Finite Group Databasebeta

← #24634

Subgroups — #24634

Order 256.

476 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#463C211yesmaximal, characteristic, complemented
2C2×C4×D8C211yesmaximal, characteristic, complemented
3#2544C211yesmaximal, characteristic, complemented
4#2357C211yesmaximal, characteristic, complemented
5#2055C211yesmaximal, characteristic, complemented
6#2054C211yesmaximal, characteristic, complemented
7#2012C211yesmaximal, characteristic, complemented
8#2011C211yesmaximal, characteristic
9#1983C211yesmaximal, characteristic, complemented
10#1982C211yesmaximal, characteristic, complemented
11#1958C211yesmaximal, characteristic, complemented
12#1939C211yesmaximal, characteristic, complemented
13#1928C211yesmaximal, characteristic, complemented
14#1925C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24#463—24no—
25C4×D8—24no—
26C2×#107C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#394—21no—
42#394—21no—
43#391C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#388—21no—
48#388—21no—
49#386—21no—
50#386—21no—
51#382C2211yescharacteristic
52#381—21no—
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56C4×C22⋊C4—21no—
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—
68C23×C4—21no—
69C2×SD16—22no—
70C2×SD16—41no—
71C2×SD16—41no—
72C2×D8—22no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—22no—
76C2×D8—41no—
77C2×D8—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82#127—22no—
83#127—41no—
84#124—22no—
85#124—22no—
86#124—22no—
87#124—41no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4D414yescomplemented
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C22⋊C4—21no—
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×C22⋊C4—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114C2×C42C2311yescharacteristic
115#107—22no—
116C4⋊C8—22no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119#103—22no—
120#103—22no—
121#103—41no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—41no—
127#102—41no—
128#102—41no—
129C4×C8—22no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24—21no—
140C24—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×D4—21no—
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—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4C2×D412yes—
170C2×D4#4112yes—
171C22×C4—21no—
172C22×C4—24no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2411yescharacteristic
189SD16—44no—
190D8—28no—
191D8—44no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8#4112yes—
204C4⋊C4—22no—
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⋊C4C2×D412yes—
214C22⋊C4—24no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—81no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—81no—
231C42—81no—
232C42C2×D412yes—
233C42#4112yes—
234C23—22no—
235C23—22no—
236C23—41no—
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—
255C23C22×D411yescharacteristic
256Q8—41no—
257Q8—41no—
258Q8—41no—
259Q8—41no—
260D4—22no—
261D4—22no—
262D4—22no—
263D4—22no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—41no—
272D4—84no—
273D4—84no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
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—84no—
305C2×C4—84no—
306C2×C4#12112yes—
307C2×C4#12112yes—
308C2×C4#13512yes—
309C2×C4C22×D411yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C2×C4C2×#4111yescharacteristic
313C2×C4H(5,𝔽2)11yescharacteristic
314C2×C4H(5,𝔽2)11yescharacteristic
315C8—22no—
316C8—41no—
317C8—41no—
318C8—41no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—42no—
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—
340C22C2×#12111yescharacteristic
341C22#54611yescharacteristic
342C22#55011yescharacteristic
343C22C2×#13511yescharacteristic
344C22#57511yescharacteristic
345C22#57611yescharacteristic
346C22#57611yescharacteristic
347C4—22no—
348C4—22no—
349C4—22no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4#52511yescharacteristic
359C4#53511yescharacteristic
360C4#55311yescharacteristic
361C4#55511yescharacteristic
362C2—82no—
363C2—82no—
364C2—161no—
365C2#241411yescharacteristic
366C2#303511yescharacteristic
367C2#304111yescharacteristic
368C2#329711yescharacteristic
369C2#332211yescharacteristic
370C2#339611yescharacteristic
371C2#340311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2011C2
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
1. C2×#463C2semidirect
2. C2×C4×D8C2semidirect
3. #2544C2semidirect
4. #2357C2semidirect
5. #2055C2semidirect
6. #2054C2semidirect
7. #2012C2semidirect
9. #1983C2semidirect
10. #1982C2semidirect
11. #1958C2semidirect
12. #1939C2semidirect
13. #1928C2semidirect
14. #1925C2semidirect
15. #1804C2semidirect
91. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
C2×C4×D82
#25442
#23572
#20552
#20542
#20122
#20112
#19832
#19822
#19582
#19392
#19282
#19252
#18042