Finite Group Databasebeta

← #28573

Subgroups — #28573

Order 256.

513 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#466C211yesmaximal, characteristic
2C2×#465C211yesmaximal, characteristic, complemented
3#2850C211yesmaximal, characteristic, complemented
4C4×#121C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2073C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2012C211yesmaximal, characteristic, complemented
9#1928C211yesmaximal, characteristic, complemented
10#1926C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1892C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15C2×#336C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4C2211yescharacteristic
26C22×C42C2211yescharacteristic
27#466—24no—
28#465—24no—
29C2×#107C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39#391C2211yescharacteristic
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#388—21no—
44#388—21no—
45#388—21no—
46#386—21no—
47#386—21no—
48#386—21no—
49#382C2211yescharacteristic
50#380C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4—21no—
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×C4D411yescharacteristic
68C23×C4D411yescharacteristic
69C23×C4C2311yescharacteristic
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—41no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—41no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86#124—41no—
87#124—41no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90#121—22no—
91#121—22no—
92#121—22no—
93#121D414yes—
94C4×D4—41no—
95C4×D4—41no—
96C4×D4—44no—
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⋊C4—21no—
115C2×C42—21no—
116C2×C42—41no—
117C2×C42—41no—
118C2×C42D411yescharacteristic
119C2×C42D411yescharacteristic
120C2×C42C2311yescharacteristic
121C2×C42C2311yescharacteristic
122#107—22no—
123#106—22no—
124#103—22no—
125#103—22no—
126#103—41no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—41no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24—21no—
140C24—21no—
141C24C2×D411yescharacteristic
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×D4—24no—
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×D4C2×D412yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D412yes—
190C22×C4C2×D411yescharacteristic
191C22×C4#4112yes—
192C22×C4C2411yescharacteristic
193SD16—44no—
194D8—44no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—44no—
206C2×C8—44no—
207C2×C8#4112yes—
208C2×C8#4112yes—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4C2×D412yes—
221C22⋊C4—24no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—41no—
237C42—41no—
238C42—42no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—22no—
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—41no—
257C23#12112yes—
258C23#13512yes—
259C23AGL(1,ℤ/8)12yes—
260C23C22×D411yescharacteristic
261Q8—41no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265D4—22no—
266D4—22no—
267D4—48no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—84no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—44no—
294C2×C4—44no—
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—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—84no—
313C2×C4#12112yes—
314C2×C4#13512yes—
315C2×C4C22×D411yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4H(5,𝔽2)11yescharacteristic
320C2×C4H(5,𝔽2)11yescharacteristic
321C8—22no—
322C8—22no—
323C8—41no—
324C8—41no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22#46314yes—
346C22C2×#12111yescharacteristic
347C22#53211yescharacteristic
348C22#53411yescharacteristic
349C22C2×#13511yescharacteristic
350C22C2×#13511yescharacteristic
351C22C2×AGL(1,ℤ/8)11yescharacteristic
352C22#57511yescharacteristic
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—42no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4#54611yescharacteristic
365C4#54611yescharacteristic
366C4#55311yescharacteristic
367C4#55311yescharacteristic
368C2—24no—
369C2—82no—
370C2—161no—
371C2#254311yescharacteristic
372C2C2×#46311yescharacteristic
373C2#303511yescharacteristic
374C2#316811yescharacteristic
375C2#318011yescharacteristic
376C2#318811yescharacteristic
377C2#319111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#466C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#465C2semidirect
3. #2850C2semidirect
4. C4×#121C2semidirect
5. #2085C2semidirect
6. #2073C2semidirect
7. #2042C2semidirect
8. #2012C2semidirect
9. #1928C2semidirect
10. #1926C2semidirect
11. #1893C2semidirect
12. #1892C2semidirect
13. #1879C2semidirect
14. #1877C2semidirect
15. C2×#336C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4662
C2×#4652
#28502
C4×#1212
#20852
#20732
#20422
#20122
#19282
#19262
#18932
#18922
#18792
#18772
C2×#3362