Finite Group Databasebeta

← #24152

Subgroups — #24152

Order 256.

739 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 739 conjugacy classes of subgroups — too many to draw. Only its 143 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×#532C211yesmaximal, characteristic, complemented
2C2×#476C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic, complemented
4C4×#121C211yesmaximal, characteristic, complemented
5#2034C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1987C211yesmaximal, characteristic, complemented
8#1982C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal, complemented
10#1864C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C22×D8C2212yes—
14#532—24no—
15#532—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#124—21no—
19C2×#121—21no—
20C2×#121C2212yes—
21C2×#121C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4—21no—
24C2×C4×D4C2212yes—
25C2×C4×D4C2211yescharacteristic
26C22×C42C2211yescharacteristic
27#476—24no—
28#459—24no—
29C2×#106C2212yes—
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×C22⋊C8C2212yes—
37#390C2211yescharacteristic
38#389C2211yescharacteristic
39#388—21no—
40#386—21no—
41#380C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43C4×C22⋊C4—21no—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C2×#41—28no—
48C2×#41—44no—
49C2×#41—44no—
50C22×Q8—21no—
51C22×D4—21no—
52C22×D4—22no—
53C22×D4—21no—
54C22×D4—21no—
55C22×D4C2312yes—
56C22×D4C2311yescharacteristic
57C23×C4—22no—
58C23×C4—21no—
59C23×C4D411yescharacteristic
60C23×C4D411yescharacteristic
61C23×C4C2311yescharacteristic
62C2×D8—24no—
63C2×D8—24no—
64C2×D8—42no—
65C22×C8C2312yes—
66C22×C8C2312yes—
67C4⋊Q8—22no—
68C4⋊Q8D414yescomplemented
69#127—22no—
70#127D414yescomplemented
71#124—28no—
72#121—28no—
73#121—24no—
74#121—22no—
75#121—24no—
76#121—22no—
77C4×D4—28no—
78C4×D4—24no—
79C4×D4—24no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—44no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—21no—
99C2×C42D411yescharacteristic
100C2×C42D411yescharacteristic
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#106—24no—
104C4⋊C8—22no—
105C4⋊C8—22no—
106#102—24no—
107#102—24no—
108#102—24no—
109#102—42no—
110C22⋊C8—42no—
111#95—21no—
112#95—21no—
113#95—21no—
114C24—21no—
115C24—22no—
116C24C2×D411yescharacteristic
117#41—432no—
118C2×Q8—24no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×D4—28no—
122C2×D4—24no—
123C2×D4—24no—
124C2×D4—24no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—22no—
132C2×D4—21no—
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—44no—
139C2×D4—42no—
140C2×D4—44no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4C2×D414yes—
144C2×D4#4112yescomplemented
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—28no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2411yescharacteristic
173D8—48no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—42no—
188C4⋊C4C2×D414yes—
189C4⋊C4#4112yescomplemented
190C22⋊C4—28no—
191C22⋊C4—28no—
192C22⋊C4—24no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C42—22no—
198C42—22no—
199C42—21no—
200C42—21no—
201C42—42no—
202C42C2×D412yes—
203C42C2×D412yes—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—24no—
211C23—22no—
212C23—41no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23—42no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23—41no—
223C23—42no—
224C23AGL(1,ℤ/8)12yes—
225C23C22×D411yescharacteristic
226Q8—48no—
227D4—22no—
228D4—24no—
229D4—24no—
230D4—22no—
231D4—48no—
232D4—48no—
233D4—416no—
234D4—41no—
235D4—42no—
236D4—41no—
237D4—42no—
238D4—48no—
239C2×C4—28no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—24no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—416no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—84no—
269C2×C4#12012yes—
270C2×C4#12012yes—
271C2×C4#13512yes—
272C2×C4AGL(1,ℤ/8)12yes—
273C2×C4AGL(1,ℤ/8)12yes—
274C2×C4C22×D411yescharacteristic
275C2×C4C22×D411yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4H(5,𝔽2)11yescharacteristic
279C2×C42−1+411yescharacteristic
280C8—42no—
281C8—42no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—44no—
293C22—44no—
294C22—44no—
295C22—82no—
296C22—82no—
297C22—81no—
298C22—82no—
299C22—81no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22C2×#12011yescharacteristic
304C22#52511yescharacteristic
305C22#53711yescharacteristic
306C22C2×#13511yescharacteristic
307C22C2×AGL(1,ℤ/8)11yescharacteristic
308C22C2×AGL(1,ℤ/8)11yescharacteristic
309C22C2×AGL(1,ℤ/8)11yescharacteristic
310C4—22no—
311C4—22no—
312C4—42no—
313C4—41no—
314C4—41no—
315C4—44no—
316C4—81no—
317C4—81no—
318C4#44914yes—
319C4#54611yescharacteristic
320C4#54711yescharacteristic
321C4#54811yescharacteristic
322C4#55011yescharacteristic
323C2—42no—
324C2—82no—
325C2—84no—
326C2#240711yescharacteristic
327C2C2×#44911yescharacteristic
328C2#300311yescharacteristic
329C2#310611yescharacteristic
330C2#310911yescharacteristic
331C2#324711yescharacteristic
332C2#325311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#532C2semidirect
2. C2×#476C2semidirect
3. C2×#459C2semidirect
4. C4×#121C2semidirect
5. #2034C2semidirect
6. #1999C2semidirect
7. #1987C2semidirect
8. #1982C2semidirect
9. #1879C2semidirect
10. #1864C2semidirect
11. #1797C2semidirect
68. C4⋊Q8D4semidirect
70. #127D4semidirect
144. C2×D4#41semidirect
189. C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#4762
C2×#4592
C4×#1212
#20342
#19992
#19872
#19822
#18792
#18642
#17972