Finite Group Databasebeta

← #20995

Subgroups — #20995

Order 256.

832 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#575C211yesmaximal, characteristic
2C2×#568C211yesmaximal, characteristic, complemented
3C2×#421C211yesmaximal, characteristic, complemented
4#2896C211yesmaximal, characteristic, complemented
5C2×#413C211yesmaximal, characteristic, complemented
6#1903C214yesmaximal, complemented
7#1902C214yesmaximal, complemented
8C2×#344C211yesmaximal, characteristic, complemented
9C2×#338C211yesmaximal, 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
21#568—22no—
22#568—22no—
23#568—22no—
24#568C2214yes—
25C2×#131—22no—
26C2×#131—22no—
27C22×M16—22no—
28C22×M16C2211yescharacteristic
29C22×M16C2211yescharacteristic
30C22×M16C2211yescharacteristic
31C22×M16C2211yescharacteristic
32C2×#117C2211yescharacteristic
33#421—22no—
34#421C2214yes—
35C2×#104—22no—
36#419C2212yescomplemented
37#418C2212yescomplemented
38#417—22no—
39#416—22no—
40#413—22no—
41#413—22no—
42C2×#101C2212yescomplemented
43C2×#100C2212yescomplemented
44#344C2214yescomplemented
45#338C2214yescomplemented
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×C412yes—
65C2×#41D412yes—
66C2×#41D412yes—
67C2×#41D412yes—
68C2×#41D412yes—
69C2×#41C2312yes—
70C22×Q8C2×C411yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×D4—21no—
73C22×D4—21no—
74C22×D4C2×C411yescharacteristic
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
93#131—28no—
94#131—28no—
95#131—28no—
96#131—28no—
97C2×M16—24no—
98C2×M16—28no—
99C2×M16—21no—
100C2×M16—21no—
101C2×M16—21no—
102C2×M16—21no—
103C2×M16—21no—
104C2×M16—21no—
105C2×M16—21no—
106C2×M16—21no—
107C2×M16—21no—
108C2×M16—21no—
109C2×M16—21no—
110C2×M16—28no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16C2×C412yes—
117C2×M16D412yes—
118C2×M16D412yes—
119C2×M16D412yes—
120C2×M16C2312yes—
121C2×M16C2312yes—
122C2×M16C2312yes—
123C2×M16C2312yes—
124C22×C8—24no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8C2×C412yes—
128C22×C8D412yes—
129#117—21no—
130#117—21no—
131#117—21no—
132#117C2312yes—
133C2×C4⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C42—21no—
136#104—28no—
137#103—24no—
138#102—24no—
139#101—24no—
140#100—24no—
141C24—21no—
142C24—21no—
143C24#4111yescharacteristic
144#41—22no—
145#41—22no—
146#41—22no—
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—41no—
165#41—41no—
166#41—41no—
167#41—41no—
168#41—41no—
169#41—41no—
170#41—41no—
171#41—41no—
172#41C22⋊C414yes—
173#41C22⋊C414yes—
174#41C2×D414yes—
175#41C2×D414yes—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8C22⋊C412yes—
185C2×Q8C22⋊C412yes—
186C2×Q8C22×C412yes—
187C2×Q8C22×C412yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C22⋊C412yes—
209C2×D4C22⋊C412yes—
210C2×D4C22×C412yes—
211C2×D4C22×C412yes—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4C22×C411yescharacteristic
228C22×C4C22×C411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2×D411yescharacteristic
232C22×C4C2×D411yescharacteristic
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D411yescharacteristic
235C22×C4C2×D411yescharacteristic
236C22×C4C2×D411yescharacteristic
237C22×C4#4111yescharacteristic
238C22×C4#4111yescharacteristic
239C22×C4#4111yescharacteristic
240C22×C4C2411yescharacteristic
241Q16—28no—
242SD16—28no—
243SD16—28no—
244D8—28no—
245M16—216no—
246M16—22no—
247M16—22no—
248M16—22no—
249M16—22no—
250M16—22no—
251M16—22no—
