Finite Group Databasebeta

← #23143

Subgroups — #23143

Order 256.

505 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#469C211yesmaximal, characteristic
2#2363C211yesmaximal, characteristic, complemented
3#2361C211yesmaximal, characteristic, complemented
4#1936C211yesmaximal, characteristic, complemented
5#1934C211yesmaximal, characteristic, complemented
6#1920C211yesmaximal, characteristic, complemented
7#1920C211yesmaximal, characteristic, complemented
8#1879C211yesmaximal, characteristic, complemented
9#1879C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1874C211yesmaximal, characteristic, complemented
12#1762C211yesmaximal, characteristic, complemented
13C4×#102C211yesmaximal, characteristic, complemented
14#1753C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#469C418yescomplemented
26C2×#106C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#390—21no—
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#388—21no—
43#386—21no—
44#386—21no—
45#385—21no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4—21no—
49C4×C22⋊C4C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×D4—21no—
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×D8—41no—
69C2×D8—41no—
70C2×D8C2×C414yescomplemented
71C2×M16—41no—
72C2×M16—41no—
73C2×M16C2×C414yescomplemented
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#121—22no—
79#121—22no—
80#121C2×C414yescomplemented
81#121C2×C414yescomplemented
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—41no—
85C4×D4—41no—
86C4×D4—44no—
87#117—22no—
88#117—41no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—41no—
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⋊C4—21no—
109C2×C22⋊C4D411yescharacteristic
110C2×C22⋊C4D411yescharacteristic
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116#106C2×C414yescomplemented
117#102—22no—
118#102—22no—
119#102—41no—
120#102—41no—
121#102C2×C414yescomplemented
122#102C2×C414yescomplemented
123C22⋊C8—22no—
124#97—41no—
125C4×C8—41no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C24—21no—
134C24C2×D411yescharacteristic
135C2×D4—24no—
136C2×D4—24no—
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—42no—
148C2×D4—42no—
149C2×D4—81no—
150C2×D4—81no—
151C2×D4C22×C412yes—
152C2×D4C22×C412yes—
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—21no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4C22×C412yes—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4#4112yes—
179C22×C4C2411yescharacteristic
180D8—44no—
181M16—44no—
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×C8C22×C412yes—
193C2×C8C22×C412yes—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—81no—
205C4⋊C4C22×C412yes—
206C4⋊C4C22×C412yes—
207C4⋊C4#4112yes—
208C4⋊C4#4112yes—
209C22⋊C4—24no—
210C22⋊C4—24no—
211C22⋊C4—22no—
212C22⋊C4—41no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—81no—
219C22⋊C4—81no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—42no—
225C42—42no—
226C42—81no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—22no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C4×D412yes—
248C23C22×D411yescharacteristic
249D4—22no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—48no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—84no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
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—41no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4C4×D412yes—
298C2×C4AGL(1,ℤ/8)12yes—
299C2×C4C23×C411yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C8—22no—
306C8—22no—
307C8—41no—
308C8—41no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22C2×C4×D411yescharacteristic
331C22#54611yescharacteristic
332C22#54611yescharacteristic
333C22C2×AGL(1,ℤ/8)11yescharacteristic
334C22#57511yescharacteristic
335C22#57611yescharacteristic
336C22#57711yescharacteristic
337C4—22no—
338C4—22no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—42no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4C4×#4111yescharacteristic
349C4#51811yescharacteristic
350C4#55311yescharacteristic
351C4#55311yescharacteristic
352C2—42no—
353C2—82no—
354C2—82no—
355C2#236411yescharacteristic
356C2C4×AGL(1,ℤ/8)11yescharacteristic
357C2#294611yescharacteristic
358C2#330611yescharacteristic
359C2#330911yescharacteristic
360C2#332511yescharacteristic
361C2#332611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#469C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2363C2semidirect
3. #2361C2semidirect
4. #1936C2semidirect
5. #1934C2semidirect
6. #1920C2semidirect
7. #1920C2semidirect
8. #1879C2semidirect
9. #1879C2semidirect
10. #1875C2semidirect
11. #1874C2semidirect
12. #1762C2semidirect
13. C4×#102C2semidirect
14. #1753C2semidirect
15. #1736C2semidirect
25. #469C4semidirect
70. C2×D8C2×C4semidirect
73. C2×M16C2×C4semidirect
80. #121C2×C4semidirect
81. #121C2×C4semidirect
116. #106C2×C4semidirect
121. #102C2×C4semidirect
122. #102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4692
#23632
#23612
#19362
#19342
#19202
#18792
#18752
#18742
#17622
C4×#1022
#17532
#17362