Finite Group Databasebeta

← #17107

Subgroups — #17107

Order 256.

807 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 807 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C2×#394C212yesmaximal
2C2×#393C212yesmaximal, complemented
3#1446C211yesmaximal, characteristic, complemented
4#1445C212yesmaximal, complemented
5#1444C211yesmaximal, characteristic, complemented
6#1440C212yesmaximal, complemented
7#1440C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1438C211yesmaximal, characteristic
10C23×Q8C2211yescharacteristic
11C23×D4C2211yescharacteristic
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2211yescharacteristic
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20#394—28no—
21#393—28no—
22C2×#95C2212yes—
23C2×#95C2212yes—
24C2×#95C2212yes—
25C2×#95C2212yes—
26C2×#95C2211yescharacteristic
27C2×#95C2211yescharacteristic
28C2×#95C2211yescharacteristic
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C25C2312yes—
34C22×Q8—24no—
35C22×Q8—22no—
36C22×Q8—41no—
37C22×Q8—42no—
38C22×Q8—41no—
39C22×D4—24no—
40C22×D4—22no—
41C22×D4—41no—
42C22×D4—41no—
43C22×D4—42no—
44C23×C4C2312yes—
45C23×C4C2312yes—
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—42no—
57C2×C22⋊C4—24no—
58C2×C22⋊C4—24no—
59C2×C22⋊C4—24no—
60C2×C22⋊C4—24no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—42no—
65C2×C22⋊C4—42no—
66C2×C22⋊C4—42no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4—42no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4—42no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4D418yes—
75#95—24no—
76#95—24no—
77#95—24no—
78#95—24no—
79#95—22no—
80#95—42no—
81#95—41no—
82#95—42no—
83#95—41no—
84#95—41no—
85#95—42no—
86C24—22no—
87C24—22no—
88C24—22no—
89C24—22no—
90C24—22no—
91C24—22no—
92C24—22no—
93C24—22no—
94C24—22no—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24C2×D414yes—
101C24C2411yescharacteristic
102C2×Q8—44no—
103C2×Q8—48no—
104C2×Q8—48no—
105C2×Q8—44no—
106C2×Q8—44no—
107C2×D4—48no—
108C2×D4—44no—
109C2×D4—44no—
110C2×D4—48no—
111C2×D4—44no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
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—21no—
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×C4C2×D414yes—
158C22×C4C2×D414yes—
159C22×C4#4112yes—
160C22×C4#4112yes—
161C22×C4#4112yes—
162C4⋊C4—48no—
163C4⋊C4—44no—
164C22⋊C4—48no—
165C22⋊C4—48no—
166C22⋊C4—48no—
167C22⋊C4—48no—
168C22⋊C4—48no—
169C22⋊C4—48no—
170C23—24no—
171C23—24no—
172C23—24no—
173C23—24no—
174C23—24no—
175C23—24no—
176C23—24no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—42no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23C22×D412yes—
206C23C2×#4111yescharacteristic
207C23C2×#4111yescharacteristic
208C23C2×#4111yescharacteristic
209C23H(5,𝔽2)11yescharacteristic
210C23H(5,𝔽2)12yes—
211C23H(5,𝔽2)11yescharacteristic
212C23H(5,𝔽2)12yes—
213C23H(5,𝔽2)11yescharacteristic
214C232−1+411yescharacteristic
215C232−1+412yes—
216Q8—416no—
217D4—416no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
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—
268C22—44no—
269C22—44no—
270C22—44no—
271C22—44no—
272C22—44no—
273C22—44no—
274C22—44no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22#53412yes—
284C22#53512yes—
285C22#53612yes—
286C22#53812yes—
287C22#53811yescharacteristic
288C22#53811yescharacteristic
289C22#53911yescharacteristic
290C22#54212yes—
291C22#54212yes—
292C22D4211yescharacteristic
293C22#54612yes—
294C22#54612yes—
295C22#54712yes—
296C22#55511yescharacteristic
297C22#55611yescharacteristic
298C22#55911yescharacteristic
299C22#56011yescharacteristic
300C22H(3,𝔽4)11yescharacteristic
301C22#56211yescharacteristic
302C22#56212yes—
303C22#56212yes—
304C22#56212yes—
305C22#56311yescharacteristic
306C4—42no—
307C4—44no—
308C4—42no—
309C4—42no—
310C4—44no—
311C4—82no—
312C4—82no—
313C4—82no—
314C2—84no—
315C2#266811yescharacteristic
316C2#267011yescharacteristic
317C2#268011yescharacteristic
318C2#268612yes—
319C2#270012yes—
320C2#271012yes—
321C2#280811yescharacteristic
322C2#281612yes—
323C2#282411yescharacteristic
324C2#283011yescharacteristic
325C2#283211yescharacteristic
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×#393C2semidirect
3. #1446C2semidirect
4. #1445C2semidirect
5. #1444C2semidirect
6. #1440C2semidirect
7. #1440C2semidirect
8. #1439C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3942
C2×#3932
#14462
#14452
#14442
#14402
#14392
#14382