252M16—22no—
253M16—22no—
254M16—22no—
255M16—22no—
256M16—22no—
257M16—22no—
258M16—22no—
259M16—41no—
260M16—41no—
261M16C2×D414yes—
262C2×C8—216no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8C22×C414yes—
274C2×C8C2×D414yes—
275C4⋊C4—22no—
276C4⋊C4—41no—
277C22⋊C4—22no—
278C22⋊C4—41no—
279C42—22no—
280C42—41no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—41no—
286C23—41no—
287C23—81no—
288C23—82no—
289C23—82no—
290C23C4×D411yescharacteristic
291C23C4×D411yescharacteristic
292C23#12111yescharacteristic
293C23#12111yescharacteristic
294C23#12311yescharacteristic
295C23#12311yescharacteristic
296C23C2×#4111yescharacteristic
297Q8—22no—
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8—22no—
302Q8—41no—
303Q8—41no—
304Q8C2×C22⋊C414yes—
305D4—24no—
306D4—24no—
307D4—24no—
308D4—24no—
309D4—22no—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—22no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—42no—
318D4—41no—
319D4—41no—
320D4C2×C22⋊C414yes—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4C2×C22⋊C411yescharacteristic
338C2×C4C2×C22⋊C411yescharacteristic
339C2×C4C2×C22⋊C411yescharacteristic
340C2×C4C2×C22⋊C411yescharacteristic
341C2×C4C4×D411yescharacteristic
342C2×C4C4×D411yescharacteristic
343C2×C4C4×D411yescharacteristic
344C2×C4C4×D411yescharacteristic
345C2×C4C4×D411yescharacteristic
346C2×C4C4×D411yescharacteristic
347C2×C4#12011yescharacteristic
348C2×C4#12011yescharacteristic
349C2×C4#12011yescharacteristic
350C2×C4#12011yescharacteristic
351C2×C4#12111yescharacteristic
352C2×C4#12111yescharacteristic
353C2×C4#12111yescharacteristic
354C2×C4#12111yescharacteristic
355C2×C4#12111yescharacteristic
356C2×C4#12111yescharacteristic
357C2×C4#12311yescharacteristic
358C2×C4#12311yescharacteristic
359C2×C4C23×C411yescharacteristic
360C2×C4C22×D411yescharacteristic
361C2×C4C22×D411yescharacteristic
362C2×C4C22×D411yescharacteristic
363C2×C4C22×D411yescharacteristic
364C2×C4C2×#4111yescharacteristic
365C8—24no—
366C8—24no—
367C8—24no—
368C8—42no—
369C22—24no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—44no—
378C22—42no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22#38611yescharacteristic
385C22#38611yescharacteristic
386C22#38611yescharacteristic
387C22#38611yescharacteristic
388C22C2×C4×D411yescharacteristic
389C22C2×#12111yescharacteristic
390C22C2×#12311yescharacteristic
391C4—24no—
392C4—42no—
393C4—82no—
394C4#38611yescharacteristic
395C4#38611yescharacteristic
396C4#38611yescharacteristic
397C4#38611yescharacteristic
398C4C22×C22⋊C411yescharacteristic
399C4C2×C4×D411yescharacteristic
400C4C2×#12011yescharacteristic
401C4C2×#12111yescharacteristic
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—44no—
410C2—82no—
411C2C2×#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. C2×#568C2semidirect
3. C2×#421C2semidirect
4. #2896C2semidirect
5. C2×#413C2semidirect
6. #1903C2semidirect
7. #1902C2semidirect
8. C2×#344C2semidirect
9. C2×#338C2semidirect
16. C2×#137C4semidirect
18. C2×AGL(1,ℤ/8)C4semidirect
20. C2×#135C4semidirect
36. #419C22semidirect
37. #418C22semidirect
42. C2×#101C22semidirect
43. C2×#100C22semidirect
44. #344C22semidirect
45. #338C22semidirect
89. C2×Q16C2×C4semidirect
90. C2×SD16C2×C4semidirect
91. C2×SD16C2×C4semidirect
92. C2×D8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5752
C2×#5682
C2×#4212
#28962
C2×#4132
#19032
#19022
C2×#3442
C2×#3382