Finite Group Databasebeta

← #501964

Subgroups — #501964

Order 512.

656 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2233C212yesmaximal, complemented
2#28227C211yesmaximal, characteristic
3D4×#107C211yesmaximal, characteristic, complemented
4#13494C212yesmaximal, complemented
5#13381C212yesmaximal, complemented
6#13189C211yesmaximal, characteristic, complemented
7#13177C211yesmaximal, characteristic, complemented
8#12920C212yesmaximal, complemented
9#12906C211yesmaximal, characteristic, complemented
10#12811C211yesmaximal, characteristic, complemented
11C4×#357C211yesmaximal, characteristic, complemented
12C2×#466C2212yes—
13C2×C4×D8C2211yescharacteristic
14C2×C8×D4C2211yescharacteristic
15C22×#107C2212yes—
16#2537—21no—
17D4×C4⋊C4—21no—
18#2233—28no—
19C2×#367C2212yes—
20C2×#367C2212yes—
21C2×#367C2212yes—
22C2×C4⋊C16C2211yescharacteristic
23C2×#357C2212yes—
24C2×#357C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×C22⋊C16C2212yes—
27C2×C22⋊C16C2212yes—
28C2×C4×C16C2211yescharacteristic
29#2033—22no—
30#1947—21no—
31#1852—22no—
32#1845C2211yescharacteristic
33#1845C2211yescharacteristic
34#1808C2212yes—
35#1808C2212yes—
36#1802—21no—
37C4×#107C2211yescharacteristic
38#1380C2212yes—
39#1380C2212yes—
40#1380C2211yescharacteristic
41#1380C2211yescharacteristic
42C22×D8C2311yescharacteristic
43C23×C8C2312yes—
44C2×#121—22no—
45C2×#121—41no—
46C2×C4×D4—21no—
47C2×C4×D4—41no—
48C2×C4×D4D411yescharacteristic
49C22×C4⋊C4—22no—
50C22×C16C2312yes—
51C22×C16C2311yescharacteristic
52C22×C16C2311yescharacteristic
53#466—24no—
54#466—42no—
55C4×D8—22no—
56C4×D8—22no—
57C8×D4—22no—
58C8×D4—22no—
59C2×#107—24no—
60C2×#107—22no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107—42no—
64C2×#107C2311yescharacteristic
65C2×#107C2311yescharacteristic
66C2×#107C2312yes—
67C2×#107C2312yes—
68C2×#107C2311yescharacteristic
69C2×C4⋊C8D411yescharacteristic
70C2×#102—22no—
71C2×#102—21no—
72C2×C22⋊C8D412yes—
73C2×C4×C8C2311yescharacteristic
74#394—42no—
75#389—41no—
76#385—42no—
77#384—21no—
78#384—21no—
79#380—22no—
80#380—22no—
81#380—41no—
82C4×C4⋊C4—21no—
83#367—24no—
84#367—24no—
85#367—42no—
86C4⋊C16—22no—
87#357—24no—
88#357—22no—
89#357D414yescomplemented
90C22⋊C16—24no—
91C22⋊C16—24no—
92C4×C16—22no—
93#340—22no—
94#340—22no—
95#340—21no—
96C22×D4—21no—
97C22×D4—42no—
98C22×D4D811yescharacteristic
99C23×C4—42no—
100C23×C4—41no—
101C23×C4C2×D412yes—
102C2×D8—21no—
103C2×D8—21no—
104C2×D8—21no—
105C2×D8—21no—
106C2×D8—21no—
107C2×D8C2×D412yes—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8C2×D412yes—
115C22×C8C2×D411yescharacteristic
116C22×C8C2411yescharacteristic
117#121—44no—
118#121—82no—
119#121—84no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4—84no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—82no—
137C2×C4⋊C4D811yescharacteristic
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4D812yes—
143C2×C42—41no—
144C2×C42C2×D411yescharacteristic
145C2×C16—21no—
146C2×C16—21no—
147C2×C16—21no—
148C2×C16—21no—
149C2×C16—22no—
150C2×C16—22no—
151C2×C16—22no—
152C2×C16C2×D412yes—
153C2×C16#4112yes—
154#107—21no—
155#107—22no—
156#107—22no—
157#107—21no—
158#107—21no—
159#107—21no—
160#107—22no—
161#107—22no—
162#107—21no—
163#107—44no—
164#107—81no—
165#107—81no—
166#107C2×D412yes—
