Finite Group Databasebeta

← #497989

Subgroups — #497989

Order 512.

654 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D16C211yesmaximal, characteristic
2#22807C211yesmaximal, characteristic, complemented
3C4⋊C4×D8C211yesmaximal, characteristic, complemented
4#13149C212yesmaximal, complemented
5#13112C212yesmaximal, complemented
6#13012C212yesmaximal
7#12888C211yesmaximal, characteristic, complemented
8#12811C211yesmaximal, characteristic, complemented
9#12811C211yesmaximal, characteristic, complemented
10#12662C211yesmaximal, characteristic, complemented
11#12637C212yesmaximal, complemented
12C22×D16C2211yescharacteristic
13C2×C4×D8C2212yes—
14C2×C4×D8C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic
16#2354—21no—
17D4×C4⋊C4—21no—
18C4×D16C418yescomplemented
19C2×#366C2211yescharacteristic
20C2×C4⋊C16C2211yescharacteristic
21C2×#357C2212yes—
22C2×#357C2212yes—
23C2×#357C2211yescharacteristic
24C2×#357C2211yescharacteristic
25C2×#346C2212yes—
26C2×C4×C16C2211yescharacteristic
27#1970C2212yes—
28#1942C2212yes—
29#1925—22no—
30#1862—22no—
31#1845C2211yescharacteristic
32#1845C2211yescharacteristic
33#1802—21no—
34#1802—21no—
35C4×#107C2212yes—
36C8×C4⋊C4C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2212yes—
42C22×D8C2311yescharacteristic
43C22×D8C2311yescharacteristic
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4—22no—
47C2×#117—41no—
48C22×C4⋊C4—41no—
49C2×D16—22no—
50C2×D16—22no—
51C2×D16C2×C414yes—
52C22×C16C2312yes—
53C22×C16C2311yescharacteristic
54C22×C16C2311yescharacteristic
55C4×D8—24no—
56C4×D8—22no—
57C4×D8—22no—
58C4×D8—42no—
59C4×D8C2×C414yes—
60C4×D8C2×C414yescomplemented
61C2×#107—21no—
62C2×#107—21no—
63C2×#107C2312yes—
64C2×#107C2312yes—
65C2×#107C2311yescharacteristic
66C2×#107C2311yescharacteristic
67C2×C4⋊C8D411yescharacteristic
68C2×#102—21no—
69C2×#102—21no—
70C2×#102—22no—
71C2×#102—22no—
72C2×C4×C8C2312yes—
73C2×C4×C8C2311yescharacteristic
74#390—22no—
75#389—22no—
76#388—42no—
77#385—42no—
78#384—21no—
79#384—21no—
80#380—41no—
81#380—41no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4D411yescharacteristic
84#366C2×C414yescomplemented
85C4⋊C16—22no—
86#357—24no—
87#357—42no—
88#357C2×C414yes—
89#357C2×C414yescomplemented
90#346—42no—
91C4×C16C2×C414yescomplemented
92#340—22no—
93#340—22no—
94#336D412yes—
95C22×D4—21no—
96C22×D4—21no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—42no—
100C2×D8—21no—
101C2×D8—21no—
102C2×D8—21no—
103C2×D8—21no—
104C2×D8—21no—
105C2×D8—21no—
106C2×D8—21no—
107C2×D8—21no—
108C2×D8C22×C412yes—
109C2×D8C22×C412yes—
110C2×D8#4112yes—
111C2×D8#4112yes—
112C22×C8C2×D412yes—
113C22×C8C2×D411yescharacteristic
114C22×C8C2411yescharacteristic
115C4×D4—24no—
116C4×D4—24no—
117C4×D4—44no—
118C4×D4—42no—
119C4×D4—42no—
120C4×D4—82no—
121#117—84no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—84no—
132C2×C4⋊C4D811yescharacteristic
133C2×C4⋊C4D811yescharacteristic
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C42—42no—
140C2×C42C2×D412yes—
141C2×C42C2×D411yescharacteristic
142D16—28no—
143C2×C16—21no—
144C2×C16—21no—
145C2×C16—21no—
146C2×C16—21no—
147C2×C16—22no—
148C2×C16—22no—
149C2×C16—22no—
150C2×C16C22×C412yes—
151C2×C16C22×C412yes—
152#107—21no—
153#107—21no—
154#107—22no—
155#107—22no—
156#107—22no—
157#107—22no—
158#107—81no—
