Finite Group Databasebeta

← #8421

Subgroups — #8421

Order 256.

471 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 89 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#3499C211yesmaximal, characteristic
2#1988C211yesmaximal, characteristic, complemented
3#1806C211yesmaximal, characteristic, complemented
4#1547C211yesmaximal, characteristic, complemented
5#1546C211yesmaximal, characteristic, complemented
6#1502C211yesmaximal, characteristic, complemented
7#1477C211yesmaximal, characteristic, complemented
8C2×2−1+4—21no—
9D4×Q8—21no—
10D4×Q8C412yescomplemented
11#547—22no—
12#547—21no—
13#547—22no—
14#547C412yescomplemented
15#536—22no—
16#536—21no—
17#532—21no—
18#532C2211yescharacteristic
19C2×C4⋊Q8—21no—
20C2×C4⋊Q8C2211yescharacteristic
21C4×#41C2211yescharacteristic
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#104—21no—
26C4×C4⋊C4C2211yescharacteristic
27#340—22no—
28#339—21no—
29#332—22no—
30#332—21no—
31#331—21no—
32#327—22no—
33Q8⋊C8—21no—
34D4⋊C8—21no—
352−1+4—42no—
362−1+4—42no—
372−1+4—42no—
382−1+4—42no—
39C2×#41—22no—
40C2×#41—24no—
41C2×#41—22no—
42C2×#41—21no—
43C2×#41D411yescharacteristic
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—21no—
47C2×M16—21no—
48C22×C8—21no—
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8C2×C412yes—
55C4⋊Q8D411yescharacteristic
56C4⋊Q8D412yes—
57#127D411yescharacteristic
58#124—22no—
59#124—21no—
60#123—24no—
61#123—24no—
62#123—24no—
63C22⋊Q8—24no—
64C22⋊Q8—22no—
65C22⋊Q8—24no—
66C22⋊Q8—22no—
67#121—22no—
68#121—22no—
69#121—21no—
70#121D412yes—
71C4×Q8C2×C411yescharacteristic
72C4×Q8D411yescharacteristic
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—21no—
76C4×D4—21no—
77C4×D4—21no—
78C4×D4C2×C411yescharacteristic
79C4×D4C2×C412yes—
80C4×D4D411yescharacteristic
81#117—21no—
82#117D411yescharacteristic
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D412yes—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42C2311yescharacteristic
91C4⋊C8—21no—
92C4⋊C8—21no—
93C4⋊C8—21no—
94C4⋊C8—21no—
95C4⋊C8—21no—
96#104—42no—
97C22⋊C8—21no—
98C4×C8—21no—
99#95—21no—
100#41—21no—
101#41—21no—
102#41—44no—
103#41—44no—
104#41—48no—
105#41—48no—
106#41—44no—
107#41—44no—
108#41—41no—
109#41—41no—
110#41—41no—
111#41—41no—
112#41D812yes—
113#41SD1612yes—
114C2×Q8—24no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—44no—
119C2×Q8—44no—
120C2×Q8—44no—
121C2×Q8—44no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8C22⋊C411yescharacteristic
127C2×D4—21no—
128C2×D4—24no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4C22⋊C411yescharacteristic
133C2×D4C22⋊C412yes—
134C22×C4—21no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145M16—41no—
146M16—41no—
147C2×C8—42no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—41no—
161C4⋊C4C22⋊C412yes—
162C4⋊C4C22⋊C414yes—
163C4⋊C4C22⋊C411yescharacteristic
164C4⋊C4C22⋊C411yescharacteristic
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—22no—
170C22⋊C4—21no—
171C22⋊C4—22no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—41no—
176C42—41no—
177C42C22×C411yescharacteristic
178C42C2×D411yescharacteristic
179C42C2×D411yescharacteristic
180C42C2×D411yescharacteristic
181C23—21no—
182C23—22no—
183C23#12011yescharacteristic
184Q8—21no—
185Q8—48no—
186Q8—42no—
187Q8—42no—
188Q8—41no—
189Q8—42no—
190Q8—41no—
191Q8—42no—
192Q8#10212yes—
193D4—22no—
194D4—22no—
195D4—21no—
196D4—42no—
197D4—48no—
198D4—42no—
199D4—42no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4#10212yes—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—48no—
217C2×C4—48no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4C2×C22⋊C411yescharacteristic
228C2×C4C2×C22⋊C411yescharacteristic
229C2×C4C2×C22⋊C411yescharacteristic
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4#12011yescharacteristic
233C2×C4C2×D811yescharacteristic
234C2×C4C2×SD1611yescharacteristic
235C2×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C4AGL(1,ℤ/8)11yescharacteristic
237C8—81no—
238C8—81no—
239C8—81no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22#37911yescharacteristic
249C22#44711yescharacteristic
250C22#45011yescharacteristic
251C4—21no—
252C4—21no—
253C4—21no—
254C4—22no—
255C4—41no—
256C4—41no—
257C4—44no—
258C4—42no—
259C4—82no—
260C4C2×#10211yescharacteristic
261C4#41811yescharacteristic
262C4#44711yescharacteristic
263C4#45011yescharacteristic
264C4#45411yescharacteristic
265C4#45611yescharacteristic
266C2—21no—
267C2—21no—
268C2—42no—
269C2—82no—
270C2#178611yescharacteristic
271C2#179311yescharacteristic
272C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3499C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1988C2semidirect
3. #1806C2semidirect
4. #1547C2semidirect
5. #1546C2semidirect
6. #1502C2semidirect
7. #1477C2semidirect
10. D4×Q8C4semidirect
14. #547C4semidirect

Maximal subgroups

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

subgroupindex
#34992
#19882
#18062
#15472
#15462
#15022
#14772