Finite Group Databasebeta

← #23010

Subgroups — #23010

Order 256.

523 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#460C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic, complemented
3C2×C4×SD16C211yesmaximal, characteristic, complemented
4#2366C211yesmaximal, characteristic
5#2351C211yesmaximal, characteristic, complemented
6#1961C211yesmaximal, characteristic
7#1949C211yesmaximal, characteristic, complemented
8#1936C211yesmaximal, characteristic, complemented
9#1934C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1771C211yesmaximal, characteristic, complemented
15#1763C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25#460C418yescomplemented
26#440—24no—
27C4×SD16—24no—
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×#97C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#390—21no—
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389—21no—
46#389—21no—
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#386—21no—
50#386—21no—
51#382—21no—
52#382C2211yescharacteristic
53C4×C4⋊C4—21no—
54C4×C4⋊C4C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4—21no—
65C22×D4C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16C2×C414yescomplemented
74C2×SD16C2×C414yescomplemented
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79#127—22no—
80#127C2×C414yes—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×Q8—22no—
84C4×Q8—41no—
85C4×Q8—41no—
86C4×Q8C2×C414yescomplemented
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—41no—
90C4×D4—41no—
91C4×D4—44no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4D411yescharacteristic
98C2×C4⋊C4D411yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#107—22no—
118#106—22no—
119C4⋊C8C2×C414yescomplemented
120#103—22no—
121#103D414yescomplemented
122#102—22no—
123#102—22no—
124#102—41no—
125#102—41no—
126#102C2×C414yescomplemented
127#102C2×C414yescomplemented
128#97—22no—
129#97—22no—
130C4×C8—22no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8C22×C412yes—
144C2×Q8C2×D412yes—
145C2×Q8#4112yes—
146C2×D4—24no—
147C2×D4—24no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—81no—
161C2×D4—81no—
162C2×D4C22×C412yes—
163C2×D4C22×C412yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2411yescharacteristic
183SD16—28no—
184SD16—28no—
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—
194C2×C8C22×C412yes—
195C2×C8C22×C412yes—
196C2×C8C2×D412yes—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4C22×C412yes—
213C4⋊C4C2×D412yes—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—81no—
219C22⋊C4—81no—
220C42—24no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—42no—
226C42—42no—
227C42—42no—
228C42C22×C412yes—
229C42#4112yes—
230C23—22no—
231C23—22no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23C22×D411yescharacteristic
245Q8—22no—
246Q8—22no—
247Q8—22no—
248Q8C4×D414yescomplemented
249D4—22no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—48no—
254D4—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—84no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
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—84no—
293C2×C4—84no—
294C2×C4C4×D412yes—
295C2×C4C4×D412yes—
296C2×C4AGL(1,ℤ/8)12yes—
297C2×C4C23×C411yescharacteristic
298C2×C4C22×D411yescharacteristic
299C2×C4C2×#4111yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C2×C4H(5,𝔽2)11yescharacteristic
302C2×C42−1+411yescharacteristic
303C8—22no—
304C8—22no—
305C8—22no—
306C8—41no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C2×C4×D411yescharacteristic
322C22D4211yescharacteristic
323C22#54811yescharacteristic
324C22C2×AGL(1,ℤ/8)11yescharacteristic
325C22#57511yescharacteristic
326C22#57711yescharacteristic
327C22#57711yescharacteristic
328C4—24no—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4C2×C4×D411yescharacteristic
341C4#52011yescharacteristic
342C4#54711yescharacteristic
343C4#55311yescharacteristic
344C2—82no—
345C2—82no—
346C2#236111yescharacteristic
347C2C4×AGL(1,ℤ/8)11yescharacteristic
348C2#294911yescharacteristic
349C2#326511yescharacteristic
350C2#327111yescharacteristic
351C2#336811yescharacteristic
352C2#337111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C2×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. C2×#440C2semidirect
3. C2×C4×SD16C2semidirect
5. #2351C2semidirect
7. #1949C2semidirect
8. #1936C2semidirect
9. #1934C2semidirect
10. #1880C2semidirect
11. #1880C2semidirect
12. #1864C2semidirect
13. #1862C2semidirect
14. #1771C2semidirect
15. #1763C2semidirect
25. #460C4semidirect
73. C2×SD16C2×C4semidirect
74. C2×SD16C2×C4semidirect
86. C4×Q8C2×C4semidirect
119. C4⋊C8C2×C4semidirect
121. #103D4semidirect
126. #102C2×C4semidirect
127. #102C2×C4semidirect
248. Q8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4602
C2×#4402
C2×C4×SD162
#23662
#23512
#19612
#19492
#19362
#19342
#18802
#18642
#18622
#17712
#17632