Finite Group Databasebeta

← #25152

Subgroups — #25152

Order 256.

651 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#463C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic
4C2×C4×SD16C211yesmaximal, characteristic, complemented
5#2426C211yesmaximal, characteristic, complemented
6D4×C22⋊C4C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1923C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1879C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C2×#124C2211yescharacteristic, complemented
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic, complemented
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C22×C22⋊C4—21no—
28#463—24no—
29#450—24no—
30#447—24no—
31C4×SD16—24no—
32C2×#107C2211yescharacteristic, complemented
33C2×#106C2211yescharacteristic, complemented
34C2×C4⋊C8C2211yescharacteristic, complemented
35C2×#103C2211yescharacteristic, complemented
36C2×#103C2211yescharacteristic, complemented
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic, complemented
42C2×#102C2211yescharacteristic, complemented
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic, complemented
46#396—21no—
47#394—21no—
48#390—21no—
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#390C2211yescharacteristic, complemented
52#389C2211yescharacteristic, complemented
53#388—21no—
54#386—21no—
55#386—21no—
56#380C2211yescharacteristic
57#379—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×Q8C2311yescharacteristic
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—41no—
84C2×SD16D414yescomplemented
85C2×D8—22no—
86C2×D8—22no—
87C2×D8—22no—
88C2×D8—22no—
89C2×D8—22no—
90C2×D8—41no—
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95#124—22no—
96#124—41no—
97C22⋊Q8—22no—
98C22⋊Q8—41no—
99#121—22no—
100#121—41no—
101C4×Q8—22no—
102C4×Q8—41no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—41no—
106C4×D4—41no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—44no—
128C2×C22⋊C4D411yescharacteristic
129C2×C22⋊C4D411yescharacteristic
130C2×C42—21no—
131C2×C42C2311yescharacteristic
132C2×C42C2311yescharacteristic
133#107—41no—
134#106—22no—
135C4⋊C8—22no—
136#103—22no—
137#103—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—41no—
143#102D414yes—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C4×C8—22no—
147#95—21no—
148#95—21no—
149#95D411yescharacteristic
150#95D411yescharacteristic
151C24—21no—
152C24—24no—
153C24—21no—
154C24—21no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—42no—
159C24—81no—
160C24—81no—
161C24C2×D411yescharacteristic
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8C2×D412yes—
168C2×D4—24no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—44no—
189C2×D4—44no—
190C2×D4—42no—
191C2×D4—81no—
192C2×D4C2×D412yes—
193C2×D4C2×D412yes—
194C2×D4#4112yes—
195C2×D4#4112yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2411yescharacteristic
219SD16—28no—
220SD16—44no—
221D8—44no—
222D8—44no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8C2×D412yes—
234C2×C8C2×D412yes—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—81no—
245C4⋊C4C2×D412yes—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—81no—
259C22⋊C4—81no—
260C22⋊C4—81no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—42no—
266C23—21no—
267C23—22no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—22no—
276C23—48no—
277C23—48no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—84no—
296C23—84no—
297C23—84no—
298C23—84no—
299C23—84no—
300C23#13512yes—
301C23C22×D411yescharacteristic
302Q8—22no—
303Q8—22no—
304Q8—41no—
305Q8—41no—
306D4—22no—
307D4—22no—
308D4—22no—
309D4—22no—
310D4—22no—
311D4—48no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—84no—
319D4#12114yes—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4AGL(1,ℤ/8)12yes—
356C2×C4C22×D411yescharacteristic
357C2×C4C22×D411yescharacteristic
358C2×C4C2×#4111yescharacteristic
359C2×C4C2×#4111yescharacteristic
360C2×C4H(5,𝔽2)11yescharacteristic
361C2×C4H(5,𝔽2)11yescharacteristic
362C8—22no—
363C8—22no—
364C8—41no—
365C8—41no—
366C22—22no—
367C22—24no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—41no—
372C22—42no—
373C22—41no—
374C22—42no—
375C22—41no—
376C22—42no—
377C22—41no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—816no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22#52511yescharacteristic
396C22D4211yescharacteristic
397C22D4211yescharacteristic
398C22C2×#13511yescharacteristic
399C22C2×AGL(1,ℤ/8)11yescharacteristic
400C22#57611yescharacteristic
401C22#57711yescharacteristic
402C4—22no—
403C4—41no—
404C4—42no—
405C4—41no—
406C4—41no—
407C4—42no—
408C4—41no—
409C4—81no—
410C4—81no—
411C4—81no—
412C4C2×#12111yescharacteristic
413C4#53811yescharacteristic
414C4#54611yescharacteristic
415C4#54611yescharacteristic
416C2—42no—
417C2—44no—
418C2—82no—
419C2—161no—
420C2#242411yescharacteristic
421C2#310511yescharacteristic
422C2#311811yescharacteristic
423C2#326511yescharacteristic
424C2#327411yescharacteristic
425C2#328011yescharacteristic
426C2#328211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#463C2semidirect
2. C2×#450C2semidirect
4. C2×C4×SD16C2semidirect
5. #2426C2semidirect
6. D4×C22⋊C4C2semidirect
7. #2065C2semidirect
8. #2036C2semidirect
9. #2034C2semidirect
10. #2012C2semidirect
11. #1970C2semidirect
12. #1923C2semidirect
13. #1880C2semidirect
14. #1879C2semidirect
15. #1862C2semidirect
21. C2×#124C22semidirect
24. C2×C4×Q8C22semidirect
32. C2×#107C22semidirect
33. C2×#106C22semidirect
34. C2×C4⋊C8C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
41. C2×#102C22semidirect
42. C2×#102C22semidirect
45. C2×C4×C8C22semidirect
51. #390C22semidirect
52. #389C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect
84. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
C2×#4502
C2×#4472
C2×C4×SD162
#24262
D4×C22⋊C42
#20652
#20362
#20342
#20122
#19702
#19232
#18802
#18792
#18622