Finite Group Databasebeta

← #29814

Subgroups — #29814

Order 256.

530 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#496C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic, complemented
3#2856C211yesmaximal, characteristic, complemented
4#2850C211yesmaximal, characteristic, complemented
5#2055C212yesmaximal, complemented
6#2054C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1999C212yesmaximal, complemented
9#1961C211yesmaximal, characteristic, complemented
10#1959C211yesmaximal, characteristic, complemented
11#1836C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14C2×#127C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121—21no—
18C2×#121—21no—
19C2×#121C2211yescharacteristic
20C2×#121C2212yes—
21C22×C42C2211yescharacteristic
22#496—24no—
23#432—24no—
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×C22⋊C8C2212yes—
30C2×#97C2211yescharacteristic
31C2×#97C2211yescharacteristic
32#391C2211yescharacteristic
33#390—21no—
34#390—21no—
35#390C2211yescharacteristic
36#390C2212yes—
37#388—21no—
38#388—21no—
39#386—21no—
40#386—21no—
41#384C2211yescharacteristic
42#382C2211yescharacteristic
43#340C2212yes—
44#340C2212yes—
45C22×Q8C2311yescharacteristic
46C22×D4—22no—
47C22×D4—21no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C22×D4C2312yes—
52C23×C4D411yescharacteristic
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C2×SD16—24no—
56C2×SD16—42no—
57C2×SD16—42no—
58C2×D8—24no—
59C2×D8—42no—
60C2×D8—42no—
61C22×C8C2312yes—
62C22×C8C2312yes—
63#127—22no—
64#127—22no—
65#124—22no—
66#124—22no—
67C22⋊Q8—42no—
68#121—42no—
69#121—42no—
70#121—42no—
71#121—44no—
72#121—44no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42D411yescharacteristic
91C2×C42D411yescharacteristic
92C2×C42C2311yescharacteristic
93C2×C42C2311yescharacteristic
94C4⋊C8—24no—
95#103—42no—
96#102—42no—
97#102—42no—
98#102—42no—
99C22⋊C8—24no—
100#97—22no—
101#97D414yescomplemented
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106C24—21no—
107C24—22no—
108C24C2×D411yescharacteristic
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—42no—
131C2×D4—44no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—42no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2411yescharacteristic
155SD16—48no—
156D8—48no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8C2×D414yes—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—44no—
172C22⋊C4—42no—
173C22⋊C4—44no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—44no—
179C42—22no—
180C42—22no—
181C42—21no—
182C42—21no—
183C42C2×D412yes—
184C42#4112yes—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—41no—
192C23—42no—
193C23—41no—
194C23—42no—
195C23—42no—
196C23—41no—
197C23—42no—
198C23—42no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—42no—
204C23—42no—
205C23AGL(1,ℤ/8)12yes—
206C23C22×D411yescharacteristic
207Q8—42no—
208Q8—42no—
209D4—42no—
210D4—42no—
211D4—42no—
212D4—42no—
213D4—42no—
214D4—42no—
215D4—84no—
216D4—84no—
217D4—84no—
218D4—88no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4#12112yes—
248C2×C4#12112yes—
249C2×C4AGL(1,ℤ/8)12yes—
250C2×C4C22×D411yescharacteristic
251C2×C4C2×#4111yescharacteristic
252C2×C4H(5,𝔽2)12yes—
253C2×C4H(5,𝔽2)12yes—
254C8—24no—
255C8—42no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—81no—
264C22—82no—
265C22—81no—
266C22—82no—
267C22—81no—
268C22—82no—
269C22—82no—
270C22—81no—
271C22—82no—
272C22—81no—
273C22—82no—
274C22—82no—
275C22—81no—
276C22—81no—
277C22C2×#12111yescharacteristic
278C22#53511yescharacteristic
279C22#53511yescharacteristic
280C22C2×AGL(1,ℤ/8)11yescharacteristic
281C22C2×AGL(1,ℤ/8)11yescharacteristic
282C22#57511yescharacteristic
283C22#57511yescharacteristic
284C4—22no—
285C4—22no—
286C4—22no—
287C4—41no—
288C4—41no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—82no—
293C4#53512yes—
294C4#54012yes—
295C2—42no—
296C2—161no—
297C2—162no—
298C2#261911yescharacteristic
299C2#305511yescharacteristic
300C2#305511yescharacteristic
301C2#321311yescharacteristic
302C2#321411yescharacteristic
303C2#321711yescharacteristic
304C2#321911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#496C2semidirect
2. C2×#432C2semidirect
3. #2856C2semidirect
4. #2850C2semidirect
5. #2055C2semidirect
6. #2054C2semidirect
7. #2000C2semidirect
8. #1999C2semidirect
9. #1961C2semidirect
10. #1959C2semidirect
11. #1836C2semidirect
101. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4962
C2×#4322
#28562
#28502
#20552
#20542
#20002
#19992
#19612
#19592
#18362