Finite Group Databasebeta

← #141116

Subgroups — #141116

Order 512.

511 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 106 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#33173C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic, complemented
3C2×#1350C212yesmaximal, complemented
4C2×#1349C212yesmaximal
5C4×#341C211yesmaximal, characteristic, complemented
6C2×#495C411yescharacteristic
7C2×#467C2211yescharacteristic
8C2×#466C2211yescharacteristic
9C2×#465—21no—
10C2×C4×M16—21no—
11C22×C4×C8C2211yescharacteristic
12#2850—21no—
13C2×C4⋊C16—21no—
14C2×C22⋊C16C2212yes—
15#1970C411yescharacteristic
16#1969C411yescharacteristic
17#1968—21no—
18#1958—21no—
19#1846C411yescharacteristic
20C2×#341—21no—
21C2×#341C2212yes—
22#1374—21no—
23#1350—24no—
24#1349—24no—
25C22×Q16C2×C411yescharacteristic
26C22×SD16—21no—
27C22×D8C2×C411yescharacteristic
28C22×M16—22no—
29C23×C8D411yescharacteristic
30C23×C8C2311yescharacteristic
31C2×#124—21no—
32C2×C22⋊Q8—21no—
33C2×#121—21no—
34C22×C42D411yescharacteristic
35C22×C16—22no—
36#495—42no—
37#467—22no—
38#467—22no—
39#466—22no—
40#466—22no—
41#465—44no—
42C2×#107C2×C411yescharacteristic
43C2×#107C2×C411yescharacteristic
44C2×#106—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×#102—21no—
48C2×#102—21no—
49C4×M16—24no—
50C4×M16—42no—
51C2×#97—41no—
52C2×C4×C8—42no—
53C2×C4×C8—41no—
54C2×C4×C8C2×C411yescharacteristic
55C2×C4×C8C2×C411yescharacteristic
56C2×C4×C8D411yescharacteristic
57C2×C4×C8D411yescharacteristic
58#391—21no—
59#390—21no—
60#388—41no—
61#388—41no—
62#386—41no—
63#386—41no—
64#382—21no—
65C4⋊C16—42no—
66C22⋊C16—22no—
67C22⋊C16—22no—
68#341—24no—
69#341—22no—
70#341D414yes—
71C22×Q8—21no—
72C22×D4—21no—
73C22×D4—41no—
74C23×C4SD1611yescharacteristic
75C23×C4SD1611yescharacteristic
76C23×C4C2×D411yescharacteristic
77C2×Q16—22no—
78C2×Q16—41no—
79C2×Q16—41no—
80C2×SD16—42no—
81C2×SD16—42no—
82C2×SD16—42no—
83C2×D8—22no—
84C2×D8—41no—
85C2×D8—41no—
86C2×M16—24no—
87C2×M16—24no—
88C2×M16—24no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—42no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—42no—
98C22×C8C22⋊C411yescharacteristic
99C22×C8C22⋊C411yescharacteristic
100C22×C8C22×C411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D412yes—
103C22×C8#4112yes—
104#124—42no—
105#124—42no—
106C22⋊Q8—42no—
107C22⋊Q8—42no—
108#121—42no—
109#121—42no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42C22⋊C411yescharacteristic
122C2×C42C22⋊C411yescharacteristic
123C2×C42D811yescharacteristic
124C2×C42D811yescharacteristic
125C2×C16—42no—
126C2×C16—42no—
127C2×C16—42no—
128#107—22no—
129#107—22no—
130#106—42no—
131#103—42no—
132#103—41no—
133#103—41no—
134#102—42no—
135#102—41no—
136#102—41no—
137#97—44no—
138C4×C8—22no—
139C4×C8—22no—
140C4×C8—41no—
141C4×C8—41no—
142C4×C8—44no—
143#95—41no—
144#95—41no—
145#95—41no—
146#95—41no—
147C24—41no—
148C24C2×SD1611yescharacteristic
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—82no—
162C2×D4—82no—
163C2×D4—82no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4#10211yescharacteristic
178C22×C4#10211yescharacteristic
179C22×C4#10211yescharacteristic
180C22×C4#10211yescharacteristic
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4#12112yes—
183C22×C4C2×D811yescharacteristic
184C22×C4AGL(1,ℤ/8)12yes—
185Q16—82no—
186SD16—84no—
187D8—82no—
188M16—28no—
189M16—44no—
190M16—44no—
191M16—44no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—44no—
204C2×C8—44no—
205C2×C8—44no—
206C2×C8—44no—
207C2×C8#10412yes—
208C2×C8C4×D412yes—
209C2×C8#12411yescharacteristic
210C2×C8#12711yescharacteristic
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—82no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C22⋊C4—82no—
222C42—22no—
223C42—22no—
224C42—21no—
225C42—21no—
226C42#9912yes—
227C42#10412yes—
228C16—82no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23C2×#10211yescharacteristic
243C23#46012yes—
244Q8—81no—
245Q8—81no—
246Q8—81no—
247Q8—81no—
248D4—81no—
249D4—81no—
250D4—81no—
251D4—81no—
252D4—164no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4#32712yes—
280C2×C4#32712yes—
281C2×C4#35912yes—
282C2×C4#39011yescharacteristic
283C2×C4C2×#9911yescharacteristic
284C2×C4#41011yescharacteristic
285C2×C4#41811yescharacteristic
286C2×C4C2×#10411yescharacteristic
287C2×C4C2×#10411yescharacteristic
288C2×C4#45912yes—
289C8—21no—
290C8—22no—
291C8—21no—
292C8—22no—
293C8—48no—
294C8—41no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—161no—
303C22—161no—
304C22—161no—
305C22—161no—
306C22—161no—
307C22—161no—
308C22—161no—
309C22C2×#32711yescharacteristic
310C22#150211yescharacteristic
311C22#198211yescharacteristic
312C22C2×#35911yescharacteristic
313C22#214311yescharacteristic
314C22#214811yescharacteristic
315C22#214811yescharacteristic
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—41no—
321C4—161no—
322C4—161no—
323C4—161no—
324C4#150511yescharacteristic
325C4#150711yescharacteristic
326C4#196211yescharacteristic
327C4#196411yescharacteristic
328C2—42no—
329C2—321no—
330C2#971211yescharacteristic
331C2#1170911yescharacteristic
332C2#1171011yescharacteristic
333C2#1174911yescharacteristic
334C2#1175011yescharacteristic
335C2#1325111yescharacteristic
336C2#1325811yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12783C2semidirect
3. C2×#1350C2semidirect
5. C4×#341C2semidirect

Maximal subgroups

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

subgroupindex
#331732
#127832
C2×#13502
C2×#13492
C4×#3412