Finite Group Databasebeta

← #26472

Subgroups — #26472

Order 256.

450 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic, complemented
2C2×#476C211yesmaximal, characteristic, complemented
3#2550C211yesmaximal, characteristic, complemented
4#2366C211yesmaximal, characteristic
5#2067C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1982C211yesmaximal, characteristic
10#1961C211yesmaximal, characteristic
11#1949C211yesmaximal, characteristic, complemented
12#1940C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C4×D4—21no—
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22#478—24no—
23#476—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#97C2211yescharacteristic
40#398—21no—
41#395—21no—
42#391—21no—
43#390—21no—
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#389—21no—
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#386—21no—
50#386—21no—
51#382C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C4⋊Q8—22no—
73C4⋊Q8—41no—
74#127—22no—
75#127Q814yescomplemented
76#125—22no—
77#125—41no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—41no—
81C4×D4—41no—
82C4×D4—44no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
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×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—22no—
108#107—22no—
109#106—22no—
110#106D414yescomplemented
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114#102—22no—
115#102—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—41no—
121#102—41no—
122#97—22no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95D411yescharacteristic
130#95D411yescharacteristic
131C24—21no—
132C24—21no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—81no—
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—81no—
151C2×D4—81no—
152C2×D4C2×Q812yes—
153C2×D4C2×Q812yes—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2411yescharacteristic
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—
183C2×C8—21no—
184C2×C8C2×D412yes—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C4⋊C4C2×D412yes—
202C4⋊C4C2×D412yes—
203C4⋊C4#4112yes—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—42no—
215C42—81no—
216C42C2×Q812yes—
217C42#4112yes—
218C23—22no—
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—
232C23C22×D411yescharacteristic
233Q8—84no—
234D4—22no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—48no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
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—84no—
278C2×C4—84no—
279C2×C4C22⋊Q812yes—
280C2×C4C22⋊Q812yes—
281C2×C4AGL(1,ℤ/8)12yes—
282C2×C4C22×D411yescharacteristic
283C2×C4C22×Q811yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4H(5,𝔽2)11yescharacteristic
287C2×C42−1+411yescharacteristic
288C8—22no—
289C8—41no—
290C8—41no—
291C8—41no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22C2×C22⋊Q811yescharacteristic
307C22D4211yescharacteristic
308C22#54811yescharacteristic
309C22C2×AGL(1,ℤ/8)11yescharacteristic
310C22#57511yescharacteristic
311C22#57611yescharacteristic
312C22#57711yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—42no—
317C4—41no—
318C4—42no—
319C4—41no—
320C4—41no—
321C4—42no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4#53211yescharacteristic
326C4#54211yescharacteristic
327C4D4×Q811yescharacteristic
328C4#55411yescharacteristic
329C2—82no—
330C2—82no—
331C2#245211yescharacteristic
332C2#307711yescharacteristic
333C2#308111yescharacteristic
334C2#326511yescharacteristic
335C2#326811yescharacteristic
336C2#336611yescharacteristic
337C2#337111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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×#478C2semidirect
2. C2×#476C2semidirect
3. #2550C2semidirect
5. #2067C2semidirect
6. #2067C2semidirect
7. #2025C2semidirect
8. #2023C2semidirect
11. #1949C2semidirect
12. #1940C2semidirect
13. #1864C2semidirect
14. #1862C2semidirect
15. #1776C2semidirect
75. #127Q8semidirect
110. #106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×#4762
#25502
#23662
#20672
#20252
#20232
#19822
#19612
#19492
#19402
#18642
#18622
#17762