Finite Group Databasebeta

← #502333

Subgroups — #502333

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2206C211yesmaximal, characteristic
2C2×#2183C211yesmaximal, characteristic, complemented
3#28736C211yesmaximal, characteristic, complemented
4#28263C211yesmaximal, characteristic, complemented
5#13476C212yesmaximal, complemented
6#13448C212yesmaximal, complemented
7#13239C212yesmaximal, complemented
8#13202C211yesmaximal, characteristic, complemented
9#13177C211yesmaximal, characteristic, complemented
10#13149C212yesmaximal, complemented
11C4×#357C211yesmaximal, characteristic, complemented
12C22×D16C2212yes—
13C2×#493C2211yescharacteristic
14C2×C8×Q8C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic
16#2548—21no—
17#2538—21no—
18#2206—24no—
19#2183—24no—
20C2×#367C2212yes—
21C2×C4⋊C16C2212yes—
22C2×C4⋊C16C2211yescharacteristic
23C2×#357C2212yes—
24C2×#357C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2212yes—
27C2×#357C2212yes—
28C2×#346C2212yes—
29C2×C4×C16C2211yescharacteristic
30#2067—22no—
31#2054—22no—
32#1982—21no—
33#1970C2212yes—
34#1970C2212yes—
35#1970C2212yes—
36#1942C2212yes—
37#1845C2211yescharacteristic
38#1802—21no—
39C4×#107C2211yescharacteristic
40#1380C2211yescharacteristic
41#1380C2211yescharacteristic
42#1380C2212yes—
43C22×D8—21no—
44C22×D8C2312yes—
45C22×D8C2311yescharacteristic
46C2×#127—21no—
47C2×#124—41no—
48C2×C22⋊Q8—41no—
49C2×C4×Q8D411yescharacteristic
50C2×C4×D4—21no—
51C2×D16—24no—
52C2×D16—24no—
53C2×D16—42no—
54C22×C16C2311yescharacteristic
55C22×C16C2312yes—
56C22×C16C2311yescharacteristic
57#493—22no—
58#493—22no—
59C8×Q8—22no—
60C8×Q8—22no—
61C4×D8—22no—
62C4×D8—22no—
63C2×#107—21no—
64C2×#107C2312yes—
65C2×#107C2311yescharacteristic
66C2×#107C2311yescharacteristic
67C2×#107C2311yescharacteristic
68C2×C4⋊C8D412yes—
69C2×C4⋊C8D411yescharacteristic
70C2×#102—21no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—22no—
74C2×C4×C8C2312yes—
75C2×C4×C8C2311yescharacteristic
76#396—42no—
77#394—42no—
78#390—22no—
79#390—22no—
80#390—22no—
81#390—41no—
82#389—22no—
83#384—21no—
84#380—41no—
85C4×C4⋊C4—21no—
86#367—42no—
87C4⋊C16—24no—
88C4⋊C16D414yes—
89#357—22no—
90#357—24no—
91#357—22no—
92#357—24no—
93#357—42no—
94#346—24no—
95C4×C16—22no—
96#340—21no—
97#340—22no—
98C22×Q8D811yescharacteristic
99C22×D4—21no—
100C22×D4—22no—
101C22×D4—41no—
102C22×D4—41no—
103C23×C4—41no—
104C2×D8—21no—
105C2×D8—22no—
106C2×D8—21no—
107C2×D8—21no—
108C2×D8—22no—
109C2×D8—22no—
110C2×D8—22no—
111C2×D8—21no—
112C2×D8—22no—
113C2×D8—21no—
114C2×D8—22no—
115C2×D8—42no—
116C2×D8—81no—
117C2×D8—81no—
118C2×D8#4112yes—
119C22×C8C2×D412yes—
120C22×C8C2×D411yescharacteristic
121C22×C8C2411yescharacteristic
122#127—42no—
123#127—42no—
124#124—84no—
125C22⋊Q8—84no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×D4—42no—
129C4×D4—42no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4D811yescharacteristic
138C2×C4⋊C4D812yes—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C42—41no—
147C2×C42C2×D412yes—
148C2×C42C2×D411yescharacteristic
149D16—48no—
150C2×C16—21no—
151C2×C16—22no—
152C2×C16—22no—
153C2×C16—21no—
154C2×C16—21no—
155C2×C16—21no—
156C2×C16—21no—
157C2×C16C2×D414yes—
158C2×C16#4112yes—
159#107—21no—
160#107—21no—
161#107—22no—
162#107—21no—
163#107—21no—
164#107—22no—
165#107—21no—
166#107—21no—
167#107—81no—
168C4⋊C8—24no—
169C4⋊C8—22no—
170#102—42no—
171#102—82no—
172#102—82no—
173#102—82no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8C2×D412yes—
