Finite Group Databasebeta

← #32096

Subgroups — #32096

Order 256.

432 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2541C211yesmaximal, characteristic
2#2508C211yesmaximal, characteristic, complemented
3#2092C211yesmaximal, characteristic, complemented
4#2065C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2041C211yesmaximal, characteristic, complemented
9#1982C211yesmaximal, characteristic, complemented
10#1961C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1917C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic, complemented
16C2×#127C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#121—21no—
20C2×#121—21no—
21C2×C4×D4C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#97C2211yescharacteristic
38#400—21no—
39#398—21no—
40#392—21no—
41#392—21no—
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#387—21no—
49#382—21no—
50#382C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4—21no—
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73#127—22no—
74#127—22no—
75#127—41no—
76#127—41no—
77#127—41no—
78#127—41no—
79#121—44no—
80#121—44no—
81C4×D4—41no—
82C4×D4—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107—41no—
104#106—41no—
105C4⋊C8—41no—
106#102—22no—
107#102—22no—
108#102—22no—
109#102—22no—
110#102—41no—
111#102—41no—
112#102—41no—
113#102—41no—
114#102—41no—
115#102—41no—
116#102—41no—
117#102—41no—
118#97—41no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95D411yescharacteristic
127#95D411yescharacteristic
128C24—21no—
129C24—21no—
130C24—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—81no—
161C2×D4—81no—
162C2×D4—81no—
163C2×D4—81no—
164C2×D4—81no—
165C2×D4—81no—
166C2×D4#4112yes—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2411yescharacteristic
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—42no—
206C4⋊C4—81no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C4⋊C4#4112yes—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—42no—
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—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23C22×D411yescharacteristic
244D4—22no—
245D4—22no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—84no—
257D4—84no—
258D4—84no—
259D4—84no—
260D4—84no—
261D4—84no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
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—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—84no—
294C2×C4AGL(1,ℤ/8)12yes—
295C2×C4AGL(1,ℤ/8)12yes—
296C2×C4C2×#4111yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4H(5,𝔽2)11yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C2×C4H(5,𝔽2)11yescharacteristic
301C2×C42−1+411yescharacteristic
302C8—41no—
303C8—41no—
304C8—41no—
305C8—41no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22#53511yescharacteristic
328C22#54611yescharacteristic
329C22#54811yescharacteristic
330C22C2×AGL(1,ℤ/8)11yescharacteristic
331C22C2×AGL(1,ℤ/8)11yescharacteristic
332C22#57611yescharacteristic
333C22#57711yescharacteristic
334C4—22no—
335C4—22no—
336C4—42no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4#53911yescharacteristic
346C4#54011yescharacteristic
347C4#55311yescharacteristic
348C4#56211yescharacteristic
349C2—82no—
350C2—161no—
351C2—161no—
352C2#271111yescharacteristic
353C2#321711yescharacteristic
354C2#322411yescharacteristic
355C2#331011yescharacteristic
356C2#332511yescharacteristic
357C2#336611yescharacteristic
358C2#336811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2508C2semidirect
3. #2092C2semidirect
4. #2065C2semidirect
5. #2041C2semidirect
6. #2041C2semidirect
7. #2041C2semidirect
8. #2041C2semidirect
9. #1982C2semidirect
10. #1961C2semidirect
11. #1936C2semidirect
12. #1925C2semidirect
13. #1917C2semidirect
14. #1864C2semidirect
15. #1862C2semidirect

Maximal subgroups

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

subgroupindex
#25412
#25082
#20922
#20652
#20412
#19822
#19612
#19362
#19252
#19172
#18642
#18622