Finite Group Databasebeta

← #134576

Subgroups — #134576

Order 512.

430 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#495C211yesmaximal, characteristic
2#12898C211yesmaximal, characteristic, complemented
3C2×#1360C212yesmaximal
4#8805C211yesmaximal, characteristic, complemented
5#7275C212yesmaximal, complemented
6C2×#495C2211yescharacteristic
7C2×C4×Q16C411yescharacteristic
8C2×C4×SD16—21no—
9C2×C4×D8C411yescharacteristic
10C4×#124—21no—
11C2×C4⋊C16—21no—
12C2×C4⋊C16—21no—
13C2×C4⋊C16C2212yes—
14#1970C411yescharacteristic
15#1969C411yescharacteristic
16#1968—21no—
17#1958—21no—
18#1846C2211yescharacteristic
19#1842—21no—
20C42×C8C2211yescharacteristic
21C2×#334—21no—
22C2×#334C2212yes—
23#1360—24no—
24C22×Q16C2×C411yescharacteristic
25C22×SD16—21no—
26C22×D8C2×C411yescharacteristic
27C2×#124—21no—
28C2×C4×Q8—21no—
29C2×C4×D4—21no—
30C22×C16—22no—
31#495—21no—
32#495—21no—
33#495D412yes—
34#495Q812yes—
35C4×Q16—42no—
36C4×SD16—42no—
37C4×SD16—42no—
38C4×D8—42no—
39C2×#107C2×C411yescharacteristic
40C2×#107C2×C411yescharacteristic
41C2×#106—21no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—41no—
44C2×C4⋊C8—41no—
45C2×#103—21no—
46C2×#103—21no—
47C2×#102—21no—
48C2×#102—21no—
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8C2311yescharacteristic
55#391—21no—
56#390—21no—
57#388—41no—
58#388—41no—
59#382—21no—
60C4×C22⋊C4—41no—
61C4×C22⋊C4—41no—
62C43D411yescharacteristic
63C4⋊C16—22no—
64C4⋊C16—22no—
65C4⋊C16—42no—
66C4⋊C16—42no—
67#334—22no—
68#334—22no—
69#334—42no—
70C22×Q8—21no—
71C22×D4—21no—
72C23×C4—41no—
73C2×Q16—21no—
74C2×Q16—41no—
75C2×Q16—41no—
76C2×Q16C4⋊C412yes—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—41no—
80C2×SD16—41no—
81C2×SD16—41no—
82C2×SD16—41no—
83C2×D8—21no—
84C2×D8—41no—
85C2×D8—41no—
86C2×D8C4⋊C412yes—
87C22×C8—42no—
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22×C411yescharacteristic
91C22×C8C2×D411yescharacteristic
92#124—22no—
93#124—22no—
94#124—41no—
95#124—41no—
96C4×Q8—42no—
97C4×Q8—42no—
98C4×D4—42no—
99C4×D4—42no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C42—41no—
109C2×C42C22⋊C411yescharacteristic
110C2×C42C22⋊C411yescharacteristic
111C2×C42D811yescharacteristic
112C2×C42D811yescharacteristic
113C2×C42SD1611yescharacteristic
114C2×C42SD1611yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C16—42no—
117C2×C16—42no—
118C2×C16—42no—
119#107—41no—
120#107—41no—
121#106—41no—
122#106—41no—
123C4⋊C8—42no—
124C4⋊C8—42no—
125C4⋊C8—82no—
126C4⋊C8—82no—
127#103—42no—
128#103—42no—
129#102—42no—
130#102—42no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—22no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—41no—
137C4×C8D812yes—
138C4×C8SD1612yes—
139C4×C8C2×D411yescharacteristic
140C4×C8C2×Q811yescharacteristic
141C4×C8#4111yescharacteristic
142C4×C8#4111yescharacteristic
143#95—41no—
144#95—41no—
145#95—41no—
146#95—41no—
147C24—41no—
148C2×Q8—22no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×D4—22no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4#10211yescharacteristic
167C22×C4#10211yescharacteristic
168C22×C4#10211yescharacteristic
169C22×C4#10211yescharacteristic
170C22×C4C2×C22⋊C411yescharacteristic
171C22×C4C2×D811yescharacteristic
172C22×C4C2×SD1611yescharacteristic
173Q16—42no—
174Q16—42no—
175SD16—42no—
176SD16—42no—
177SD16—42no—
178SD16—42no—
179D8—42no—
180D8—42no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—82no—
191C2×C8#10212yes—
192C2×C8C2×C4⋊C411yescharacteristic
193C2×C8#11711yescharacteristic
194C2×C8#12111yescharacteristic
195C2×C8C22⋊Q811yescharacteristic
196C2×C8C2×D811yescharacteristic
197C2×C8C2×SD1611yescharacteristic
198C2×C8AGL(1,ℤ/8)11yescharacteristic
199C2×C8#13711yescharacteristic
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—82no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C42—22no—
213C42—22no—
214C42—22no—
215C42—21no—
216C42—21no—
217C42—41no—
218C42—82no—
219C42#10012yes—
220C42#10212yes—
221C42#12111yescharacteristic
222C42C22⋊Q811yescharacteristic
223C42AGL(1,ℤ/8)11yescharacteristic
224C42#13711yescharacteristic
225C16—82no—
226C23—42no—
227C23—81no—
228C23—81no—
229C23—81no—
230C23—81no—
231C23—81no—
232C23—81no—
233C23C2×#10211yescharacteristic
234Q8—42no—
235Q8—42no—
236Q8—81no—
237Q8—81no—
238D4—42no—
239D4—42no—
240D4—81no—
241D4—81no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—164no—
263C2×C4#33112yes—
264C2×C4#33112yes—
265C2×C4#35912yes—
266C2×C4#36012yes—
267C2×C4#36012yes—
268C2×C4#38011yescharacteristic
269C2×C4C2×#10011yescharacteristic
270C2×C4#41311yescharacteristic
271C2×C4C2×#10211yescharacteristic
272C2×C4#41711yescharacteristic
273C2×C4#41711yescharacteristic
274C2×C4#45911yescharacteristic
275C2×C4#46111yescharacteristic
276C2×C4#47411yescharacteristic
277C2×C4#47611yescharacteristic
278C8—21no—
279C8—21no—
280C8—22no—
281C8—41no—
282C8—162no—
283C8#45911yescharacteristic
284C8#46111yescharacteristic
285C8#47411yescharacteristic
286C8#47611yescharacteristic
287C22—82no—
288C22—82no—
289C22—82no—
290C22—161no—
291C22—161no—
292C22—161no—
293C22—161no—
294C22C2×#33111yescharacteristic
295C22#154711yescharacteristic
296C22#180211yescharacteristic
297C22C2×#35911yescharacteristic
298C22#214311yescharacteristic
299C22C2×#36011yescharacteristic
300C22C2×#36011yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—82no—
309C4—161no—
310C4—161no—
311C4#136014yescomplemented
312C4#154711yescharacteristic
313C4#156011yescharacteristic
314C4#180011yescharacteristic
315C4#180211yescharacteristic
316C2—162no—
317C2#858511yescharacteristic
318C2C2×#136011yescharacteristic
319C2#1192511yescharacteristic
320C2#1197711yescharacteristic
321C2#1198011yescharacteristic
322C2#1281511yescharacteristic
323C2#1281611yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12898C2semidirect
4. #8805C2semidirect
5. #7275C2semidirect
311. C4#1360semidirect

Maximal subgroups

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

subgroupindex
C4×#4952
#128982
C2×#13602
#88052
#72752