331C2—21no—
332C2—21no—
333C2—82no—
334C2#2936811yescharacteristic
335C2#2938411yescharacteristic
336C2#2938611yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2598
prime radicalC22×D4#41
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21736C2semidirect
3. #10688C2semidirect
5. #10548C2semidirect
7. #9579C2semidirect
10. #8881C2semidirect
11. #8800C2semidirect
173. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#217362
#212652
#106882
#106612
#105482
#100362
#95792
#92312
#90502
#88812
#88002