167C4⋊C8—22no—
168#102—44no—
169#102—42no—
170C22⋊C8—24no—
171C4×C8#4112yes—
172C4×C8#4112yes—
173#95—41no—
174#95—42no—
175#95—42no—
176C24—41no—
177C24C2×D812yes—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—84no—
187C2×D4—84no—
188C2×D4—162no—
189C2×D4SD3214yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—84no—
202C22×C4—84no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4—162no—
207C22×C4C2×D812yes—
208C22×C4C2×D811yescharacteristic
209C22×C4C2×D811yescharacteristic
210C22×C4C22×D411yescharacteristic
211D8—22no—
212D8—22no—
213D8—41no—
214D8—41no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8#12312yes—
225C2×C8C22×D411yescharacteristic
226C2×C8C2×#4111yescharacteristic
227C2×C8C2×#4111yescharacteristic
228C2×C8C2×#4111yescharacteristic
229C2×C8H(5,𝔽2)11yescharacteristic
230C2×C82−1+411yescharacteristic
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—84no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—161no—
246C4⋊C4—161no—
247C22⋊C4—24no—
248C22⋊C4—84no—
249C22⋊C4—84no—
250C22⋊C4—84no—
251C22⋊C4—82no—
252C42—82no—
253C42#12312yes—
254C42#13712yes—
255C16—22no—
256C16—22no—
257C16—42no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—42no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23C2×SD3214yes—
272C23C22×D811yescharacteristic
273D4—28no—
274D4—42no—
275D4—42no—
276D4—81no—
277D4—81no—
278D4—168no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—168no—
304C2×C4—164no—
305C2×C4#48012yes—
306C2×C4#48012yes—
307C2×C4C2×SD3212yes—
308C2×C4#50912yes—
309C2×C4C2×#12311yescharacteristic
310C2×C4D4211yescharacteristic
311C2×C4#54711yescharacteristic
312C2×C4C2×#13711yescharacteristic
313C2×C4#57611yescharacteristic
314C2×C4#57811yescharacteristic
315C8—22no—
316C8—42no—
317C8#53211yescharacteristic
318C8#54211yescharacteristic
319C8#54711yescharacteristic
320C8#55311yescharacteristic
321C22—24no—
322C22—24no—
323C22—24no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—161no—
332C22—161no—
333C22—161no—
334C22—161no—
335C22C2×#48011yescharacteristic
336C22D4×D811yescharacteristic
337C22#333411yescharacteristic
338C22C22×SD3211yescharacteristic
339C22C2×#50911yescharacteristic
340C22#341611yescharacteristic
341C22#341611yescharacteristic
342C4—22no—
343C4—42no—
344C4—82no—
345C4—162no—
346C4—162no—
347C4—161no—
348C4—161no—
349C4#223314yescomplemented
350C4#242211yescharacteristic
351C4#309811yescharacteristic
352C4#327611yescharacteristic
353C4#335711yescharacteristic
354C2—44no—
355C2—162no—
356C2#2501011yescharacteristic
357C2C2×#223311yescharacteristic
358C2#3365311yescharacteristic
359C2#3383811yescharacteristic
360C2D4×SD3211yescharacteristic
361C2#3390211yescharacteristic
362C2#3392111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#28227C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2233C2semidirect
3. D4×#107C2semidirect
4. #13494C2semidirect
5. #13381C2semidirect
6. #13189C2semidirect
7. #13177C2semidirect
8. #12920C2semidirect
9. #12906C2semidirect
10. #12811C2semidirect
11. C4×#357C2semidirect
89. #357D4semidirect
349. C4#2233semidirect

Maximal subgroups

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

subgroupindex
C2×#22332
#282272
D4×#1072
#134942
#133812
#131892
#131772
#129202
#129062
#128112
C4×#3572