Finite Group Databasebeta

← #26257

Subgroups — #26257

Order 256.

473 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×#496C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic, complemented
3#2448C211yesmaximal, characteristic, complemented
4#2445C211yesmaximal, characteristic, complemented
5#2055C211yesmaximal, characteristic, complemented
6#2054C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2041C211yesmaximal, characteristic, complemented
9#1973C211yesmaximal, characteristic, complemented
10#1972C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1943C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1879C211yesmaximal, characteristic, complemented
15#1771C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C2×#127C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#123—21no—
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25#496—24no—
26C4×SD16—24no—
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#97C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#399—21no—
42#397—21no—
43#394—21no—
44#394—21no—
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#389C2211yescharacteristic
52#389C2211yescharacteristic
53#388—21no—
54#388—21no—
55#386—21no—
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—41no—
73C2×SD16D414yes—
74C2×D8—22no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—22no—
79C2×D8—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84#127—22no—
85#127—41no—
86#124—22no—
87#124—41no—
88#123—44no—
89C4×Q8—22no—
90C4×Q8—41no—
91C4×D4—22no—
92C4×D4—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4D411yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
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×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#107—22no—
120#106—22no—
121C4⋊C8—22no—
122#103—22no—
123#103—41no—
124#102—22no—
125#102—41no—
126#102—41no—
127#102—41no—
128#102—41no—
129#102—41no—
130#97—22no—
131#97—22no—
132C4×C8—22no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
141C24—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8C2×D412yes—
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—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—81no—
170C2×D4—81no—
171C2×D4C2×D412yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—81no—
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2411yescharacteristic
187SD16—28no—
188SD16—44no—
189D8—44no—
190D8—44no—
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×C8C2×D412yes—
200C2×C8#4112yes—
201C2×C8#4112yes—
202C2×C8#4112yes—
203C4⋊C4—22no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—81no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—42no—
236C23—22no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
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—41no—
257C23C22×D411yescharacteristic
258Q8—22no—
259Q8—22no—
260Q8—41no—
261Q8—41no—
262D4—22no—
263D4—22no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—84no—
275D4—84no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
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—21no—
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×C4AGL(1,ℤ/8)12yes—
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—41no—
318C8—41no—
319C8#12314yescomplemented
320C22—42no—
321C22—42no—
322C22—42no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#53211yescharacteristic
342C22D4211yescharacteristic
343C22#55011yescharacteristic
344C22C2×AGL(1,ℤ/8)11yescharacteristic
345C22#57511yescharacteristic
346C22#57611yescharacteristic
347C22#57711yescharacteristic
348C4—22no—
349C4—41no—
350C4—41no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4C2×#12311yescharacteristic
360C4#54011yescharacteristic
361C4#54611yescharacteristic
362C4#55911yescharacteristic
363C2—82no—
364C2—161no—
365C2—161no—
366C2#244511yescharacteristic
367C2#317111yescharacteristic
368C2#317911yescharacteristic
369C2#326511yescharacteristic
370C2#326811yescharacteristic
371C2#339911yescharacteristic
372C2#340111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#496C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×SD16C2semidirect
3. #2448C2semidirect
4. #2445C2semidirect
5. #2055C2semidirect
6. #2054C2semidirect
7. #2042C2semidirect
8. #2041C2semidirect
9. #1973C2semidirect
10. #1972C2semidirect
11. #1970C2semidirect
12. #1943C2semidirect
13. #1880C2semidirect
14. #1879C2semidirect
15. #1771C2semidirect
319. C8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#4962
C2×C4×SD162
#24482
#24452
#20552
#20542
#20422
#20412
#19732
#19722
#19702
#19432
#18802
#18792
#17712