Finite Group Databasebeta

← #25506

Subgroups — #25506

Order 256.

601 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#529C211yesmaximal, characteristic, complemented
2C2×#469C211yesmaximal, characteristic
3C2×#468C211yesmaximal, characteristic, complemented
4#2854C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic, complemented
8#2012C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1894C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1865C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1728C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20#529—24no—
21#529—24no—
22C2×#126—21no—
23C2×#125C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#123—21no—
26C2×#123—21no—
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×D4—21no—
32C2×C4×D4C2211yescharacteristic
33C2×#117—21no—
34C22×C42C2211yescharacteristic
35#469—24no—
36#468—24no—
37C2×#107C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#388—21no—
52#388—21no—
53#386—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4D411yescharacteristic
70C23×C4D411yescharacteristic
71C23×C4C2311yescharacteristic
72C2×SD16—22no—
73C2×SD16—22no—
74C2×SD16—41no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—41no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—41no—
83C2×M16—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88#126—28no—
89#125—22no—
90#125—22no—
91#124—22no—
92#124—22no—
93#123—28no—
94#123—44no—
95C22⋊Q8—22no—
96C22⋊Q8D414yes—
97#121—22no—
98#121—22no—
99#121—22no—
100#121D414yes—
101C4×Q8—22no—
102C4×Q8—22no—
103C4×D4—28no—
104C4×D4—22no—
105C4×D4—22no—
106#117—28no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C42—21no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42D411yescharacteristic
133C2×C42D411yescharacteristic
134C2×C42C2311yescharacteristic
135C2×C42C2311yescharacteristic
136#107—22no—
137#106—22no—
138#103—22no—
139#103—22no—
140#103—41no—
141#102—22no—
142#102—22no—
143#102—22no—
144#102—22no—
145#102—41no—
146#95—21no—
147#95—21no—
148#95—21no—
149C24—21no—
150C24—21no—
151C24C2×D411yescharacteristic
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8C2×D412yes—
158C2×D4—24no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4C2×D412yes—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D412yes—
200C22×C4C2×D412yes—
201C22×C4C2×D411yescharacteristic
202C22×C4C2411yescharacteristic
203SD16—44no—
204D8—44no—
205M16—44no—
206M16—44no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8#4112yes—
218C2×C8#4112yes—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4C2×D412yes—
234C4⋊C4C2×D412yes—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—24no—
247C42—22no—
248C42—22no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23#12012yes—
274C23C22×D411yescharacteristic
275Q8—22no—
276Q8—22no—
277Q8—41no—
278Q8—41no—
279D4—22no—
280D4—22no—
281D4—48no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—84no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—44no—
310C2×C4—48no—
311C2×C4—44no—
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—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4#12012yes—
328C2×C4AGL(1,ℤ/8)12yes—
329C2×C4C22×D411yescharacteristic
330C2×C4C22×D411yescharacteristic
331C2×C4C2×#4111yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C2×C42−1+411yescharacteristic
335C8—22no—
336C8—22no—
337C8—41no—
338C8—41no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22C2×#12011yescharacteristic
361C22#53211yescharacteristic
362C22#53711yescharacteristic
363C22C2×AGL(1,ℤ/8)11yescharacteristic
364C22#57511yescharacteristic
365C22#57511yescharacteristic
366C22#57511yescharacteristic
367C4—22no—
368C4—22no—
369C4—42no—
370C4—41no—
371C4—41no—
372C4—42no—
373C4—41no—
374C4—42no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4#54611yescharacteristic
379C4#54611yescharacteristic
380C4#54711yescharacteristic
381C4#54711yescharacteristic
382C2—42no—
383C2—82no—
384C2—161no—
385C2#243011yescharacteristic
386C2#300511yescharacteristic
387C2#300611yescharacteristic
388C2#317111yescharacteristic
389C2#318311yescharacteristic
390C2#325111yescharacteristic
391C2#325611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#469C2
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
1. C2×#529C2semidirect
3. C2×#468C2semidirect
4. #2854C2semidirect
5. #2059C2semidirect
6. #2042C2semidirect
7. #2025C2semidirect
8. #2012C2semidirect
9. #1895C2semidirect
10. #1894C2semidirect
11. #1880C2semidirect
12. #1879C2semidirect
13. #1865C2semidirect
14. #1862C2semidirect
15. #1728C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4692
C2×#4682
#28542
#20592
#20422
#20252
#20122
#18952
#18942
#18802
#18792
#18652
#18622
#17282