Finite Group Databasebeta

← #20074

Subgroups — #20074

Order 256.

306 conjugacy classes of subgroups.

The subgroup lattice has 283 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3386C211yesmaximal, characteristic
2#3374C211yesmaximal, characteristic, complemented
3#3359C211yesmaximal, characteristic
4D4×Q16C211yesmaximal, characteristic
5#1723C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1690C211yesmaximal, characteristic, complemented
8#1679C211yesmaximal, characteristic
9#1674C211yesmaximal, characteristic, complemented
10#1653C211yesmaximal, characteristic, complemented
11#1651C211yesmaximal, characteristic
12#1649C211yesmaximal, characteristic
13#1648C211yesmaximal, characteristic, complemented
14#1587C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic
16C22×Q16—21no—
17#557C2211yescharacteristic
18#557C2211yescharacteristic
19D4×Q8C2211yescharacteristic
20D4×Q8C2211yescharacteristic
21#501—21no—
22#500C2211yescharacteristic
23#500C2211yescharacteristic
24#498C2211yescharacteristic
25#494C2211yescharacteristic
26#491—21no—
27#487—21no—
28#487C2211yescharacteristic
29#487C2211yescharacteristic
30#479—21no—
31#478—21no—
32#477—21no—
33#475—21no—
34#475C2211yescharacteristic
35#475C2211yescharacteristic
36#475C2211yescharacteristic
37#474C2211yescharacteristic
38#467—21no—
39#462—21no—
40#462C2211yescharacteristic
41#462C2211yescharacteristic
42#462C2211yescharacteristic
43#461—21no—
44#461C2211yescharacteristic
45#451—21no—
46#451—21no—
47#446C2211yescharacteristic
48#446C2211yescharacteristic
49C8×Q8C2211yescharacteristic
50#441—21no—
51#440—21no—
52C4×Q16C2211yescharacteristic
53C4×SD16—21no—
54C4×SD16C2211yescharacteristic
55C8×D4C2211yescharacteristic
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C22×Q8—21no—
68C22×Q8—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×Q16—44no—
75C2×Q16D411yescharacteristic
76C2×SD16—21no—
77C2×SD16D411yescharacteristic
78C22×C8—21no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#125—21no—
88#125—21no—
89#125—41no—
90#125—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8C2311yescharacteristic
100C4×Q8C2311yescharacteristic
101C4×Q8C2311yescharacteristic
102C4×D4—21no—
103C4×D4—21no—
104C4×D4C2311yescharacteristic
105#107—21no—
106#107—21no—
107#107—21no—
108#107—41no—
109#107D411yescharacteristic
110#106—21no—
111#106—21no—
112#106—21no—
113#106—41no—
114#106D411yescharacteristic
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—21no—
131#103—21no—
132#103—21no—
133#103D411yescharacteristic
134#103D411yescharacteristic
135#103D411yescharacteristic
136#102—21no—
137#102D411yescharacteristic
138C22⋊C8—21no—
139#97—21no—
140#97—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—81no—
154C2×Q8—81no—
155C2×Q8C2×D411yescharacteristic
156C2×Q8C2×D411yescharacteristic
157C2×Q8C2×D411yescharacteristic
158C2×D4C2×D411yescharacteristic
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—41no—
162Q16—22no—
163Q16—42no—
164Q16—42no—
165Q16—42no—
166Q16—42no—
167Q16—81no—
168SD16—22no—
169SD16—42no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—42no—
176C2×C8C2×D411yescharacteristic
177C2×C8C2×D411yescharacteristic
178C2×C8C2×D411yescharacteristic
179C2×C8C2×D411yescharacteristic
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—41no—
211C42C2411yescharacteristic
212C23—21no—
213Q8—21no—
214Q8—21no—
215Q8—21no—
216Q8—21no—
217Q8—21no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—82no—
227Q8—82no—
228Q8AGL(1,ℤ/8)12yes—
229D4—21no—
230D4#13712yes—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4C22×D411yescharacteristic
247C2×C4C22×D411yescharacteristic
248C2×C4H(5,𝔽2)11yescharacteristic
249C8—22no—
250C8—21no—
251C8—21no—
252C8—21no—
253C8—21no—
254C8—41no—
255C8—41no—
256C8—41no—
257C8—41no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22D4211yescharacteristic
262C4—22no—
263C4—21no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4C2×AGL(1,ℤ/8)11yescharacteristic
275C4C2×#13711yescharacteristic
276C4#57711yescharacteristic
277C4#57811yescharacteristic
278C2—42no—
279C2#326511yescharacteristic
280C2#327011yescharacteristic
281C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radicalC2×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3374C2semidirect
5. #1723C2semidirect
6. #1700C2semidirect
7. #1690C2semidirect
9. #1674C2semidirect
10. #1653C2semidirect
13. #1648C2semidirect
14. #1587C2semidirect

Maximal subgroups

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

subgroupindex
#33862
#33742
#33592
D4×Q162
#17232
#17002
#16902
#16792
#16742
#16532
#16512
#16492
#16482
#15872
#15852