Finite Group Databasebeta

← #19494

Subgroups — #19494

Order 256.

312 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 109 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#3206C211yesmaximal, characteristic, complemented
2#3106C211yesmaximal, characteristic
3#3079C211yesmaximal, characteristic, complemented
4#2991C211yesmaximal, characteristic, complemented
5#1703C212yesmaximal, complemented
6#1685C212yesmaximal, complemented
7#1684C212yesmaximal, complemented
8#1666C212yesmaximal, complemented
9#1574C211yesmaximal, characteristic, complemented
10#1560C211yesmaximal, characteristic, complemented
11#1546C211yesmaximal, characteristic, complemented
12#532C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C4×#41C2211yescharacteristic
15#492C2212yes—
16#487C2212yes—
17#482—21no—
18#477—21no—
19#477C2211yescharacteristic
20#476—21no—
21#476C2211yescharacteristic
22#475C2212yes—
23#474C2212yes—
24#461—21no—
25#461—21no—
26#460C2211yescharacteristic
27#459C2211yescharacteristic
28#449—21no—
29#448—22no—
30C8×Q8C2212yes—
31C4×SD16—21no—
32C4×D8—21no—
33#436—22no—
34#436C2212yes—
35#431—21no—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#428—21no—
39#423—21no—
40#423C2211yescharacteristic
41#423C2211yescharacteristic
42#417—21no—
43#335C2212yes—
44#334C2212yes—
45#333C2212yes—
46#332C2212yes—
47#332C2212yes—
48#331C2212yes—
49C2×#41—21no—
50C2×#41D411yescharacteristic
51C22×Q8—21no—
52C2×SD16—21no—
53C2×SD16—22no—
54C2×SD16—22no—
55C2×D8—21no—
56C2×M16—21no—
57C2×M16—21no—
58C2×M16—22no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8C2312yes—
63C4⋊Q8C2311yescharacteristic
64#127C2311yescharacteristic
65#124—21no—
66#121—21no—
67#121—21no—
68C4×Q8D411yescharacteristic
69C4×Q8C2311yescharacteristic
70C4×D4—21no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4D411yescharacteristic
75C4×D4C2311yescharacteristic
76#117—21no—
77#117D411yescharacteristic
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C42—21no—
81C2×C42C2311yescharacteristic
82#107—22no—
83#107—21no—
84#106—22no—
85#106—21no—
86C4⋊C8—22no—
87C4⋊C8—21no—
88C4⋊C8—21no—
89C4⋊C8D411yescharacteristic
90C4⋊C8D411yescharacteristic
91C4⋊C8C2312yes—
92C4⋊C8C2312yes—
93C4⋊C8C2311yescharacteristic
94C4⋊C8C2311yescharacteristic
95#103—22no—
96#103—21no—
97#103—22no—
98#103—22no—
99#102—22no—
100#102—21no—
101#102—21no—
102#102—21no—
103C22⋊C8—21no—
104C22⋊C8—22no—
105C22⋊C8D412yes—
106C4×C8—22no—
107C4×C8—21no—
108C4×C8C2312yes—
109#41—22no—
110#41—41no—
111#41—82no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—41no—
115C2×Q8—81no—
116C2×Q8—81no—
117C2×Q8C2×D411yescharacteristic
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—41no—
123C2×D4C2×D411yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4#4111yescharacteristic
132SD16—42no—
133SD16—82no—
134SD16—82no—
135D8—42no—
136M16—42no—
137M16—82no—
138M16—81no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8C2×D412yes—
144C2×C8C2×D412yes—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4C2×D411yescharacteristic
154C4⋊C4C2×D411yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—21no—
157C22⋊C4—21no—
158C22⋊C4—41no—
159C22⋊C4—41no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42C2×D411yescharacteristic
164C42C2×D411yescharacteristic
165C42#4111yescharacteristic
166C42C2411yescharacteristic
167C23—21no—
168C23—41no—
169C23#12111yescharacteristic
170Q8—21no—
171Q8—42no—
172Q8—42no—
173Q8—81no—
174Q8—82no—
175Q8#13712yes—
176D4—21no—
177D4—21no—
178D4—42no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—41no—
183D4—81no—
184D4—82no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4#12111yescharacteristic
202C2×C4#12111yescharacteristic
203C2×C4#12111yescharacteristic
204C2×C4C22×D411yescharacteristic
205C2×C4C22×D411yescharacteristic
206C2×C4C2×#4111yescharacteristic
207C8—22no—
208C8—22no—
209C8—41no—
210C8—42no—
211C8—41no—
212C8—42no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22C2×#12111yescharacteristic
223C4—21no—
224C4—21no—
225C4—22no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—82no—
230C4—81no—
231C4#47112yes—
232C4C2×#13711yescharacteristic
233C4#57511yescharacteristic
234C4#57611yescharacteristic
235C4#57711yescharacteristic
236C2—41no—
237C2—81no—
238C2—161no—
239C2#304511yescharacteristic
240C2#305611yescharacteristic
241C2C2×#47111yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3106C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3206C2semidirect
3. #3079C2semidirect
4. #2991C2semidirect
5. #1703C2semidirect
6. #1685C2semidirect
7. #1684C2semidirect
8. #1666C2semidirect
9. #1574C2semidirect
10. #1560C2semidirect
11. #1546C2semidirect

Maximal subgroups

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

subgroupindex
#32062
#31062
#30792
#29912
#17032
#16852
#16842
#16662
#15742
#15602
#15462