Finite Group Databasebeta

← #26939

Subgroups — #26939

Order 256.

516 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#459C211yesmaximal, characteristic, complemented
3#2428C211yesmaximal, characteristic, complemented
4#2363C211yesmaximal, characteristic, complemented
5#2087C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#1999C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1970C211yesmaximal, characteristic, complemented
10#1926C211yesmaximal, characteristic, complemented
11#1923C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#482—24no—
27#459—24no—
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#394—21no—
43#392—21no—
44#390—21no—
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#388—21no—
50#386—21no—
51#386—21no—
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C2×D8—22no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—22no—
76C2×D8—41no—
77C2×D8—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82#127—22no—
83#127—41no—
84#121—22no—
85#121—22no—
86#121—22no—
87#121—22no—
88#121—44no—
89#121—41no—
90#121—41no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—41no—
94C4×D4—41no—
95#117—22no—
96#117—41no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
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×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4D411yescharacteristic
115C2×C22⋊C4D411yescharacteristic
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120#107—22no—
121#106—22no—
122C4⋊C8—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—41no—
129#102—41no—
130#102D414yescomplemented
131C22⋊C8—22no—
132C22⋊C8—41no—
133C4×C8—41no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
141C24—21no—
142C24C2×D411yescharacteristic
143C2×D4—24no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
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—21no—
159C2×D4—21no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4C2×D412yes—
175C2×D4#4112yes—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2411yescharacteristic
199D8—44no—
200D8—44no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8C2×D412yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—42no—
221C4⋊C4—81no—
222C4⋊C4C2×D412yes—
223C4⋊C4#4112yes—
224C4⋊C4#4112yes—
225C22⋊C4—24no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—42no—
242C42—81no—
243C23—22no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23C22×D411yescharacteristic
272D4—22no—
273D4—22no—
274D4—22no—
275D4—22no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—41no—
284D4—84no—
285D4—84no—
286D4—84no—
287D4—84no—
288D4—84no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4#13512yes—
324C2×C4AGL(1,ℤ/8)12yes—
325C2×C4C22×D411yescharacteristic
326C2×C4C2×#4111yescharacteristic
327C2×C4C2×#4111yescharacteristic
328C2×C4C2×#4111yescharacteristic
329C2×C4H(5,𝔽2)11yescharacteristic
330C2×C4H(5,𝔽2)11yescharacteristic
331C8—22no—
332C8—41no—
333C8—41no—
334C8—41no—
335C22—41no—
336C22—42no—
337C22—41no—
338C22—42no—
339C22—41no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22#52511yescharacteristic
364C22D4211yescharacteristic
365C22#54611yescharacteristic
366C22C2×#13511yescharacteristic
367C22C2×AGL(1,ℤ/8)11yescharacteristic
368C22#57611yescharacteristic
369C22#57611yescharacteristic
370C4—22no—
371C4—22no—
372C4—42no—
373C4—41no—
374C4—41no—
375C4—42no—
376C4—41no—
377C4—42no—
378C4—81no—
379C4—81no—
380C4#53211yescharacteristic
381C4#54011yescharacteristic
382C4#54611yescharacteristic
383C4#55311yescharacteristic
384C2—81no—
385C2—82no—
386C2—82no—
387C2—161no—
388C2#246411yescharacteristic
389C2#310611yescharacteristic
390C2#312211yescharacteristic
391C2#327411yescharacteristic
392C2#328011yescharacteristic
393C2#330511yescharacteristic
394C2#331811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×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
1. C2×#482C2semidirect
2. C2×#459C2semidirect
3. #2428C2semidirect
4. #2363C2semidirect
5. #2087C2semidirect
6. #2041C2semidirect
7. #1999C2semidirect
8. #1999C2semidirect
9. #1970C2semidirect
10. #1926C2semidirect
11. #1923C2semidirect
12. #1879C2semidirect
13. #1875C2semidirect
14. #1864C2semidirect
15. C4×#102C2semidirect
130. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4592
#24282
#23632
#20872
#20412
#19992
#19702
#19262
#19232
#18792
#18752
#18642
C4×#1022