Finite Group Databasebeta

← #26214

Subgroups — #26214

Order 256.

445 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2414C211yesmaximal, characteristic
2D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#2100C211yesmaximal, characteristic, complemented
4#2077C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#1926C211yesmaximal, characteristic, complemented
8#1921C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1874C211yesmaximal, characteristic, complemented
11#1874C211yesmaximal, characteristic, complemented
12#1870C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1823C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×D4—21no—
20C2×C4×D4C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33#394—21no—
34#392—21no—
35#390—21no—
36#390C2211yescharacteristic
37#389—21no—
38#389C2211yescharacteristic
39#388—21no—
40#385—21no—
41#385—21no—
42#384C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4—21no—
61C22×D4D411yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C2×M16—22no—
68C2×M16—41no—
69C2×M16—41no—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74#121—22no—
75#121—22no—
76#121—22no—
77#121—41no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—44no—
81#117—22no—
82#117—41no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4D411yescharacteristic
109C2×C22⋊C4D411yescharacteristic
110C2×C42—21no—
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113#107—22no—
114#106—41no—
115#102—22no—
116#102—41no—
117#102—41no—
118#102—41no—
119C22⋊C8—22no—
120C22⋊C8—41no—
121C22⋊C8—41no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127C24—21no—
128C24C2×D411yescharacteristic
129C24C2×D411yescharacteristic
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—81no—
143C2×D4—81no—
144C2×D4—81no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
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—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4#4112yes—
173C22×C4#4112yes—
174C22×C4C2411yescharacteristic
175M16—44no—
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—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C4⋊C4#4112yes—
207C4⋊C4#4112yes—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—81no—
217C22⋊C4—81no—
218C42—21no—
219C42—21no—
220C42—42no—
221C42—42no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23#12312yes—
242C23AGL(1,ℤ/8)12yes—
243C23C22×D411yescharacteristic
244D4—28no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—84no—
250D4—84no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
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—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4#12312yes—
293C2×C4#13712yes—
294C2×C4C2×#4111yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C2×C4C2×#4111yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4H(5,𝔽2)11yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C8—41no—
301C8—41no—
302C8—41no—
303C8—41no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C2×#12311yescharacteristic
326C22#54611yescharacteristic
327C22#54611yescharacteristic
328C22C2×AGL(1,ℤ/8)11yescharacteristic
329C22C2×#13711yescharacteristic
330C22#57611yescharacteristic
331C22#57711yescharacteristic
332C4—22no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—42no—
337C4—42no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4#52911yescharacteristic
344C4#53811yescharacteristic
345C4#55311yescharacteristic
346C4#55311yescharacteristic
347C2—42no—
348C2—42no—
349C2—161no—
350C2#244411yescharacteristic
351C2#309211yescharacteristic
352C2#309611yescharacteristic
353C2#330611yescharacteristic
354C2#331811yescharacteristic
355C2#332911yescharacteristic
356C2#333311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2414C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×C4⋊C4C2semidirect
3. #2100C2semidirect
4. #2077C2semidirect
5. #2071C2semidirect
6. #2033C2semidirect
7. #1926C2semidirect
8. #1921C2semidirect
9. #1895C2semidirect
10. #1874C2semidirect
11. #1874C2semidirect
12. #1870C2semidirect
13. #1854C2semidirect
14. #1823C2semidirect
15. #1736C2semidirect

Maximal subgroups

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

subgroupindex
#24142
D4×C4⋊C42
#21002
#20772
#20712
#20332
#19262
#19212
#18952
#18742
#18702
#18542
#18232
#17362