Finite Group Databasebeta

← #23604

Subgroups — #23604

Order 256.

501 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 167 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×#430C211yesmaximal, characteristic, complemented
2C2×#428C211yesmaximal, characteristic, complemented
3#2905C211yesmaximal, characteristic, complemented
4#2896C211yesmaximal, characteristic
5C2×#417C211yesmaximal, characteristic
6C2×#413C211yesmaximal, characteristic
7#2064C214yesmaximal, complemented
8#2063C214yesmaximal, complemented
9C2×#343C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C22×M16C2211yescharacteristic
12C22×M16C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C22×M16C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17#430—24no—
18C2×#108C2214yes—
19#428—22no—
20#428C2214yes—
21C2×#107—21no—
22C2×#106—21no—
23#424—22no—
24#424C2214yes—
25#423—22no—
26#419C2212yes—
27#418C2212yes—
28#417—22no—
29#417—22no—
30#417—21no—
31#417—21no—
32#416—22no—
33C2×#103C2212yes—
34C2×#102C2212yes—
35#413—24no—
36C2×#101C2212yes—
37C2×#100C2212yes—
38#343C2214yes—
39C2×#41—22no—
40C2×#41—21no—
41C2×#41—21no—
42C2×#41—21no—
43C2×#41—21no—
44C2×#41—22no—
45C2×#41—22no—
46C2×#41Q814yes—
47C22×Q8C2311yescharacteristic
48C22×D4—21no—
49C22×D4C2311yescharacteristic
50C23×C4—21no—
51C23×C4—21no—
52C23×C4C2311yescharacteristic
53C2×M16—21no—
54C2×M16—22no—
55C2×M16—22no—
56C2×M16—22no—
57C2×M16—21no—
58C2×M16—21no—
59C2×M16—22no—
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—22no—
63C2×M16D414yes—
64C2×M16D412yes—
65C2×M16D412yes—
66C2×M16C2312yes—
67C2×M16C2312yes—
68C2×M16C2312yes—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8C2312yes—
73#117—21no—
74#117—21no—
75#117D412yes—
76#117C2312yes—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C22⋊C4—21no—
86C2×C42—21no—
87#108—28no—
88#107—24no—
89#106—24no—
90C4⋊C8—28no—
91#103—24no—
92#103—22no—
93#103—22no—
94#102—24no—
95#102—22no—
96#102—22no—
97#101—24no—
98#100—24no—
99C24—21no—
100C24#4111yescharacteristic
101#41—24no—
102#41—24no—
103#41—24no—
104#41—24no—
105#41—42no—
106#41—42no—
107#41—42no—
108#41—42no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8C2×Q812yes—
115C2×D4—24no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—42no—
124C2×D4C2×Q812yes—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—42no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—82no—
136C22×C4C2×D412yes—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×Q812yes—
142C22×C4#4111yescharacteristic
143C22×C4#4112yes—
144C22×C4#4112yes—
145C22×C4C2411yescharacteristic
146M16—24no—
147M16—24no—
148M16—24no—
149M16—22no—
150M16—22no—
151M16—22no—
152M16—22no—
153M16—22no—
154M16—42no—
155M16—42no—
156M16C2×D414yes—
157C2×C8—24no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8C2×D414yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4C2×D412yes—
173C4⋊C4C2×D412yes—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C42—22no—
177C42—22no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—81no—
184C23—81no—
185C23#12111yescharacteristic
186C23#12111yescharacteristic
187C23C22⋊Q812yes—
188C23#12612yes—
189C23C2×#4111yescharacteristic
190Q8—22no—
191Q8—22no—
192Q8—41no—
193Q8—41no—
194D4—22no—
195D4—22no—
196D4—44no—
197D4—44no—
198D4—41no—
199D4—41no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4#12112yes—
214C2×C4#12112yes—
215C2×C4#12112yes—
216C2×C4#12112yes—
217C2×C4#12111yescharacteristic
218C2×C4#12111yescharacteristic
219C2×C4C22⋊Q812yes—
220C2×C4C22⋊Q812yes—
221C2×C4C22⋊Q812yes—
222C2×C4C22⋊Q812yes—
223C2×C4C22⋊Q812yes—
224C2×C4#12612yes—
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C2×C4C22×D411yescharacteristic
228C2×C4C22×Q811yescharacteristic
229C2×C4C2×#4111yescharacteristic
230C2×C4C2×#4111yescharacteristic
231C8—24no—
232C8—24no—
233C8—42no—
234C8—42no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—82no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22#39612yes—
249C22#39612yes—
250C22C2×#12111yescharacteristic
251C22C2×C22⋊Q811yescharacteristic
252C22C2×#12611yescharacteristic
253C4—42no—
254C4—42no—
255C4—42no—
256C4—82no—
257C4#39612yes—
258C4#39612yes—
259C4C2×#12111yescharacteristic
260C4C2×#12111yescharacteristic
261C4C2×C22⋊Q811yescharacteristic
262C4C2×C22⋊Q811yescharacteristic
263C2—21no—
264C2—21no—
265C2—22no—
266C2—22no—
267C2—82no—
268C2C2×#39611yescharacteristic
269C2#306811yescharacteristic
270C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#430C2semidirect
2. C2×#428C2semidirect
3. #2905C2semidirect
7. #2064C2semidirect
8. #2063C2semidirect
9. C2×#343C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4302
C2×#4282
#29052
#28962
C2×#4172
C2×#4132
#20642
#20632
C2×#3432