Finite Group Databasebeta

← #29672

Subgroups — #29672

Order 256.

548 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 127 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×#542C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic, complemented
4#2621C211yesmaximal, characteristic, complemented
5#2024C212yesmaximal, complemented
6#2015C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#2000C212yesmaximal
9#1805C211yesmaximal, characteristic, complemented
10#1804C211yesmaximal, characteristic, complemented
11#1798C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14#542—24no—
15#542—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#126—21no—
19C2×#124—21no—
20C2×#124C2211yescharacteristic
21C2×#123—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2212yes—
25C2×#121C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117C2211yescharacteristic
29#462—24no—
30#461—24no—
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#103C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2212yes—
38#398—21no—
39#398—21no—
40#397—21no—
41#397—21no—
42#391C2212yes—
43#382C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8C2312yes—
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C2×Q16—24no—
55C2×Q16—42no—
56C2×Q16—42no—
57C2×SD16—24no—
58C2×SD16—42no—
59C2×SD16—42no—
60C22×C8C2312yes—
61C22×C8C2312yes—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—22no—
65C4⋊Q8—22no—
66#126—28no—
67#124—28no—
68#124—22no—
69#124—22no—
70#123—28no—
71C22⋊Q8—28no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—42no—
75C22⋊Q8—42no—
76#121—22no—
77#121—22no—
78C4×Q8—22no—
79C4×Q8D414yes—
80C4×D4—22no—
81C4×D4D414yes—
82#117—22no—
83#117—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106C4⋊C8—22no—
107C4⋊C8—22no—
108#103—24no—
109#103—24no—
110#103—42no—
111#102—42no—
112C22⋊C8—42no—
113#95—21no—
114#95—21no—
115#95—22no—
116#95—22no—
117C24—21no—
118C24C2×D411yescharacteristic
119C2×Q8—24no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—21no—
130C2×Q8—22no—
131C2×Q8—82no—
132C2×Q8C2×D412yes—
133C2×D4—24no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—82no—
140C2×D4C2×D412yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—82no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2411yescharacteristic
160Q16—48no—
161SD16—48no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C4⋊C4—24no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4C2×D412yes—
181C4⋊C4C2×D412yes—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—44no—
190C42—24no—
191C42—21no—
192C42—21no—
193C42C2×D412yes—
194C42C2×D412yes—
195C23—21no—
196C23—22no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—22no—
201C23—41no—
202C23—41no—
203C23—42no—
204C23—42no—
205C23C22×D411yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—41no—
209Q8—41no—
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—42no—
214Q8—84no—
215D4—22no—
216D4—22no—
217D4—41no—
218D4—41no—
219D4—84no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4#12012yes—
248C2×C4#12012yes—
249C2×C4C22×D411yescharacteristic
250C2×C4C22×D411yescharacteristic
251C2×C4H(5,𝔽2)11yescharacteristic
252C2×C4H(5,𝔽2)11yescharacteristic
253C2×C4H(5,𝔽2)11yescharacteristic
254C2×C42−1+411yescharacteristic
255C8—42no—
256C8—42no—
257C22—41no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—82no—
266C22—82no—
267C22C2×#12011yescharacteristic
268C22#53511yescharacteristic
269C22#53711yescharacteristic
270C22#57712yes—
271C22#57812yes—
272C4—22no—
273C4—22no—
274C4—41no—
275C4—42no—
276C4—41no—
277C4—42no—
278C4—42no—
279C4—41no—
280C4—82no—
281C4—82no—
282C4#53511yescharacteristic
283C4#53511yescharacteristic
284C4#53711yescharacteristic
285C4#53711yescharacteristic
286C2—81no—
287C2—82no—
288C2#261511yescharacteristic
289C2#300911yescharacteristic
290C2#301111yescharacteristic
291C2#322512yes—
292C2#325812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#462C2semidirect
3. C2×#461C2semidirect
4. #2621C2semidirect
5. #2024C2semidirect
6. #2015C2semidirect
7. #2001C2semidirect
9. #1805C2semidirect
10. #1804C2semidirect
11. #1798C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4622
C2×#4612
#26212
#20242
#20152
#20012
#20002
#18052
#18042
#17982