Finite Group Databasebeta

← #23482

Subgroups — #23482

Order 256.

825 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2C2×#428C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic, complemented
4C2×#417C211yesmaximal, characteristic, complemented
5C2×#407C211yesmaximal, characteristic, complemented
6C2×#380C211yesmaximal, characteristic, complemented
7#2036C212yesmaximal, complemented
8#2035C212yesmaximal, complemented
9#2034C212yesmaximal, complemented
10#2033C212yesmaximal, complemented
11#1736C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C22×M16C2211yescharacteristic
15C22×M16C2211yescharacteristic
16#525—24no—
17#525—24no—
18C2×#123—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#120—21no—
24C2×C4×D4—21no—
25C2×C4×D4—21no—
26C2×#117C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28C22×C4⋊C4C2211yescharacteristic
29C22×C22⋊C4—21no—
30#428—24no—
31C2×#107C2212yescomplemented
32C2×#106C2212yescomplemented
33#417—24no—
34#417—24no—
35C2×#103C2212yescomplemented
36C2×#103C2212yescomplemented
37C2×#102C2212yescomplemented
38C2×#102C2212yescomplemented
39#407—24no—
40C2×C22⋊C8C2212yes—
41C2×C22⋊C8C2212yes—
42#380—21no—
43#380—21no—
44#380C2212yes—
45#380C2212yes—
46C2×#95—21no—
47#340C2212yescomplemented
48#340C2212yescomplemented
49C25D411yescharacteristic
50C2×#41—22no—
51C2×#41—22no—
52C2×#41—22no—
53C2×#41—22no—
54C2×#41—22no—
55C2×#41D414yes—
56C22×Q8C2311yescharacteristic
57C22×D4—21no—
58C22×D4—21no—
59C22×D4—21no—
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4D411yescharacteristic
71C23×C4D411yescharacteristic
72C23×C4D411yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16D414yescomplemented
82C22×C8C2312yes—
83C22×C8C2312yes—
84#123—28no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89#121—22no—
90#121—22no—
91#121—22no—
92#121—22no—
93#120—28no—
94C4×D4—28no—
95C4×D4—28no—
96#117—22no—
97#117D414yes—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4C2312yes—
107C2×C4⋊C4C2312yes—
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—44no—
118C2×C42—21no—
119#107—24no—
120#106—24no—
121#103—24no—
122#103—24no—
123#102—24no—
124#102—24no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127#95—22no—
128#95—22no—
129C24—22no—
130C24—22no—
131C24—22no—
132C24—22no—
133C24—22no—
134C24—21no—
135C24—41no—
136C24—41no—
137C24C2×D411yescharacteristic
138C24C2×D411yescharacteristic
139C24C2×D411yescharacteristic
140#41—28no—
141#41—28no—
142#41—44no—
143#41—44no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8C2×D412yes—
150C2×D4—24no—
151C2×D4—24no—
152C2×D4—24no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4C2×D412yes—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D412yes—
208C22×C4C2×D412yes—
209C22×C4C2×D412yes—
210C22×C4#4112yes—
211C22×C4#4112yes—
212C22×C4#4112yes—
213C22×C4C2411yescharacteristic
214M16—28no—
215M16—44no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8C2×D414yes—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4C2×D412yes—
233C4⋊C4C2×D412yes—
234C22⋊C4—24no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C42—24no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—28no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—44no—
261C23—44no—
262C23—44no—
263C23—44no—
264C23—44no—
265C23—44no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#12012yes—
273C23#12112yes—
274C23#12312yes—
275C23AGL(1,ℤ/8)12yes—
276C23#13712yes—
277C23C22×D411yescharacteristic
278Q8—22no—
279Q8—22no—
280Q8—41no—
281Q8—41no—
282D4—22no—
283D4—22no—
284D4—48no—
285D4—48no—
286D4—48no—
287D4—41no—
288D4—41no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—48no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4#12012yes—
325C2×C4#12114yes—
326C2×C4#12114yes—
327C2×C4#12112yes—
328C2×C4#12314yes—
329C2×C4#12312yes—
330C2×C4#12414yes—
331C2×C4C22×D411yescharacteristic
332C2×C4C22×D411yescharacteristic
333C2×C4C22×D411yescharacteristic
334C2×C4C2×#4111yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C8—24no—
338C8—42no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—416no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22#39414yes—
368C22C2×#12011yescharacteristic
369C22C2×#12111yescharacteristic
370C22C2×#12311yescharacteristic
371C22C2×AGL(1,ℤ/8)11yescharacteristic
372C22C2×#13711yescharacteristic
373C22#57511yescharacteristic
374C22#57511yescharacteristic
375C4—41no—
376C4—41no—
377C4—42no—
378C4—42no—
379C4—42no—
380C4—82no—
381C4—82no—
382C4#39414yes—
383C4C2×#12111yescharacteristic
384C4C2×#12111yescharacteristic
385C4C2×#12311yescharacteristic
386C4C2×#12411yescharacteristic
387C2—24no—
388C2—42no—
389C2—42no—
390C2—82no—
391C2C2×#39411yescharacteristic
392C2#300611yescharacteristic
393C2#300711yescharacteristic
394C2#305511yescharacteristic
395C2#305611yescharacteristic
396C2#309211yescharacteristic
397C2#309311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#525C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#428C2semidirect
3. C2×#417C2semidirect
4. C2×#417C2semidirect
5. C2×#407C2semidirect
6. C2×#380C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #2034C2semidirect
10. #2033C2semidirect
11. #1736C2semidirect
31. C2×#107C22semidirect
32. C2×#106C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
37. C2×#102C22semidirect
38. C2×#102C22semidirect
47. #340C22semidirect
48. #340C22semidirect
81. C2×M16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4282
C2×#4172
C2×#4072
C2×#3802
#20362
#20352
#20342
#20332
#17362