Finite Group Databasebeta

← #16359

Subgroups — #16359

Order 256.

951 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123C211yesmaximal, characteristic
2C2×#398C211yesmaximal, characteristic, complemented
3C2×#397C211yesmaximal, characteristic, complemented
4C2×#380C211yesmaximal, characteristic, complemented
5#1442C212yesmaximal, complemented
6#1441C211yesmaximal, characteristic, complemented
7#1439C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1439C211yesmaximal, characteristic, complemented
10#1438C212yesmaximal
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C2×#123—24no—
15C2×#123—24no—
16C2×#123—24no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2212yes—
21C22×C4⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2212yes—
28C22×C22⋊C4C2212yes—
29#398—24no—
30#397—24no—
31#380—24no—
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C25C2311yescharacteristic
43C22×D4—22no—
44C22×D4—22no—
45C22×D4—24no—
46C22×D4—22no—
47C22×D4—42no—
48C23×C4—24no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—42no—
52C23×C4D414yes—
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2312yes—
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
62#123—232no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4D414yes—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4D414yes—
98#95—24no—
99#95—22no—
100#95—24no—
101#95—22no—
102#95—24no—
103#95—24no—
104#95—42no—
105#95—42no—
106#95—42no—
107C24—21no—
108C24—21no—
109C24—22no—
110C24—21no—
111C24—21no—
112C24—21no—
113C24—22no—
114C24—22no—
115C24—21no—
116C24—21no—
117C24—21no—
118C24—22no—
119C24—22no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—22no—
124C24—21no—
125C24—22no—
126C24C2×D412yes—
127C24C2×D412yes—
128C24#4114yes—
129C24C2411yescharacteristic
130C2×D4—28no—
131C2×D4—48no—
132C2×D4—48no—
133C2×D4—44no—
134C2×D4—44no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—28no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—28no—
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—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—44no—
185C22×C4—48no—
186C22×C4—48no—
187C22×C4C2×D412yes—
188C22×C4C2×D412yes—
189C22×C4C2×D412yes—
190C22×C4C2×D412yes—
191C22×C4C2×D412yes—
192C22×C4C2×D412yes—
193C22×C4C2×D412yes—
194C22×C4#4114yes—
195C4⋊C4—28no—
196C4⋊C4—28no—
197C4⋊C4—44no—
198C4⋊C4—48no—
199C4⋊C4—44no—
200C22⋊C4—28no—
201C22⋊C4—28no—
202C22⋊C4—28no—
203C22⋊C4—48no—
204C22⋊C4—48no—
205C22⋊C4—48no—
206C22⋊C4—48no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—24no—
214C23—24no—
215C23—24no—
216C23—24no—
217C23—24no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—24no—
223C23—24no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23#12314yes—
253C23C22×D411yescharacteristic
254C23C22×D411yescharacteristic
255C23C22×D411yescharacteristic
256C23C2×#4112yes—
257C23C2×#4112yes—
258C23H(5,𝔽2)12yes—
259C23H(5,𝔽2)11yescharacteristic
260C23H(5,𝔽2)11yescharacteristic
261C23H(5,𝔽2)11yescharacteristic
262C232−1+412yes—
263C232−1+411yescharacteristic
264D4—416no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—416no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
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×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4#12014yes—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—44no—
328C22—44no—
329C22—44no—
330C22—44no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—44no—
336C22—44no—
337C22—81no—
338C22—81no—
339C22—82no—
340C22—82no—
341C22—81no—
342C22—81no—
343C22C2×#12011yescharacteristic
344C22C2×#12311yescharacteristic
345C22#53211yescharacteristic
346C22#53411yescharacteristic
347C22#53511yescharacteristic
348C22#53511yescharacteristic
349C22#53511yescharacteristic
350C22#53611yescharacteristic
351C22#53711yescharacteristic
352C22#53711yescharacteristic
353C22#53711yescharacteristic
354C22#53812yes—
355C22#53912yes—
356C22#53912yes—
357C22#54012yes—
358C22#54612yes—
359C22#54612yes—
360C22#54712yes—
361C22#54812yes—
362C22#55112yes—
363C22#55212yes—
364C22#56212yes—
365C22#56312yes—
366C4—24no—
367C4—42no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—44no—
372C4—82no—
373C4—82no—
374C4—82no—
375C2—44no—
376C2—82no—
377C2#240511yescharacteristic
378C2#243211yescharacteristic
379C2#249811yescharacteristic
380C2#261511yescharacteristic
381C2#262311yescharacteristic
382C2#266111yescharacteristic
383C2#269512yes—
384C2#269612yes—
385C2#270612yes—
386C2#270712yes—
387C2#281411yescharacteristic
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×#398C2semidirect
3. C2×#397C2semidirect
4. C2×#380C2semidirect
5. #1442C2semidirect
6. #1441C2semidirect
7. #1439C2semidirect
8. #1439C2semidirect
9. #1439C2semidirect
11. C22⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1232
C2×#3982
C2×#3972
C2×#3802
#14422
#14412
#14392
#14382
C22⋊#952