Finite Group Databasebeta

← #15784

Subgroups — #15784

Order 256.

999 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 999 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124C211yesmaximal, characteristic
2C2×#397C212yesmaximal, complemented
3C2×#386C211yesmaximal, characteristic
4#1446C212yesmaximal, complemented
5#1440C212yesmaximal, complemented
6#1439C212yesmaximal, complemented
7#1438C211yesmaximal, characteristic
8#1436C212yesmaximal, complemented
9C22⋊#95C212yesmaximal, complemented
10C23×Q8C2211yescharacteristic
11C23×D4C2211yescharacteristic
12C24×C4C2211yescharacteristic
13C2×#124—24no—
14C2×#124—28no—
15C22×C4⋊C4C2212yes—
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C42C2211yescharacteristic
24#397—28no—
25#386—24no—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C25C2311yescharacteristic
36C25C2311yescharacteristic
37C22×Q8—24no—
38C22×Q8—22no—
39C22×Q8—24no—
40C22×Q8—41no—
41C22×Q8—41no—
42C22×D4—24no—
43C22×D4—24no—
44C22×D4—41no—
45C22×D4—41no—
46C22×D4D414yes—
47C23×C4—22no—
48C23×C4—24no—
49C23×C4—41no—
50C23×C4—41no—
51C23×C4—42no—
52C23×C4C2312yes—
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2312yes—
56C23×C4C2312yes—
57C23×C4C2312yes—
58C23×C4C2311yescharacteristic
59#124—232no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—42no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4D418yes—
81C2×C42—24no—
82C2×C42—22no—
83#95—24no—
84#95—24no—
85#95—24no—
86#95—24no—
87#95—24no—
88#95—24no—
89#95—42no—
90#95—42no—
91#95—42no—
92C24—22no—
93C24—22no—
94C24—22no—
95C24—22no—
96C24—21no—
97C24—21no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—21no—
105C24—21no—
106C24—22no—
107C24C2×D412yes—
108C24C2×D414yes—
109C24C2×D412yes—
110C24C2411yescharacteristic
111C2×Q8—28no—
112C2×Q8—48no—
113C2×Q8—48no—
114C2×Q8—44no—
115C2×Q8—44no—
116C2×D4—28no—
117C2×D4—28no—
118C2×D4—44no—
119C2×D4—48no—
120C2×D4—48no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
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—44no—
163C22×C4—48no—
164C22×C4—48no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4C2×D414yes—
168C22×C4C2×D414yes—
169C22×C4C2×D412yes—
170C22×C4#4114yes—
171C22×C4#4114yes—
172C22×C4#4114yes—
173C4⋊C4—48no—
174C22⋊C4—216no—
175C22⋊C4—216no—
176C22⋊C4—44no—
177C22⋊C4—48no—
178C22⋊C4—48no—
179C22⋊C4—44no—
180C22⋊C4—44no—
181C42—28no—
182C23—22no—
183C23—24no—
184C23—22no—
185C23—24no—
186C23—24no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—24no—
191C23—24no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—24no—
196C23—24no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—41no—
202C23—41no—
203C23—42no—
204C23—42no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23—42no—
220C23—42no—
221C23—41no—
222C23—41no—
223C23#12014yes—
224C23C22×D411yescharacteristic
225C23C22×D412yes—
226C23C2×#4112yes—
227C23C2×#4112yes—
228C23C2×#4112yes—
229C23H(5,𝔽2)11yescharacteristic
230C23H(5,𝔽2)11yescharacteristic
231C23H(5,𝔽2)12yes—
232C232−1+412yes—
233Q8—416no—
234D4—416no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—416no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4#12314yes—
283C22—24no—
284C22—28no—
285C22—42no—
286C22—44no—
287C22—44no—
288C22—44no—
289C22—42no—
290C22—44no—
291C22—44no—
292C22—42no—
293C22—42no—
294C22—44no—
295C22—44no—
296C22—42no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22C2×#12011yescharacteristic
302C22C2×#12311yescharacteristic
303C22#52912yes—
304C22#53212yes—
305C22#53411yescharacteristic
306C22#53712yes—
307C22#53812yes—
308C22#54212yes—
309C22#54212yes—
310C22#54612yes—
311C22#54612yes—
312C22#54612yes—
313C22#54612yes—
314C22#54712yes—
315C22#54712yes—
316C22#55111yescharacteristic
317C22#55111yescharacteristic
318C22#55612yes—
319C22#55912yes—
320C22H(3,𝔽4)11yescharacteristic
321C22#56311yescharacteristic
322C4—24no—
323C4—44no—
324C4—44no—
325C4—44no—
326C4—44no—
327C4—82no—
328C4—82no—
329C2—44no—
330C2—82no—
331C2#240511yescharacteristic
332C2#243012yes—
333C2#244712yes—
334C2#252112yes—
335C2#255712yes—
336C2#268411yescharacteristic
337C2#271212yes—
338C2#271512yes—
339C2#271711yescharacteristic
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×#397C2semidirect
4. #1446C2semidirect
5. #1440C2semidirect
6. #1439C2semidirect
8. #1436C2semidirect
9. C22⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C2×#3972
C2×#3862
#14462
#14402
#14392
#14382
#14362
C22⋊#952