Finite Group Databasebeta

← #22045

Subgroups — #22045

Order 256.

548 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#449C211yesmaximal, characteristic
2C4×#121C211yesmaximal, characteristic, complemented
3#2297C211yesmaximal, characteristic, complemented
4#1970C211yesmaximal, characteristic, complemented
5#1968C211yesmaximal, characteristic, complemented
6#1934C211yesmaximal, characteristic, complemented
7#1933C211yesmaximal, characteristic, complemented
8#1928C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1875C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15C4×C22⋊C8C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C22×C42C2211yescharacteristic
24#449C418yescomplemented
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#386—21no—
47#386—21no—
48#386—21no—
49#380C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53C4×C22⋊C4—21no—
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C2×#41—22no—
61C2×#41—41no—
62C2×#41—41no—
63C2×#41—41no—
64C2×#41—41no—
65C2×#41C2×C414yescomplemented
66C22×Q8C2311yescharacteristic
67C22×D4—21no—
68C22×D4—21no—
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4D411yescharacteristic
74C23×C4D411yescharacteristic
75C23×C4C2311yescharacteristic
76C2×SD16—41no—
77C2×SD16—41no—
78C2×SD16C2×C414yescomplemented
79C2×D8—41no—
80C2×D8—41no—
81C2×D8C2×C414yescomplemented
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86#121—22no—
87#121C2×C414yes—
88C4×D4—41no—
89C4×D4—41no—
90C4×D4—44no—
91#117—41no—
92#117—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
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×C42—21no—
110C2×C42—21no—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42D411yescharacteristic
114C2×C42D411yescharacteristic
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#107—41no—
118#106—41no—
119#103—41no—
120#103—41no—
121#103C2×C414yescomplemented
122#102—41no—
123#102—41no—
124#102—41no—
125#102—41no—
126#102C2×C414yescomplemented
127C22⋊C8—22no—
128C22⋊C8C2×C414yescomplemented
129C4×C8—41no—
130C4×C8—41no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24—21no—
140C24C2×D411yescharacteristic
141#41—44no—
142#41—44no—
143#41—44no—
144#41—44no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C22×C412yes—
151C2×D4—24no—
152C2×D4—24no—
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—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4C22×C412yes—
168C2×D4C22×C412yes—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4C22×C412yes—
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4#4112yes—
198C22×C4C2411yescharacteristic
199SD16—44no—
200D8—44no—
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×C8C22×C412yes—
212C2×C8C22×C412yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—42no—
223C4⋊C4C22×C412yes—
224C22⋊C4—24no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—41no—
239C42—41no—
240C42—42no—
241C42—42no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C4×D412yes—
263C23C22×D411yescharacteristic
264Q8—22no—
265Q8—22no—
266Q8—41no—
267Q8—41no—
268D4—22no—
269D4—22no—
270D4—22no—
271D4—22no—
272D4—48no—
273D4—48no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—48no—
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—41no—
313C2×C4—41no—
314C2×C4—84no—
315C2×C4C4×D412yes—
316C2×C4#13512yes—
317C2×C4#13512yes—
318C2×C4AGL(1,ℤ/8)12yes—
319C2×C4C23×C411yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C2×C4H(5,𝔽2)11yescharacteristic
325C8—22no—
326C8—22no—
327C8—41no—
328C8—41no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C2×C4×D411yescharacteristic
351C22#53411yescharacteristic
352C22#53511yescharacteristic
353C22C2×#13511yescharacteristic
354C22C2×#13511yescharacteristic
355C22C2×AGL(1,ℤ/8)11yescharacteristic
356C22#57511yescharacteristic
357C4—22no—
358C4—22no—
359C4—22no—
360C4—22no—
361C4—42no—
362C4—41no—
363C4—42no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4#51811yescharacteristic
369C4#51811yescharacteristic
370C4#53811yescharacteristic
371C4#53811yescharacteristic
372C2—42no—
373C2—82no—
374C2—82no—
375C2#232111yescharacteristic
376C2C4×#13511yescharacteristic
377C2C4×AGL(1,ℤ/8)11yescharacteristic
378C2#318811yescharacteristic
379C2#319111yescharacteristic
380C2#320711yescharacteristic
381C2#321211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
LCSG ▷ C2×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. C4×#121C2semidirect
3. #2297C2semidirect
4. #1970C2semidirect
5. #1968C2semidirect
6. #1934C2semidirect
7. #1933C2semidirect
8. #1928C2semidirect
9. #1926C2semidirect
10. #1879C2semidirect
11. #1877C2semidirect
12. #1875C2semidirect
13. #1854C2semidirect
14. #1788C2semidirect
15. C4×C22⋊C8C2semidirect
24. #449C4semidirect
65. C2×#41C2×C4semidirect
78. C2×SD16C2×C4semidirect
81. C2×D8C2×C4semidirect
121. #103C2×C4semidirect
126. #102C2×C4semidirect
128. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4492
C4×#1212
#22972
#19702
#19682
#19342
#19332
#19282
#19262
#18792
#18772
#18752
#18542
#17882
C4×C22⋊C82