159#107—81no—
160#107C22×C412yes—
161#107C22×C412yes—
162C4⋊C8—22no—
163#102—24no—
164#102—24no—
165#102—44no—
166#102—82no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8C22×C412yes—
170C4×C8#4112yes—
171#95—42no—
172#95—42no—
173#95D812yes—
174C24—41no—
175C24—41no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—84no—
195C22×C4—84no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—162no—
201C22×C4C2×D812yes—
202C22×C4C2×D812yes—
203C22×C4C2×D811yescharacteristic
204C22×C4C2×D811yescharacteristic
205C22×C4C22×D411yescharacteristic
206D8—22no—
207D8—22no—
208D8—22no—
209D8—22no—
210D8—22no—
211D8—41no—
212D8—41no—
213D8#11714yescomplemented
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8C4×D412yes—
220C2×C8C23×C411yescharacteristic
221C2×C8C2×#4111yescharacteristic
222C2×C8C2×#4111yescharacteristic
223C2×C8C2×#4111yescharacteristic
224C2×C8H(5,𝔽2)11yescharacteristic
225C2×C82−1+411yescharacteristic
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—84no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—161no—
240C4⋊C4—161no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—84no—
244C22⋊C4—84no—
245C22⋊C4—162no—
246C42—22no—
247C42—22no—
248C42—84no—
249C42C4×D412yes—
250C42#13512yes—
251C16—22no—
252C16—22no—
253C16—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23C22×D811yescharacteristic
269D4—24no—
270D4—42no—
271D4—42no—
272D4—42no—
273D4—42no—
274D4—42no—
275D4—81no—
276D4—81no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—88no—
292C2×C4—88no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—168no—
300C2×C4C4×D812yes—
301C2×C4C4×D812yes—
302C2×C4#50912yes—
303C2×C4#50912yes—
304C2×C4C2×C4×D411yescharacteristic
305C2×C4#54611yescharacteristic
306C2×C4#54811yescharacteristic
307C2×C4C2×#13511yescharacteristic
308C2×C4#57711yescharacteristic
309C2×C4#57711yescharacteristic
310C8—22no—
311C8—42no—
312C8C2×#11711yescharacteristic
313C8#52011yescharacteristic
314C8#55211yescharacteristic
315C8#55311yescharacteristic
316C22—44no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22—161no—
327C22—161no—
328C22—161no—
329C22—161no—
330C22C2×C4×D811yescharacteristic
331C22#329411yescharacteristic
332C22#335811yescharacteristic
333C22C2×#50911yescharacteristic
334C22C2×#50911yescharacteristic
335C22#341511yescharacteristic
336C22#341711yescharacteristic
337C4—22no—
338C4—42no—
339C4—42no—
340C4—82no—
341C4—82no—
342C4—162no—
343C4—162no—
344C4#217714yes—
345C4#235211yescharacteristic
346C4#295211yescharacteristic
347C4#330311yescharacteristic
348C4#336011yescharacteristic
349C2—84no—
350C2—162no—
351C2#2276611yescharacteristic
352C2C2×#217711yescharacteristic
353C2#3345911yescharacteristic
354C2#3389211yescharacteristic
355C2#3389711yescharacteristic
356C2#3393311yescharacteristic
357C2#3393411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22807C2semidirect
3. C4⋊C4×D8C2semidirect
4. #13149C2semidirect
5. #13112C2semidirect
7. #12888C2semidirect
8. #12811C2semidirect
9. #12811C2semidirect
10. #12662C2semidirect
11. #12637C2semidirect
18. C4×D16C4semidirect
60. C4×D8C2×C4semidirect
84. #366C2×C4semidirect
89. #357C2×C4semidirect
91. C4×C16C2×C4semidirect
213. D8#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D162
#228072
C4⋊C4×D82
#131492
#131122
#130122
#128882
#128112
#126622
#126372