Finite Group Databasebeta

← #508512

Subgroups — #508512

Order 512.

527 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 136 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#33135C211yesmaximal, characteristic
2C2×#1849C211yesmaximal, characteristic, complemented
3#13541C212yesmaximal, complemented
4#13535C212yesmaximal, complemented
5#13510C212yesmaximal, complemented
6#13487C212yesmaximal, complemented
7#13004C212yesmaximal
8#12928C212yesmaximal, complemented
9#12846C211yesmaximal, characteristic, complemented
10C2×#466C2211yescharacteristic
11C22×#108—21no—
12C22×#108C2212yes—
13C2×#423—21no—
14C22×#97C2211yescharacteristic
15#2854—21no—
16C2×#365C2214yes—
17C2×#361C2214yes—
18#2112C2212yes—
19#1972C2211yescharacteristic
20#1972C2211yescharacteristic
21#1944C2211yescharacteristic
22#1944C2211yescharacteristic
23#1936—21no—
24#1936—21no—
25#1893—21no—
26#1849—21no—
27#1849—21no—
28#1849C2212yes—
29#1849C2212yes—
30#1807C2211yescharacteristic
31#1387C2214yes—
32#1383C2214yes—
33#1377C2212yes—
34#1377C2212yes—
35C22×D8C2311yescharacteristic
36C22×M16—22no—
37C23×C8D411yescharacteristic
38C23×C8C2311yescharacteristic
39C2×#123—41no—
40C2×#121—21no—
41C22×C42D411yescharacteristic
42C2×M32C2314yes—
43#466—22no—
44#466—22no—
45C2×#108—22no—
46C2×#108—22no—
47C2×#108—22no—
48C2×#108—44no—
49C2×#108—44no—
50C2×#108C2314yes—
51C2×#107C2311yescharacteristic
52C2×#106C2311yescharacteristic
53C2×#106C2311yescharacteristic
54#423—22no—
55#423—22no—
56#423—42no—
57C2×C4⋊C8—42no—
58C2×#102—21no—
59C2×#102—21no—
60C2×#102—21no—
61C2×#97—22no—
62C2×#97—22no—
63C2×#97D411yescharacteristic
64C2×#97D411yescharacteristic
65C2×#97C2311yescharacteristic
66C2×#97C2311yescharacteristic
67#390—21no—
68#390—21no—
69#389—21no—
70#389—21no—
71#388—41no—
72#388—41no—
73#386—41no—
74#380—21no—
75#365—44no—
76#361—44no—
77C22⋊C16—22no—
78C22⋊C16—22no—
79#340—21no—
80#340—21no—
81#340—21no—
82C22×D4—21no—
83C22×D4—41no—
84C23×C4SD1611yescharacteristic
85C23×C4SD1611yescharacteristic
86C23×C4C2×D411yescharacteristic
87C2×D8—22no—
88C2×D8—22no—
89C2×D8—22no—
90C2×M16—24no—
91C2×M16—42no—
92C2×M16—44no—
93C2×M16—44no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—42no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8#4112yes—
105C22×C8#4112yes—
106C22×C8C2411yescharacteristic
107#123—84no—
108#121—41no—
109#121—41no—
110#121—41no—
111#121—41no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42SD1611yescharacteristic
128C2×C42SD1611yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131M32—24no—
132M32—24no—
133C2×C16—24no—
134#108—24no—
135#108—24no—
136#108—44no—
137#108—44no—
138#108—44no—
139#107—22no—
140#106—22no—
141#106—22no—
142C4⋊C8—44no—
143C4⋊C8—44no—
144#102—42no—
145#102—42no—
146#102—42no—
147#97—22no—
148#97—22no—
149#97—22no—
150#97—22no—
151#95—41no—
152#95—41no—
153#95—41no—
154C24—41no—
155C24C2×SD1611yescharacteristic
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—82no—
163C2×D4—82no—
164C2×D4—81no—
165C2×D4—81no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—81no—
179C22×C4#12311yescharacteristic
180C22×C4#12311yescharacteristic
181C22×C4C2×SD1611yescharacteristic
182C22×C4C2×SD1611yescharacteristic
183C22×C4C2×SD1611yescharacteristic
184C22×C4C2×SD1611yescharacteristic
185C22×C4C2×SD1611yescharacteristic
186C22×C4AGL(1,ℤ/8)11yescharacteristic
187C22×C4AGL(1,ℤ/8)11yescharacteristic
188C22×C4C22×D411yescharacteristic
189D8—44no—
190M16—44no—
191M16—44no—
192M16—84no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—44no—
204C2×C8—84no—
205C2×C8#12312yes—
206C2×C8AGL(1,ℤ/8)12yes—
207C2×C8#13712yes—
208C2×C8C2×#4112yes—
209C2×C8H(5,𝔽2)12yes—
210C2×C82−1+412yes—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C22⋊C4—82no—
223C22⋊C4—82no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C16—44no—
235C23—21no—
236C23—21no—
237C23—22no—
238C23—41no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23#48111yescharacteristic
247C23#48111yescharacteristic
248C23C22×SD1611yescharacteristic
249D4—81no—
250D4—81no—
251D4—81no—
252D4—81no—
253D4—162no—
254D4—162no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4#48111yescharacteristic
287C2×C4#48111yescharacteristic
288C2×C4C2×#12311yescharacteristic
289C2×C4#53511yescharacteristic
290C2×C4#53611yescharacteristic
291C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
292C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
293C2×C4C2×#13711yescharacteristic
294C8—22no—
295C8—22no—
296C8—22no—
297C8—84no—
298C8#53914yes—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22—161no—
309C22—161no—
310C22—161no—
311C22—161no—
312C22—161no—
313C22—161no—
314C22—161no—
315C22#223812yes—
316C22C2×#48111yescharacteristic
317C22#320611yescharacteristic
318C22#322711yescharacteristic
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—161no—
326C4—161no—
327C4—161no—
328C4#262311yescharacteristic
329C4#309211yescharacteristic
330C4#321711yescharacteristic
331C4#323811yescharacteristic
332C2—21no—
333C2—21no—
334C2—22no—
335C2—41no—
336C2—321no—
337C2#2990711yescharacteristic
338C2C2×#223811yescharacteristic
339C2#3365611yescharacteristic
subgroupG/H
center, socleC22C2×#481
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1849C2semidirect
3. #13541C2semidirect
4. #13535C2semidirect
5. #13510C2semidirect
6. #13487C2semidirect
8. #12928C2semidirect
9. #12846C2semidirect

Maximal subgroups

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

subgroupindex
#331352
C2×#18492
#135412
#135352
#135102
#134872
#130042
#129282
#128462