Finite Group Databasebeta

← #14043

Subgroups — #14043

Order 256.

1183 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
1C23×C4⋊C4C211yesmaximal, characteristic
2C2×#389C212yesmaximal, complemented
3C2×#385C211yesmaximal, characteristic
4C2×#385C211yesmaximal, characteristic
5C2×#385C211yesmaximal, characteristic
6C2×C4×C22⋊C4C211yesmaximal, characteristic
7#1444C212yesmaximal, complemented
8#1441C212yesmaximal, complemented
9#1436C212yesmaximal, complemented
10C22⋊#95C211yesmaximal, characteristic
11C22⋊#95C211yesmaximal, characteristic
12C24×C4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C22×C4⋊C4—24no—
16C22×C4⋊C4—28no—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2212yes—
20C22×C4⋊C4C2212yes—
21C22×C4⋊C4C2212yes—
22C22×C4⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C42C2212yes—
28#389—28no—
29#385—24no—
30#385—24no—
31#385C418yescomplemented
32C4×C22⋊C4—24no—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2211yescharacteristic
39C2×#95C2212yes—
40C2×#95C2212yes—
41C2×#95C2212yes—
42C25C2311yescharacteristic
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—24no—
46C23×C4—24no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—24no—
51C23×C4—24no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—42no—
55C23×C4—42no—
56C23×C4C2×C414yes—
57C23×C4D414yes—
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2312yes—
64C23×C4C2312yes—
65C23×C4C2312yes—
66C23×C4C2312yes—
67C23×C4C2311yescharacteristic
68C2×C4⋊C4—232no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—24no—
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—24no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4C2×C418yescomplemented
84C2×C4⋊C4D418yes—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4C2×C414yes—
93C2×C22⋊C4C2×C414yes—
94C2×C22⋊C4D414yes—
95C2×C22⋊C4Q814yes—
96C2×C22⋊C4Q814yes—
97C2×C42—24no—
98C2×C42—24no—
99C2×C42—42no—
100#95—24no—
101#95—22no—
102#95—22no—
103#95—24no—
104#95—24no—
105#95—24no—
106#95—24no—
107#95—42no—
108#95C2×C418yescomplemented
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24C22×C412yes—
114C24C2×D412yes—
115C24C2×D412yes—
116C24C2×Q812yes—
117C24C2×Q812yes—
118C24#4112yes—
119C24#4112yes—
120C24C2411yescharacteristic
121C22×C4—28no—
122C22×C4—216no—
123C22×C4—28no—
124C22×C4—28no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—216no—
137C22×C4—28no—
138C22×C4—28no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—22no—
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—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
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—28no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—44no—
175C22×C4—48no—
176C22×C4—48no—
177C22×C4—48no—
178C22×C4—48no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4C22×C412yes—
182C22×C4C22×C412yes—
183C22×C4C22×C414yes—
184C22×C4C22×C414yes—
185C22×C4C2×D412yes—
186C22×C4C2×D414yes—
187C22×C4C2×D414yes—
188C22×C4C2×D414yes—
189C22×C4C2×D412yes—
190C22×C4C2×D414yes—
191C22×C4C2×Q814yes—
192C22×C4C2×Q814yes—
193C22×C4#4114yes—
194C4⋊C4—216no—
195C4⋊C4—216no—
196C4⋊C4—216no—
197C4⋊C4—48no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C22⋊C4—28no—
201C22⋊C4—28no—
202C22⋊C4—44no—
203C22⋊C4C4⋊C4116yescomplemented
204C42—48no—
205C23—24no—
206C23—24no—
207C23—22no—
208C23—22no—
209C23—24no—
210C23—22no—
211C23—22no—
212C23—24no—
213C23—24no—
214C23—24no—
215C23—24no—
216C23—24no—
217C23—24no—
218C23—24no—
219C23—24no—
220C23—24no—
221C23C2×C4⋊C414yes—
222C23C4×D414yes—
223C23C4×Q814yes—
224C23#12114yes—
225C23C22⋊Q814yes—
226C23#12514yes—
227C23C4⋊Q814yes—
228C23C23×C411yescharacteristic
229C23C22×D411yescharacteristic
230C23C22×D411yescharacteristic
231C23C22×D412yes—
232C23C22×Q811yescharacteristic
233C23C22×Q811yescharacteristic
234C23C2×#4111yescharacteristic
235C23C2×#4112yes—
236C23C2×#4111yescharacteristic
237C23H(5,𝔽2)12yes—
238C232−1+412yes—
239C2×C4—232no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—416no—
270C2×C4—416no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
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—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4C2×C4⋊C418yescomplemented
298C2×C4C4×D418yes—
299C2×C4#12014yes—
300C22—24no—
301C22—24no—
302C22—28no—
303C22—28no—
304C22—24no—
305C22—24no—
306C22—24no—
307C22—28no—
308C22—28no—
309C22—24no—
310C22#38918yescomplemented
311C22C22×C4⋊C411yescharacteristic
312C22C2×C4×D411yescharacteristic
313C22C2×C4×D412yes—
314C22C2×C4×Q811yescharacteristic
315C22#52012yes—
316C22C2×#12011yescharacteristic
317C22C2×#12111yescharacteristic
318C22C2×C22⋊Q811yescharacteristic
319C22#52511yescharacteristic
320C22#52512yes—
321C22C2×#12511yescharacteristic
322C22C2×C4⋊Q811yescharacteristic
323C22#53411yescharacteristic
324C22#53611yescharacteristic
325C22#53912yes—
326C22D4212yes—
327C22#54612yes—
328C22#54712yes—
329C22D4×Q812yes—
330C22D4×Q812yes—
331C22#55212yes—
332C22#55412yes—
333C22#55412yes—
334C4—28no—
335C4—24no—
336C4—28no—
337C4—42no—
338C4—42no—
339C4—44no—
340C4—44no—
341C4—42no—
342C4—81no—
343C4—81no—
344C2—28no—
345C2C2×#38911yescharacteristic
346C2C4×#12011yescharacteristic
347C2#232211yescharacteristic
348C2D4×C4⋊C412yes—
349C2#236012yes—
350C2#241011yescharacteristic
351C2#241311yescharacteristic
352C2#243411yescharacteristic
353C2#244912yes—
354C2#247312yes—
355C2#258311yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC25C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#389C2semidirect
7. #1444C2semidirect
8. #1441C2semidirect
9. #1436C2semidirect
31. #385C4semidirect
83. C2×C4⋊C4C2×C4semidirect
108. #95C2×C4semidirect
203. C22⋊C4C4⋊C4semidirect
297. C2×C4C2×C4⋊C4semidirect
310. C22#389semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C42
C2×#3892
C2×#3852
C2×C4×C22⋊C42
#14442
#14412
#14362
C22⋊#952