Finite Group Databasebeta

← #14401

Subgroups — #14401

Order 256.

1059 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#390C211yesmaximal, characteristic
2C2×#386C211yesmaximal, characteristic
3C2×#386C211yesmaximal, characteristic
4C2×#385C212yesmaximal, complemented
5C2×#380C212yesmaximal, complemented
6#1444C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1436C212yesmaximal, complemented
10C23×D4C2211yescharacteristic
11C24×C4C2211yescharacteristic
12C24×C4C2211yescharacteristic
13C22×C4⋊C4C2212yes—
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C42C2211yescharacteristic
24#390—24no—
25#386—24no—
26#386C418yescomplemented
27#385—28no—
28#380—28no—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C25C2311yescharacteristic
37C25C2311yescharacteristic
38C22×D4—41no—
39C22×D4—41no—
40C22×D4—41no—
41C22×D4—41no—
42C22×D4C2×C414yes—
43C22×D4D414yes—
44C22×D4Q814yes—
45C23×C4—24no—
46C23×C4—24no—
47C23×C4—22no—
48C23×C4—24no—
49C23×C4—41no—
50C23×C4—41no—
51C23×C4—42no—
52C23×C4—41no—
53C23×C4—41no—
54C23×C4C2×C414yescomplemented
55C23×C4C2312yes—
56C23×C4C2312yes—
57C23×C4C2312yes—
58C23×C4C2312yes—
59C23×C4C2312yes—
60C23×C4C2311yescharacteristic
61C23×C4C2312yes—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4C2×C418yes—
91C2×C22⋊C4C2×C414yescomplemented
92C2×C42—22no—
93C2×C42—41no—
94C2×C42—41no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—42no—
99#95—42no—
100#95C2×C418yescomplemented
101#95D418yescomplemented
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—21no—
109C24—21no—
110C24—21no—
111C24—21no—
112C24—21no—
113C24—21no—
114C24—21no—
115C24—21no—
116C24—22no—
117C24—22no—
118C24C22×C412yes—
119C24C22×C412yes—
120C24C2×D412yes—
121C24C2×D412yes—
122C24C2×Q812yes—
123C24C2×Q812yes—
124C24C2411yescharacteristic
125C2×D4—44no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—44no—
131C2×D4C4⋊C4116yescomplemented
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
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—216no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—48no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—48no—
179C22×C4—48no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4C22×C414yes—
183C22×C4C22×C412yes—
184C22×C4C22×C414yes—
185C22×C4C2×D414yes—
186C22×C4C2×D414yes—
187C22×C4C2×D412yes—
188C22×C4C2×D414yes—
189C22×C4C2×Q812yes—
190C22×C4#4114yes—
191C4⋊C4—48no—
192C4⋊C4—48no—
193C4⋊C4—44no—
194C4⋊C4—48no—
195C22⋊C4—216no—
196C22⋊C4—44no—
197C22⋊C4—48no—
198C22⋊C4—48no—
199C22⋊C4—48no—
200C22⋊C4—44no—
201C42—44no—
202C23—24no—
203C23—24no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—24no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—24no—
219C23—24no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—24no—
225C23—24no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23—42no—
237C23—42no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23C2×C4⋊C414yes—
247C23C2×C4⋊C414yes—
248C23C23×C411yescharacteristic
249C23C22×D411yescharacteristic
250C23C22×D412yes—
251C23C22×Q811yescharacteristic
252C23C2×#4112yes—
253C23H(5,𝔽2)11yescharacteristic
254C23H(5,𝔽2)11yescharacteristic
255C23H(5,𝔽2)11yescharacteristic
256C23H(5,𝔽2)11yescharacteristic
257C23H(5,𝔽2)12yes—
258C232−1+412yes—
259D4—416no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—416no—
282C2×C4—416no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4C2×C4⋊C414yes—
310C2×C4C4×D418yes—
311C22—24no—
312C22—24no—
313C22—28no—
314C22—28no—
315C22—44no—
316C22—44no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22—44no—
328C22—44no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22C22×C4⋊C411yescharacteristic
334C22C2×C4×D412yes—
335C22#51811yescharacteristic
336C22#51811yescharacteristic
337C22#52012yes—
338C22#53411yescharacteristic
339C22#53411yescharacteristic
340C22#53512yes—
341C22#53712yes—
342C22#54012yes—
343C22#54212yes—
344C22#54311yescharacteristic
345C22#54311yescharacteristic
346C22D4212yes—
347C22D4211yescharacteristic
348C22#54612yes—
349C22#54712yes—
350C22D4×Q812yes—
351C22#55111yescharacteristic
352C22#55412yes—
353C22#56011yescharacteristic
354C22#56212yes—
355C22#56311yescharacteristic
356C4—24no—
357C4—28no—
358C4—44no—
359C4—44no—
360C4—44no—
361C4—82no—
362C4—82no—
363C2—44no—
364C2—44no—
365C2#231511yescharacteristic
366C2#232412yes—
367C2D4×C4⋊C412yes—
368C2#235811yescharacteristic
369C2#237611yescharacteristic
370C2#266911yescharacteristic
371C2#267112yes—
372C2#268311yescharacteristic
373C2#269212yes—
374C2#271711yescharacteristic
375C2#274811yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC23×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#385C2semidirect
5. C2×#380C2semidirect
6. #1444C2semidirect
7. #1439C2semidirect
8. #1439C2semidirect
9. #1436C2semidirect
26. #386C4semidirect
54. C23×C4C2×C4semidirect
91. C2×C22⋊C4C2×C4semidirect
100. #95C2×C4semidirect
101. #95D4semidirect
131. C2×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3902
C2×#3862
C2×#3852
C2×#3802
#14442
#14392
#14362