Finite Group Databasebeta

← #24205

Subgroups — #24205

Order 256.

572 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic
3C2×#440C211yesmaximal, characteristic, complemented
4#2538C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2013C211yesmaximal, characteristic, complemented
8#2012C211yesmaximal, characteristic
9#1985C211yesmaximal, characteristic, complemented
10#1959C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic
13#1863C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1803C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19#556—24no—
20#556—24no—
21C2×C4⋊Q8C2211yescharacteristic
22C2×#126—21no—
23C2×#126—21no—
24C2×#124C2211yescharacteristic
25C2×#124C2211yescharacteristic
26C2×C22⋊Q8—21no—
27C2×C22⋊Q8—21no—
28C2×C4×Q8C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33C2×#117—21no—
34#463—24no—
35#440—24no—
36C2×#107C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#97C2211yescharacteristic
49#396—21no—
50#396—21no—
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#389C2211yescharacteristic
54#389C2211yescharacteristic
55#384C2211yescharacteristic
56#380—21no—
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8D411yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—41no—
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85#126—28no—
86#126—28no—
87#124—22no—
88#124—22no—
89#124—22no—
90#124D414yes—
91C22⋊Q8—28no—
92C22⋊Q8—44no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8D414yes—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103#117—28no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4D411yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
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—21no—
131C2×C42C2311yescharacteristic
132C2×C42C2311yescharacteristic
133C2×C42C2311yescharacteristic
134#107—22no—
135C4⋊C8—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#103—41no—
140#103—41no—
141#103—41no—
142#102—22no—
143#102—22no—
144#102—22no—
145#102—41no—
146#97—22no—
147#95—21no—
148#95—21no—
149C24—21no—
150C24—21no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—41no—
164C2×Q8C2×D412yes—
165C2×Q8C2×D412yes—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4C2×D412yes—
177C2×D4#4112yes—
178C22×C4—24no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2411yescharacteristic
197SD16—28no—
198SD16—44no—
199D8—44no—
200C2×C8—21no—
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#4112yes—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—81no—
230C4⋊C4C2×D412yes—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—24no—
235C22⋊C4—24no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C42—24no—
242C42—24no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42C2×D412yes—
248C42C2×D412yes—
249C23—22no—
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—
263C23C22×D411yescharacteristic
264Q8—22no—
265Q8—22no—
266Q8—22no—
267Q8—22no—
268Q8—44no—
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273D4—22no—
274D4—22no—
275D4—22no—
276D4—22no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—48no—
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—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—84no—
316C2×C4#12012yes—
317C2×C4#12012yes—
318C2×C4AGL(1,ℤ/8)12yes—
319C2×C4C22×D411yescharacteristic
320C2×C4C22×D411yescharacteristic
321C2×C4C2×#4111yescharacteristic
322C2×C4C2×#4111yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C2×C42−1+411yescharacteristic
325C8—22no—
326C8—41no—
327C8—41no—
328C8—41no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C2×#12011yescharacteristic
344C22#54811yescharacteristic
345C22#55011yescharacteristic
346C22C2×AGL(1,ℤ/8)11yescharacteristic
347C22#57511yescharacteristic
348C22#57711yescharacteristic
349C22#57811yescharacteristic
350C4—22no—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4—41no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4#52511yescharacteristic
363C4#53711yescharacteristic
364C4#54611yescharacteristic
365C4#54711yescharacteristic
366C2—82no—
367C2—82no—
368C2#240711yescharacteristic
369C2#300611yescharacteristic
370C2#301311yescharacteristic
371C2#336711yescharacteristic
372C2#337611yescharacteristic
373C2#339711yescharacteristic
374C2#340211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#556C2semidirect
3. C2×#440C2semidirect
4. #2538C2semidirect
5. #2043C2semidirect
6. #2042C2semidirect
7. #2013C2semidirect
9. #1985C2semidirect
10. #1959C2semidirect
11. #1880C2semidirect
13. #1863C2semidirect
14. #1862C2semidirect
15. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4632
C2×#4402
#25382
#20432
#20422
#20132
#20122
#19852
#19592
#18802
#18792
#18632
#18622
#18032