Finite Group Databasebeta

← #14758

Subgroups — #14758

Order 256.

1121 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic
2C2×#395C211yesmaximal, characteristic, complemented
3C2×#385C211yesmaximal, characteristic
4C2×#380C212yesmaximal
5C22×#95C211yesmaximal, characteristic
6#1446C211yesmaximal, characteristic, complemented
7#1444C212yesmaximal, complemented
8#1444C211yesmaximal, characteristic, complemented
9#1441C211yesmaximal, characteristic, complemented
10#1441C212yesmaximal, complemented
11C22⋊#95C212yesmaximal
12C23×Q8C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2212yes—
15C2×C22⋊Q8—24no—
16C2×C22⋊Q8—28no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2212yes—
20C22×C4⋊C4C2212yes—
21C22×C4⋊C4C2212yes—
22C22×C4⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26#395—24no—
27#385—24no—
28#380—28no—
29C2×#95—24no—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2211yescharacteristic
41C25C2311yescharacteristic
42C22×Q8—24no—
43C22×Q8—22no—
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—42no—
47C23×C4—24no—
48C23×C4—24no—
49C23×C4—24no—
50C23×C4—22no—
51C23×C4—24no—
52C23×C4—22no—
53C23×C4—24no—
54C23×C4—42no—
55C23×C4—42no—
56C23×C4—42no—
57C23×C4D414yes—
58C23×C4Q818yescomplemented
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2312yes—
62C23×C4C2312yes—
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2312yes—
67C23×C4C2311yescharacteristic
68C23×C4C2312yes—
69C22⋊Q8—232no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4D418yes—
93C2×C22⋊C4Q814yescomplemented
94#95—22no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—22no—
99#95—22no—
100#95—24no—
101#95—24no—
102#95—42no—
103#95—42no—
104#95D414yes—
105C24—21no—
106C24—21no—
107C24—22no—
108C24—22no—
109C24—21no—
110C24—21no—
111C24C2×D412yes—
112C24C2×D414yes—
113C24C2×Q812yes—
114C24C2×Q814yes—
115C24#4112yes—
116C24C2411yescharacteristic
117C2×Q8—28no—
118C2×Q8—48no—
119C2×Q8—48no—
120C2×Q8—44no—
121C2×Q8—44no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—28no—
129C22×C4—216no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
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—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
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—21no—
162C22×C4—28no—
163C22×C4—216no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—44no—
175C22×C4—48no—
176C22×C4—48no—
177C22×C4—48no—
178C22×C4—48no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—48no—
182C22×C4C2×D414yes—
183C22×C4C2×D414yes—
184C22×C4C2×D412yes—
185C22×C4C2×D412yes—
186C22×C4C2×D412yes—
187C22×C4C2×D414yes—
188C22×C4C2×Q812yes—
189C22×C4C2×Q812yes—
190C22×C4C2×Q814yes—
191C22×C4C2×Q814yes—
192C22×C4#4112yes—
193C22×C4#4114yes—
194C4⋊C4—28no—
195C4⋊C4—216no—
196C4⋊C4—48no—
197C4⋊C4—48no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C22⋊C4—216no—
201C22⋊C4—28no—
202C22⋊C4—44no—
203C23—22no—
204C23—24no—
205C23—22no—
206C23—24no—
207C23—22no—
208C23—22no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—22no—
213C23—24no—
214C23—24no—
215C23—24no—
216C23—22no—
217C23—24no—
218C23—22no—
219C23—24no—
220C23—22no—
221C23#12014yes—
222C23C22⋊Q814yes—
223C23C22⋊Q818yes—
224C23C4⋊Q814yes—
225C23C4⋊Q818yes—
226C23C22×D411yescharacteristic
227C23C22×D412yes—
228C23C22×D411yescharacteristic
229C23C22×Q812yes—
230C23C22×Q811yescharacteristic
231C23C2×#4111yescharacteristic
232C23C2×#4111yescharacteristic
233C23C2×#4112yes—
234C23H(5,𝔽2)12yes—
235C23H(5,𝔽2)11yescharacteristic
236C232−1+411yescharacteristic
237Q8—416no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
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—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—416no—
266C2×C4—432no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
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—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4#12114yes—
299C2×C4C22⋊Q818yescomplemented
300C22—24no—
301C22—24no—
302C22—24no—
303C22—24no—
304C22—24no—
305C22—24no—
306C22—28no—
307C22—28no—
308C22—28no—
309C22—28no—
310C22#39518yescomplemented
311C22C2×#12011yescharacteristic
312C22C2×#12111yescharacteristic
313C22C2×C22⋊Q811yescharacteristic
314C22C2×C22⋊Q812yes—
315C22C2×C22⋊Q812yes—
316C22C2×C4⋊Q811yescharacteristic
317C22C2×C4⋊Q812yes—
318C22#53211yescharacteristic
319C22#53312yes—
320C22#53411yescharacteristic
321C22#53711yescharacteristic
322C22#54312yes—
323C22#54412yes—
324C22D4212yes—
325C22#54612yes—
326C22#54611yescharacteristic
327C22#54612yes—
328C22#54712yes—
329C22D4×Q811yescharacteristic
330C22#55112yes—
331C22#55211yescharacteristic
332C22#55412yes—
333C22#55411yescharacteristic
334C4—24no—
335C4—28no—
336C4—42no—
337C4—42no—
338C4—44no—
339C4—42no—
340C4—44no—
341C4—44no—
342C4—81no—
343C4—81no—
344C2—28no—
345C2C2×#39511yescharacteristic
346C2#241011yescharacteristic
347C2#241711yescharacteristic
348C2#243011yescharacteristic
349C2#243712yes—
350C2#243812yes—
351C2#244012yes—
352C2#254912yes—
353C2#256411yescharacteristic
354C2#265911yescharacteristic
355C2#285711yescharacteristic
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
2. C2×#395C2semidirect
6. #1446C2semidirect
7. #1444C2semidirect
8. #1444C2semidirect
9. #1441C2semidirect
10. #1441C2semidirect
58. C23×C4Q8semidirect
93. C2×C22⋊C4Q8semidirect
299. C2×C4C22⋊Q8semidirect
310. C22#395semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#3952
C2×#3852
C2×#3802
C22×#952
#14462
#14442
#14412
C22⋊#952