Finite Group Databasebeta

← #29832

Subgroups — #29832

Order 256.

426 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic
3#2861C211yesmaximal, characteristic, complemented
4#2293C211yesmaximal, characteristic, complemented
5#2068C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#2055C211yesmaximal, characteristic, complemented
10#2055C211yesmaximal, characteristic, complemented
11#2054C211yesmaximal, characteristic
12#2054C211yesmaximal, characteristic
13#1989C211yesmaximal, characteristic, complemented
14#1983C211yesmaximal, characteristic, complemented
15#1842C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117—21no—
22#478—24no—
23#463—24no—
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#388—21no—
49#388—21no—
50#388—21no—
51#388—21no—
52#388—21no—
53#388—21no—
54C4×C4⋊C4C2211yescharacteristic
55C4×C22⋊C4—21no—
56C43C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C2×SD16—22no—
66C2×SD16—41no—
67C2×SD16—41no—
68C2×D8—22no—
69C2×D8—41no—
70C2×D8—41no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#125—22no—
76#125—22no—
77#124—22no—
78#124—22no—
79C4×D4—22no—
80C4×D4D414yes—
81#117—44no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—21no—
103C2×C42D411yescharacteristic
104C2×C42D411yescharacteristic
105C2×C42D411yescharacteristic
106C2×C42D411yescharacteristic
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#107—22no—
111#106—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118#103—22no—
119#103—41no—
120#103—41no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—41no—
125#102—41no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24—21no—
134C24—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4C2×D412yes—
153C22×C4—21no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2411yescharacteristic
169SD16—44no—
170D8—44no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4C2×D412yes—
197C22⋊C4—24no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—22no—
209C42—22no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—41no—
215C42—41no—
216C42—42no—
217C42C2×D412yes—
218C42#4112yes—
219C23—22no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23C22×D411yescharacteristic
234Q8—41no—
235Q8—41no—
236Q8—41no—
237Q8—41no—
238D4—22no—
239D4—22no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—48no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4#12112yes—
280C2×C4#12112yes—
281C2×C4#13512yes—
282C2×C4AGL(1,ℤ/8)12yes—
283C2×C4C22×D411yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4H(5,𝔽2)11yescharacteristic
286C2×C4H(5,𝔽2)11yescharacteristic
287C2×C42−1+411yescharacteristic
288C2×C42−1+411yescharacteristic
289C8—41no—
290C8—41no—
291C8—41no—
292C8—41no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22C2×#12111yescharacteristic
308C22#53711yescharacteristic
309C22#53711yescharacteristic
310C22C2×#13511yescharacteristic
311C22C2×AGL(1,ℤ/8)11yescharacteristic
312C22#57511yescharacteristic
313C22#57511yescharacteristic
314C4—22no—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—42no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4#53511yescharacteristic
327C4#53611yescharacteristic
328C4#54211yescharacteristic
329C4#54211yescharacteristic
330C2—82no—
331C2—161no—
332C2#262011yescharacteristic
333C2#303511yescharacteristic
334C2#303611yescharacteristic
335C2#324711yescharacteristic
336C2#325011yescharacteristic
337C2#325111yescharacteristic
338C2#325611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#478C2semidirect
3. #2861C2semidirect
4. #2293C2semidirect
5. #2068C2semidirect
6. #2067C2semidirect
7. #2066C2semidirect
8. #2065C2semidirect
9. #2055C2semidirect
10. #2055C2semidirect
13. #1989C2semidirect
14. #1983C2semidirect
15. #1842C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×#4632
#28612
#22932
#20682
#20672
#20662
#20652
#20552
#20542
#19892
#19832
#18422