Finite Group Databasebeta

← #23680

Subgroups — #23680

Order 256.

478 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×SD16C211yesmaximal, characteristic, complemented
2C2×C4×D8C211yesmaximal, characteristic
3C4×#124C211yesmaximal, characteristic, complemented
4#2293C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2073C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#2055C211yesmaximal, characteristic, complemented
10#2054C211yesmaximal, characteristic
11#2042C211yesmaximal, characteristic, complemented
12#2012C211yesmaximal, characteristic
13C4×C4⋊C8C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117—21no—
23C4×SD16—24no—
24C4×D8—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#391C2211yescharacteristic
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#388—21no—
47#388—21no—
48#388—21no—
49#382C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53C4×C22⋊C4—21no—
54C43C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C2×SD16—22no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×D8—22no—
70C2×D8—22no—
71C2×D8—22no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76#124—22no—
77#124—22no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4D414yes—
84#117—44no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42D411yescharacteristic
106C2×C42D411yescharacteristic
107C2×C42D411yescharacteristic
108C2×C42D411yescharacteristic
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#107—22no—
113#106—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116#103—22no—
117#103—22no—
118#103—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124C4×C8—22no—
125C4×C8—22no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24—21no—
134C24—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8#4112yes—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4C2×D412yes—
152C2×D4#4112yes—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—24no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172SD16—28no—
173D8—28no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8#4112yes—
185C2×C8#4112yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4C2×D412yes—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—24no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C42—22no—
210C42—22no—
211C42—22no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—42no—
218C42—42no—
219C42C2×D412yes—
220C42#4112yes—
221C23—22no—
222C23—22no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23C22×D411yescharacteristic
236Q8—22no—
237Q8—22no—
238Q8—41no—
239Q8—41no—
240D4—22no—
241D4—22no—
242D4—22no—
243D4—22no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—48no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—84no—
281C2×C4#12112yes—
282C2×C4#12112yes—
283C2×C4#13512yes—
284C2×C4#13512yes—
285C2×C4AGL(1,ℤ/8)12yes—
286C2×C4C22×D411yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C8—22no—
293C8—22no—
294C8—41no—
295C8—41no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22C2×#12111yescharacteristic
311C22#52511yescharacteristic
312C22#53211yescharacteristic
313C22C2×#13511yescharacteristic
314C22C2×#13511yescharacteristic
315C22C2×AGL(1,ℤ/8)11yescharacteristic
316C22#57511yescharacteristic
317C4—22no—
318C4—22no—
319C4—41no—
320C4—41no—
321C4—42no—
322C4—41no—
323C4—42no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4#46314yes—
329C4#52511yescharacteristic
330C4#52511yescharacteristic
331C4#52911yescharacteristic
332C4#52911yescharacteristic
333C2—82no—
334C2—82no—
335C2#239211yescharacteristic
336C2C2×#46311yescharacteristic
337C2#303511yescharacteristic
338C2#310211yescharacteristic
339C2#310611yescharacteristic
340C2#316811yescharacteristic
341C2#317111yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutator, Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×SD16C2semidirect
3. C4×#124C2semidirect
4. #2293C2semidirect
5. #2085C2semidirect
6. #2073C2semidirect
7. #2068C2semidirect
8. #2065C2semidirect
9. #2055C2semidirect
11. #2042C2semidirect
13. C4×C4⋊C8C2semidirect
14. C4×#103C2semidirect
15. C4×#102C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×SD162
C2×C4×D82
C4×#1242
#22932
#20852
#20732
#20682
#20652
#20552
#20542
#20422
#20122
C4×C4⋊C82
C4×#1032
C4×#1022