Finite Group Databasebeta

← #16518

Subgroups — #16518

Order 256.

715 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#397C211yesmaximal, characteristic
2C2×#396C212yesmaximal
3C2×#389C211yesmaximal, characteristic, complemented
4C2×#388C211yesmaximal, characteristic
5C2×#382C211yesmaximal, characteristic, complemented
6#1445C211yesmaximal, characteristic, complemented
7#1444C211yesmaximal, characteristic, complemented
8#1441C212yesmaximal, complemented
9#1441C212yesmaximal, complemented
10#1440C212yesmaximal
11#1439C211yesmaximal, characteristic
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C42C2211yescharacteristic
24#397—24no—
25#396—28no—
26#389—24no—
27#388—24no—
28#382—24no—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2211yescharacteristic
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C25C2311yescharacteristic
41C23×C4C2312yes—
42C23×C4C2311yescharacteristic
43C23×C4C2312yes—
44C23×C4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C23×C4C2312yes—
47C23×C4C2311yescharacteristic
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C2×C4⋊C4—22no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4D414yes—
81C2×C42—22no—
82C2×C42—22no—
83C2×C42D414yes—
84#95—24no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—24no—
89#95—24no—
90#95—24no—
91#95—42no—
92#95—42no—
93#95—42no—
94#95D414yes—
95C24—21no—
96C24—21no—
97C24—21no—
98C24—22no—
99C24—21no—
100C24—21no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—21no—
105C24C2×D412yes—
106C24C2411yescharacteristic
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
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—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
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—21no—
157C22×C4C2×D414yes—
158C22×C4C2×D412yes—
159C22×C4C2×D412yes—
160C22×C4C2×D412yes—
161C22×C4C2×D412yes—
162C22×C4C2×D412yes—
163C22×C4C2×D412yes—
164C22×C4#4114yes—
165C22×C4#4112yes—
166C22×C4#4112yes—
167C4⋊C4—28no—
168C4⋊C4—48no—
169C4⋊C4—48no—
170C4⋊C4—44no—
171C4⋊C4—48no—
172C4⋊C4—44no—
173C22⋊C4—28no—
174C22⋊C4—48no—
175C22⋊C4—48no—
176C22⋊C4—44no—
177C22⋊C4—44no—
178C42—28no—
179C23—22no—
180C23—24no—
181C23—22no—
182C23—24no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—41no—
197C23—42no—
198C23—41no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23C22×D411yescharacteristic
203C23C22×D411yescharacteristic
204C23C22×D411yescharacteristic
205C23C2×#4111yescharacteristic
206C23C2×#4111yescharacteristic
207C23C2×#4112yes—
208C23H(5,𝔽2)12yes—
209C23H(5,𝔽2)11yescharacteristic
210C232−1+412yes—
211C232−1+412yes—
212C232−1+411yescharacteristic
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
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#12114yes—
272C2×C4#12114yes—
273C22—42no—
274C22—42no—
275C22—44no—
276C22—44no—
277C22—42no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—82no—
283C22—82no—
284C22C2×#12111yescharacteristic
285C22C2×#12111yescharacteristic
286C22#52511yescharacteristic
287C22#53511yescharacteristic
288C22#53611yescharacteristic
289C22#53611yescharacteristic
290C22#53612yes—
291C22#53711yescharacteristic
292C22#53712yes—
293C22#54112yes—
294C22#54112yes—
295C22#54212yes—
296C22#54212yes—
297C22D4211yescharacteristic
298C22#54612yes—
299C22#54712yes—
300C22#54711yescharacteristic
301C22#54811yescharacteristic
302C22#55212yes—
303C22#55311yescharacteristic
304C22#55312yes—
305C22#56212yes—
306C22#56312yes—
307C4—24no—
308C4—24no—
309C4—42no—
310C4—44no—
311C4—42no—
312C4—44no—
313C4—81no—
314C4—81no—
315C4—82no—
316C4—82no—
317C2—82no—
318C2#241311yescharacteristic
319C2#241611yescharacteristic
320C2#253511yescharacteristic
321C2#262011yescharacteristic
322C2#262111yescharacteristic
323C2#263711yescharacteristic
324C2#267212yes—
325C2#269812yes—
326C2#272512yes—
327C2#277212yes—
328C2#282011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC25C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#389C2semidirect
5. C2×#382C2semidirect
6. #1445C2semidirect
7. #1444C2semidirect
8. #1441C2semidirect
9. #1441C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3972
C2×#3962
C2×#3892
C2×#3882
C2×#3822
#14452
#14442
#14412
#14402
#14392