Finite Group Databasebeta

← #16361

Subgroups — #16361

Order 256.

1113 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#121C211yesmaximal, characteristic
2C2×#394C212yesmaximal, complemented
3C2×#381C211yesmaximal, characteristic, complemented
4#1444C211yesmaximal, characteristic, complemented
5#1442C211yesmaximal, characteristic, complemented
6#1440C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1438C212yesmaximal, complemented
9#1438C211yesmaximal, characteristic, complemented
10#1438C211yesmaximal, characteristic, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C23×D4C2212yes—
14C24×C4C2211yescharacteristic
15C2×#121—28no—
16C2×#121—24no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2212yes—
27#394—28no—
28#381—24no—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C25C2312yes—
40C25C2311yescharacteristic
41C22×D4—24no—
42C22×D4—24no—
43C22×D4—24no—
44C22×D4—22no—
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—42no—
48C22×D4—42no—
49C22×D4—42no—
50C22×D4—42no—
51C22×D4—42no—
52C22×D4D418yes—
53C23×C4—24no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—42no—
57C23×C4D414yes—
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C23×C4C2312yes—
63C23×C4C2311yescharacteristic
64C23×C4C2312yes—
65C23×C4C2312yes—
66C23×C4C2311yescharacteristic
67#121—232no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—22no—
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—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95#95—22no—
96#95—22no—
97#95—22no—
98#95—22no—
99#95—24no—
100#95—22no—
101#95—22no—
102#95—42no—
103#95—42no—
104#95—42no—
105C24—22no—
106C24—21no—
107C24—21no—
108C24—21no—
109C24—22no—
110C24—21no—
111C24—21no—
112C24—22no—
113C24—21no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—22no—
122C24—22no—
123C24—22no—
124C24—22no—
125C24—22no—
126C24—22no—
127C24—22no—
128C24C2×D414yes—
129C24C2×D414yes—
130C24C2×D412yes—
131C24C2411yescharacteristic
132C2×D4—28no—
133C2×D4—216no—
134C2×D4—48no—
135C2×D4—48no—
136C2×D4—48no—
137C2×D4—48no—
138C2×D4—44no—
139C2×D4—48no—
140C2×D4—48no—
141C2×D4—48no—
142C2×D4—48no—
143C2×D4—44no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—28no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
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—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—44no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4C2×D412yes—
195C22×C4C2×D414yes—
196C22×C4C2×D412yes—
197C22×C4#4114yes—
198C22×C4#4112yes—
199C22×C4#4112yes—
200C4⋊C4—28no—
201C4⋊C4—48no—
202C22⋊C4—216no—
203C22⋊C4—48no—
204C22⋊C4—48no—
205C22⋊C4—48no—
206C22⋊C4—44no—
207C22⋊C4—48no—
208C22⋊C4—48no—
209C22⋊C4—44no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—24no—
214C23—24no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—24no—
219C23—24no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—24no—
225C23—24no—
226C23—24no—
227C23—24no—
228C23—24no—
229C23—42no—
230C23—41no—
231C23—41no—
232C23—42no—
233C23—41no—
234C23—41no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23#12014yes—
268C23C22×D411yescharacteristic
269C23C22×D412yes—
270C23C2×#4111yescharacteristic
271C23C2×#4111yescharacteristic
272C23C2×#4112yes—
273C23H(5,𝔽2)11yescharacteristic
274C23H(5,𝔽2)11yescharacteristic
275C23H(5,𝔽2)12yes—
276C23H(5,𝔽2)12yes—
277C23H(5,𝔽2)11yescharacteristic
278C232−1+411yescharacteristic
279D4—416no—
280D4—432no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—416no—
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—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4#12314yes—
333C22—24no—
334C22—24no—
335C22—24no—
336C22—42no—
337C22—42no—
338C22—44no—
339C22—44no—
340C22—44no—
341C22—44no—
342C22—42no—
343C22—42no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—82no—
352C22—82no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—82no—
357C22—82no—
358C22—82no—
359C22C2×#12011yescharacteristic
360C22C2×#12311yescharacteristic
361C22#53211yescharacteristic
362C22#53412yes—
363C22#53412yes—
364C22#53411yescharacteristic
365C22#53511yescharacteristic
366C22#53511yescharacteristic
367C22#53711yescharacteristic
368C22#53812yes—
369C22#54011yescharacteristic
370C22#54011yescharacteristic
371C22#54012yes—
372C22#54211yescharacteristic
373C22#54211yescharacteristic
374C22#54612yes—
375C22#54612yes—
376C22#54612yes—
377C22#54712yes—
378C22#55312yes—
379C22#55512yes—
380C22#56012yes—
381C22#56212yes—
382C4—24no—
383C4—44no—
384C4—42no—
385C4—44no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—82no—
390C4—81no—
391C2—44no—
392C2—84no—
393C2#240511yescharacteristic
394C2#243011yescharacteristic
395C2#250011yescharacteristic
396C2#261311yescharacteristic
397C2#262411yescharacteristic
398C2#266111yescharacteristic
399C2#268612yes—
400C2#268812yes—
401C2#269912yes—
402C2#270112yes—
403C2#281311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC22×#121C2
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×#394C2semidirect
3. C2×#381C2semidirect
4. #1444C2semidirect
5. #1442C2semidirect
6. #1440C2semidirect
7. #1439C2semidirect
8. #1438C2semidirect
9. #1438C2semidirect
10. #1438C2semidirect
11. C22⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1212
C2×#3942
C2×#3812
#14442
#14422
#14402
#14392
#14382
C22⋊#952