177C4×C8#4112yes—
178#95—41no—
179#95—42no—
180C24—41no—
181C24—42no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×D4—22no—
186C2×D4—41no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—41no—
190C2×D4—42no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—41no—
194C2×D4—42no—
195C2×D4—41no—
196C2×D4—42no—
197C2×D4—82no—
198C2×D4—82no—
199C2×D4—82no—
200C2×D4—82no—
201C2×D4—161no—
202C2×D4—161no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4C2×D812yes—
211C22×C4C2×D812yes—
212C22×C4C2×D811yescharacteristic
213C22×C4C2×D811yescharacteristic
214C22×C4C22×D411yescharacteristic
215D8—22no—
216D8—22no—
217D8—42no—
218D8—41no—
219D8—42no—
220D8—42no—
221D8—41no—
222D8—42no—
223D8—84no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8#12112yes—
230C2×C8C22×D411yescharacteristic
231C2×C8C2×#4111yescharacteristic
232C2×C8C2×#4111yescharacteristic
233C2×C8C2×#4111yescharacteristic
234C2×C8H(5,𝔽2)11yescharacteristic
235C2×C82−1+411yescharacteristic
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—84no—
247C4⋊C4—82no—
248C4⋊C4—161no—
249C22⋊C4—84no—
250C22⋊C4—84no—
251C22⋊C4—84no—
252C22⋊C4—82no—
253C22⋊C4—162no—
254C22⋊C4—162no—
255C22⋊C4—162no—
256C42—22no—
257C42—22no—
258C42—82no—
259C42#12112yes—
260C42#13512yes—
261C16—24no—
262C16—22no—
263C16—41no—
264C23—42no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—82no—
272C23—82no—
273C23—82no—
274C23—82no—
275C23—82no—
276C23—82no—
277C23—82no—
278C23C22×D811yescharacteristic
279Q8—44no—
280D4—42no—
281D4—42no—
282D4—82no—
283D4—81no—
284D4—82no—
285D4—82no—
286D4—81no—
287D4—82no—
288D4—164no—
289D4—164no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—164no—
311C2×C4#46612yes—
312C2×C4#46612yes—
313C2×C4#50912yes—
314C2×C4#50912yes—
315C2×C4C2×#12111yescharacteristic
316C2×C4#54811yescharacteristic
317C2×C4#55011yescharacteristic
318C2×C4C2×#13511yescharacteristic
319C2×C4#57611yescharacteristic
320C2×C4#57711yescharacteristic
321C8—22no—
322C8—42no—
323C8#52911yescharacteristic
324C8#54211yescharacteristic
325C8#54611yescharacteristic
326C8#54811yescharacteristic
327C22—82no—
328C22—82no—
329C22—82no—
330C22—161no—
331C22—161no—
332C22—161no—
333C22—161no—
334C22—162no—
335C22—162no—
336C22—162no—
337C22—162no—
338C22—162no—
339C22—162no—
340C22—162no—
341C22C2×#46611yescharacteristic
342C22#335811yescharacteristic
343C22#339111yescharacteristic
344C22C2×#50911yescharacteristic
345C22C2×#50911yescharacteristic
346C22#341511yescharacteristic
347C22#341611yescharacteristic
348C4—22no—
349C4—42no—
350C4—42no—
351C4—161no—
352C4—161no—
353C4—161no—
354C4—162no—
355C4#221814yes—
356C4#242611yescharacteristic
357C4#306211yescharacteristic
358C4#336111yescharacteristic
359C4#339411yescharacteristic
360C2—162no—
361C2—322no—
362C2#2529211yescharacteristic
363C2C2×#221811yescharacteristic
364C2#3361211yescharacteristic
365C2#3393311yescharacteristic
366C2#3394011yescharacteristic
367C2#3396411yescharacteristic
368C2#3396611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#2206C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2183C2semidirect
3. #28736C2semidirect
4. #28263C2semidirect
5. #13476C2semidirect
6. #13448C2semidirect
7. #13239C2semidirect
8. #13202C2semidirect
9. #13177C2semidirect
10. #13149C2semidirect
11. C4×#357C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22062
C2×#21832
#287362
#282632
#134762
#134482
#132392
#132022
#131772
#131492
C4×#3572