Finite Group Databasebeta

← #15464

Subgroups — #15464

Order 256.

967 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123C211yesmaximal, characteristic
2C2×#400C211yesmaximal, characteristic, complemented
3C2×#398C211yesmaximal, characteristic, complemented
4C2×#390C211yesmaximal, characteristic
5#1440C212yesmaximal, complemented
6#1439C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1438C211yesmaximal, characteristic
9#1436C211yesmaximal, characteristic, complemented
10#1436C211yesmaximal, characteristic, complemented
11C22⋊#95C212yesmaximal, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C2×#123—24no—
15C2×#123—24no—
16C2×#123—24no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2212yes—
28C22×C42C2211yescharacteristic
29#400—24no—
30#398—24no—
31#390—24no—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C25C2311yescharacteristic
43C22×D4—24no—
44C22×D4—22no—
45C22×D4—22no—
46C22×D4—42no—
47C22×D4D414yes—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—24no—
52C23×C4—42no—
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2312yes—
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
62#123—232no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4D414yes—
74C2×C4⋊C4D414yes—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C42—22no—
97C2×C42—42no—
98#95—22no—
99#95—24no—
100#95—22no—
101#95—24no—
102#95—24no—
103#95—24no—
104#95—24no—
105#95—42no—
106#95—42no—
107C24—21no—
108C24—21no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—21no—
115C24—21no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—22no—
122C24C2×D412yes—
123C24C2×D412yes—
124C24#4114yes—
125C24C2411yescharacteristic
126C2×D4—28no—
127C2×D4—28no—
128C2×D4—44no—
129C2×D4—48no—
130C2×D4—48no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—28no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—48no—
177C22×C4—48no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4C2×D412yes—
181C22×C4C2×D412yes—
182C22×C4C2×D412yes—
183C22×C4C2×D412yes—
184C22×C4C2×D412yes—
185C22×C4C2×D412yes—
186C22×C4C2×D412yes—
187C22×C4#4114yes—
188C22×C4#4114yes—
189C4⋊C4—28no—
190C4⋊C4—28no—
191C4⋊C4—48no—
192C4⋊C4—44no—
193C22⋊C4—28no—
194C22⋊C4—28no—
195C22⋊C4—28no—
196C22⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—48no—
199C22⋊C4—48no—
200C42—44no—
201C23—22no—
202C23—24no—
203C23—22no—
204C23—24no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—24no—
215C23—24no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23#12314yes—
243C23C22×D411yescharacteristic
244C23C22×D411yescharacteristic
245C23C22×D411yescharacteristic
246C23C2×#4112yes—
247C23C2×#4112yes—
248C23C2×#4112yes—
249C23H(5,𝔽2)12yes—
250C23H(5,𝔽2)12yes—
251C232−1+412yes—
252D4—416no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—416no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4#12014yes—
307C22—24no—
308C22—28no—
309C22—42no—
310C22—42no—
311C22—44no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—44no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22C2×#12011yescharacteristic
326C22C2×#12311yescharacteristic
327C22#52912yes—
328C22#53211yescharacteristic
329C22#53211yescharacteristic
330C22#53411yescharacteristic
331C22#53511yescharacteristic
332C22#53611yescharacteristic
333C22#53812yes—
334C22#53912yes—
335C22#53912yes—
336C22#54612yes—
337C22#54612yes—
338C22#54612yes—
339C22#54712yes—
340C22#54812yes—
341C22#55012yes—
342C22#55112yes—
343C22#55212yes—
344C22#55312yes—
345C22#56212yes—
346C4—24no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—44no—
352C4—44no—
353C4—82no—
354C4—82no—
355C2—44no—
356C2—82no—
357C2#240511yescharacteristic
358C2#241912yes—
359C2#243111yescharacteristic
360C2#243211yescharacteristic
361C2#255412yes—
362C2#255812yes—
363C2#269412yes—
364C2#269612yes—
365C2#270812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC23×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#400C2semidirect
3. C2×#398C2semidirect
5. #1440C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
9. #1436C2semidirect
10. #1436C2semidirect
11. C22⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1232
C2×#4002
C2×#3982
C2×#3902
#14402
#14392
#14382
#14362
C22⋊#952