Finite Group Databasebeta

← #27282

Subgroups — #27282

Order 256.

528 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic, complemented
2C2×#417C211yesmaximal, characteristic, complemented
3#2440C211yesmaximal, characteristic, complemented
4#2431C211yesmaximal, characteristic
5#2072C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1957C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic, complemented
11#1933C211yesmaximal, characteristic, complemented
12#1895C211yesmaximal, characteristic, complemented
13#1874C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8—21no—
22C2×#121C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#489—24no—
25#417—24no—
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#398—21no—
42#395—21no—
43#392—21no—
44#392—21no—
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#386—21no—
50#386—21no—
51#385—21no—
52#384C2211yescharacteristic
53#380C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8—21no—
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—21no—
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C2311yescharacteristic
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C4⋊Q8—22no—
86C4⋊Q8—41no—
87#127—22no—
88#127—41no—
89#124—22no—
90#124—41no—
91C22⋊Q8—44no—
92#121—22no—
93#121—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
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⋊C4D411yescharacteristic
114C2×C22⋊C4D411yescharacteristic
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#103—22no—
118#103—22no—
119#103—41no—
120#103D414yescomplemented
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—41no—
127#102—41no—
128#102—41no—
129C22⋊C8—22no—
130#97—22no—
131C4×C8—41no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
141C24C2×D411yescharacteristic
142#41—28no—
143#41—44no—
144#41—44no—
145#41—44no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—81no—
154C2×Q8C2×D412yes—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—81no—
175C2×D4—81no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4#4112yes—
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—21no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—81no—
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4#4112yes—
201C22×C4#4112yes—
202C22×C4C2411yescharacteristic
203M16—44no—
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×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8C2×D412yes—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—81no—
231C4⋊C4C2×D412yes—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—41no—
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#12412yes—
265C23C22×D411yescharacteristic
266Q8—22no—
267Q8—22no—
268Q8—41no—
269Q8—41no—
270Q8—84no—
271D4—22no—
272D4—22no—
273D4—48no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—84no—
281D4—84no—
282D4—84no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—48no—
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—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—84no—
319C2×C4#12412yes—
320C2×C4AGL(1,ℤ/8)12yes—
321C2×C4#13712yes—
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4C2×#4111yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C2×C4H(5,𝔽2)11yescharacteristic
328C8—22no—
329C8—41no—
330C8—41no—
331C8—41no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22C2×#12411yescharacteristic
354C22D4211yescharacteristic
355C22#54611yescharacteristic
356C22C2×AGL(1,ℤ/8)11yescharacteristic
357C22C2×#13711yescharacteristic
358C22#57611yescharacteristic
359C22#57711yescharacteristic
360C4—22no—
361C4—22no—
362C4—22no—
363C4—42no—
364C4—41no—
365C4—42no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4#52511yescharacteristic
372C4#53811yescharacteristic
373C4#54611yescharacteristic
374C4#55111yescharacteristic
375C2—42no—
376C2—82no—
377C2—161no—
378C2#247211yescharacteristic
379C2#313611yescharacteristic
380C2#314111yescharacteristic
381C2#327411yescharacteristic
382C2#327811yescharacteristic
383C2#330611yescharacteristic
384C2#331411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2431C2
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×#489C2semidirect
2. C2×#417C2semidirect
3. #2440C2semidirect
5. #2072C2semidirect
6. #2042C2semidirect
7. #2041C2semidirect
8. #2023C2semidirect
9. #1957C2semidirect
10. #1936C2semidirect
11. #1933C2semidirect
12. #1895C2semidirect
13. #1874C2semidirect
14. #1788C2semidirect
15. #1753C2semidirect
120. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4172
#24402
#24312
#20722
#20422
#20412
#20232
#19572
#19362
#19332
#18952
#18742
#17882
#17532