Finite Group Databasebeta

← #20998

Subgroups — #20998

Order 256.

852 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#575C211yesmaximal, characteristic
2#3428C211yesmaximal, characteristic, complemented
3C2×#421C211yesmaximal, characteristic, complemented
4#2896C211yesmaximal, characteristic, complemented
5C2×#410C211yesmaximal, characteristic, complemented
6#1905C214yesmaximal, complemented
7#1904C214yesmaximal, complemented
8C2×#344C211yesmaximal, characteristic, complemented
9C2×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C22×#41C2211yescharacteristic
12#575—22no—
13#575—22no—
14#575—22no—
15#575—22no—
16C2×#137C412yescomplemented
17C2×#137C2212yes—
18C2×AGL(1,ℤ/8)C412yescomplemented
19C2×AGL(1,ℤ/8)C2212yes—
20C2×#135C414yescomplemented
21C22×M16C2211yescharacteristic
22C22×M16C2211yescharacteristic
23#520—22no—
24#520—22no—
25#519C2212yescomplemented
26#518—22no—
27#518C2212yescomplemented
28C4×#41—22no—
29C4×#41—22no—
30C2×#117—22no—
31C2×#117C2211yescharacteristic
32C2×#117C2211yescharacteristic
33C2×#117C2211yescharacteristic
34#421—22no—
35#421C2214yes—
36C2×#104—22no—
37#419C2212yescomplemented
38#418C2212yescomplemented
39#417—22no—
40#416—22no—
41#410—22no—
42#410C2214yes—
43C2×#99—22no—
44#344C2214yescomplemented
45H(3,ℤ/4)C2214yescomplemented
46C2×#41—21no—
47C2×#41—21no—
48C2×#41—21no—
49C2×#41—21no—
50C2×#41—21no—
51C2×#41—21no—
52C2×#41—21no—
53C2×#41—21no—
54C2×#41—21no—
55C2×#41—21no—
56C2×#41—21no—
57C2×#41—21no—
58C2×#41—21no—
59C2×#41—21no—
60C2×#41—21no—
61C2×#41—21no—
62C2×#41C2×C412yes—
63C2×#41C2×C412yes—
64C2×#41C2×C412yescomplemented
65C2×#41D412yes—
66C2×#41D412yes—
67C2×#41D412yes—
68C2×#41D412yes—
69C2×#41C2312yes—
70C22×Q8C2×C411yescharacteristic, complemented
71C22×Q8C2311yescharacteristic
72C22×D4—21no—
73C22×D4—21no—
74C22×D4C2×C411yescharacteristic, complemented
75C22×D4C2311yescharacteristic
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79#137—24no—
80#137—24no—
81#137—24no—
82#137—24no—
83AGL(1,ℤ/8)—24no—
84AGL(1,ℤ/8)—24no—
85AGL(1,ℤ/8)—24no—
86AGL(1,ℤ/8)—24no—
87#135—28no—
88#135—28no—
89C2×Q16C2×C414yescomplemented
90C2×SD16C2×C414yescomplemented
91C2×SD16C2×C414yescomplemented
92C2×D8C2×C414yescomplemented
93C2×M16—21no—
94C2×M16—21no—
95C2×M16—21no—
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16—21no—
100C2×M16—21no—
101C2×M16—21no—
102C2×M16C2×C412yescomplemented
103C2×M16C2312yes—
104C2×M16C2312yes—
105C22×C8—21no—
106C22×C8C2×C412yescomplemented
107C4×Q8—24no—
108C4×Q8—24no—
109C4×D4—28no—
110C4×D4—28no—
111C4×D4—24no—
112C4×D4—24no—
113#117—24no—
114#117—24no—
115#117—21no—
116#117—21no—
117#117—21no—
118#117—21no—
119#117—24no—
120#117—21no—
121#117—21no—
122#117—21no—
123#117D412yes—
124#117D412yes—
125#117C2312yes—
126#117C2312yes—
127#117C2312yes—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4D412yes—
136C2×C42—24no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42D412yes—
140#104—28no—
141#103—24no—
142#102—24no—
143#99—28no—
144C24—21no—
145C24—21no—
146C24C2×D411yescharacteristic
147#41—22no—
148#41—22no—
149#41—22no—
150#41—22no—
151#41—22no—
152#41—22no—
153#41—22no—
154#41—22no—
155#41—22no—
156#41—22no—
157#41—22no—
158#41—22no—
159#41—22no—
160#41—22no—
161#41—22no—
162#41—22no—
163#41—22no—
164#41—22no—
165#41—22no—
166#41—22no—
167#41—41no—
168#41—41no—
169#41—41no—
170#41—41no—
171#41—41no—
172#41—41no—
173#41—41no—
174#41—41no—
175#41C22⋊C414yes—
176#41C22⋊C414yes—
177#41C22⋊C414yes—
178#41C22⋊C414yes—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8C22×C412yes—
188C2×Q8C22×C412yes—
189C2×Q8C2×D412yes—
190C2×Q8C2×D412yes—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4C22×C412yes—
212C2×D4C22×C412yes—
213C2×D4C2×D412yes—
214C2×D4C2×D412yes—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—28no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4C22×C411yescharacteristic
234C22×C4C22×C411yescharacteristic
235C22×C4C2×D411yescharacteristic
236C22×C4C2×D411yescharacteristic
237C22×C4C2×D411yescharacteristic
238C22×C4C2×D411yescharacteristic
