Finite Group Databasebeta

← #28586

Subgroups — #28586

Order 256.

467 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic
3#2620C211yesmaximal, characteristic, complemented
4#2324C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2012C211yesmaximal, characteristic, complemented
9#1928C211yesmaximal, characteristic, complemented
10#1925C211yesmaximal, characteristic, complemented
11#1895C211yesmaximal, characteristic, complemented
12#1892C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1729C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#468—24no—
27#466—24no—
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38#396—21no—
39#396—21no—
40#394—21no—
41#394—21no—
42#391C2211yescharacteristic
43#390—21no—
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#386—21no—
49#385—21no—
50#384C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—41no—
71C2×D8—22no—
72C2×D8—22no—
73C2×D8—41no—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—41no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84#124—41no—
85#124—41no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88#121—22no—
89#121—22no—
90#121—22no—
91#121D414yes—
92C4×D4—41no—
93C4×D4—41no—
94#117—41no—
95#117—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4D411yescharacteristic
101C2×C4⋊C4D411yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4D411yescharacteristic
119C2×C22⋊C4D411yescharacteristic
120C2×C42—21no—
121C2×C42C2311yescharacteristic
122C2×C42C2311yescharacteristic
123#107—22no—
124#107—22no—
125#103—22no—
126#103—22no—
127#103—41no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—22no—
132#102—41no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24—21no—
140C24C2×D411yescharacteristic
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
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—21no—
159C2×D4—42no—
160C2×D4—81no—
161C2×D4—81no—
162C2×D4—81no—
163C2×D4—81no—
164C2×D4C2×D412yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—42no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D412yes—
184C22×C4#4112yes—
185C22×C4C2411yescharacteristic
186SD16—44no—
187D8—44no—
188M16—44no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—44no—
200C2×C8#4112yes—
201C2×C8#4112yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4C2×D412yes—
218C22⋊C4—24no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—81no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—22no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#12112yes—
257C23C22×D411yescharacteristic
258Q8—41no—
259Q8—41no—
260Q8—41no—
261Q8—41no—
262D4—22no—
263D4—22no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—84no—
271D4—84no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—84no—
309C2×C4#12112yes—
310C2×C4C22×D411yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C2×C4C2×#4111yescharacteristic
313C2×C4C2×#4111yescharacteristic
314C2×C4H(5,𝔽2)11yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C8—22no—
317C8—22no—
318C8—41no—
319C8—41no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22C2×#12111yescharacteristic
342C22#53211yescharacteristic
343C22#53411yescharacteristic
344C22#57611yescharacteristic
345C22#57611yescharacteristic
346C22#57711yescharacteristic
347C22#57811yescharacteristic
348C4—22no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—42no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4#54611yescharacteristic
360C4#54611yescharacteristic
361C4#55311yescharacteristic
362C4#55311yescharacteristic
363C2—42no—
364C2—82no—
365C2—161no—
366C2#254311yescharacteristic
367C2#304111yescharacteristic
368C2#304611yescharacteristic
369C2#317511yescharacteristic
370C2#317911yescharacteristic
371C2#319811yescharacteristic
372C2#320311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#466C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#468C2semidirect
3. #2620C2semidirect
4. #2324C2semidirect
5. #2083C2semidirect
6. #2071C2semidirect
7. #2042C2semidirect
8. #2012C2semidirect
9. #1928C2semidirect
10. #1925C2semidirect
11. #1895C2semidirect
12. #1892C2semidirect
13. #1879C2semidirect
14. #1877C2semidirect
15. #1729C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×#4662
#26202
#23242
#20832
#20712
#20422
#20122
#19282
#19252
#18952
#18922
#18792
#18772
#17292