239C22×C4C2×D411yescharacteristic
240C22×C4C2×D411yescharacteristic
241C22×C4C2×D411yescharacteristic
242C22×C4#4111yescharacteristic
243C22×C4#4111yescharacteristic
244C22×C4#4111yescharacteristic
245C22×C4#4111yescharacteristic
246C22×C4C2411yescharacteristic
247Q16—28no—
248SD16—28no—
249SD16—28no—
250D8—28no—
251M16—22no—
252M16—22no—
253M16—22no—
254M16—22no—
255M16—22no—
256M16—22no—
257M16—41no—
258M16—41no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8C22×C414yes—
265C4⋊C4—28no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—41no—
272C22⋊C4—28no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—41no—
278C22⋊C4C2×D414yes—
279C42—28no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42C2×D414yes—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—41no—
291C23—41no—
292C23—81no—
293C23—82no—
294C23—82no—
295C23C4×D411yescharacteristic
296C23C4×D411yescharacteristic
297C23#12011yescharacteristic
298C23#12011yescharacteristic
299C23#12111yescharacteristic
300C23#12111yescharacteristic
301C23C22×D411yescharacteristic
302Q8—22no—
303Q8—22no—
304Q8—22no—
305Q8—22no—
306Q8—22no—
307Q8—41no—
308Q8—41no—
309Q8C2×C22⋊C414yes—
310D4—24no—
311D4—24no—
312D4—24no—
313D4—24no—
314D4—22no—
315D4—22no—
316D4—22no—
317D4—22no—
318D4—22no—
319D4—42no—
320D4—42no—
321D4—42no—
322D4—42no—
323D4—41no—
324D4—41no—
325D4C2×C22⋊C414yes—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—88no—
346C2×C4—82no—
347C2×C4C2×C22⋊C411yescharacteristic
348C2×C4C2×C22⋊C411yescharacteristic
349C2×C4C2×C22⋊C411yescharacteristic
350C2×C4C2×C22⋊C411yescharacteristic
351C2×C4C4×D411yescharacteristic
352C2×C4C4×D411yescharacteristic
353C2×C4C4×D411yescharacteristic
354C2×C4C4×D411yescharacteristic
355C2×C4C4×D411yescharacteristic
356C2×C4C4×D411yescharacteristic
357C2×C4#12011yescharacteristic
358C2×C4#12011yescharacteristic
359C2×C4#12111yescharacteristic
360C2×C4#12111yescharacteristic
361C2×C4#12111yescharacteristic
362C2×C4#12111yescharacteristic
363C2×C4#12111yescharacteristic
364C2×C4#12111yescharacteristic
365C2×C4#12311yescharacteristic
366C2×C4#12311yescharacteristic
367C2×C4#12311yescharacteristic
368C2×C4#12311yescharacteristic
369C2×C4C23×C411yescharacteristic
370C2×C4C22×D411yescharacteristic
371C2×C4C22×D411yescharacteristic
372C2×C4C22×D411yescharacteristic
373C2×C4C2×#4111yescharacteristic
374C2×C4C2×#4111yescharacteristic
375C8—24no—
376C8—42no—
377C22—24no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—44no—
386C22—42no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22#38611yescharacteristic
393C22#38611yescharacteristic
394C22#38611yescharacteristic
395C22#38611yescharacteristic
396C22C2×C4×D411yescharacteristic
397C22C2×#12011yescharacteristic
398C22C2×#12111yescharacteristic
399C4—24no—
400C4—44no—
401C4—44no—
402C4—42no—
403C4—82no—
404C4#38611yescharacteristic
405C4#38611yescharacteristic
406C4#38611yescharacteristic
407C4#38611yescharacteristic
408C4C22×C22⋊C411yescharacteristic
409C4C2×C4×D411yescharacteristic
410C4C2×#12111yescharacteristic
411C4C2×#12311yescharacteristic
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—44no—
420C2—82no—
421C2C2×#38611yescharacteristic
subgroupG/H
centerC4#386
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC2C2×#386
prime radicalC2×#575C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3428C2semidirect
3. C2×#421C2semidirect
4. #2896C2semidirect
5. C2×#410C2semidirect
6. #1905C2semidirect
7. #1904C2semidirect
8. C2×#344C2semidirect
9. C2×H(3,ℤ/4)C2semidirect
16. C2×#137C4semidirect
18. C2×AGL(1,ℤ/8)C4semidirect
20. C2×#135C4semidirect
25. #519C22semidirect
27. #518C22semidirect
37. #419C22semidirect
38. #418C22semidirect
44. #344C22semidirect
45. H(3,ℤ/4)C22semidirect
64. C2×#41C2×C4semidirect
70. C22×Q8C2×C4semidirect
74. C22×D4C2×C4semidirect
89. C2×Q16C2×C4semidirect
90. C2×SD16C2×C4semidirect
91. C2×SD16C2×C4semidirect
92. C2×D8C2×C4semidirect
102. C2×M16C2×C4semidirect
106. C22×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5752
#34282
C2×#4212
#28962
C2×#4102
#19052
#19042
C2×#3442
C2×H(3,ℤ/4)2