Finite Group Databasebeta

← #20939

Subgroups — #20939

Order 256.

834 conjugacy classes of subgroups.

The subgroup lattice has 410 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
5#2879C211yesmaximal, characteristic, complemented
6#1881C218yesmaximal, complemented
7C2×#344C211yesmaximal, characteristic, complemented
8#1738C211yesmaximal, characteristic, complemented
9C22×#41C2211yescharacteristic
10C22×#41C2211yescharacteristic
11#575—22no—
12#575—22no—
13#575C414yescomplemented
14#575C2214yes—
15C2×#137—21no—
16C2×#137C412yescomplemented
17C2×AGL(1,ℤ/8)—21no—
18C2×AGL(1,ℤ/8)C412yescomplemented
19C2×#135—22no—
20#568—22no—
21#568—22no—
22#568—22no—
23#568C2214yescomplemented
24C2×#131—22no—
25C2×#131—22no—
26C22×M16—22no—
27C22×M16C2211yescharacteristic
28C22×M16C2211yescharacteristic
29C22×M16C2211yescharacteristic
30C22×M16C2211yescharacteristic
31C2×#117C2211yescharacteristic
32#421—22no—
33#421C2214yescomplemented
34C2×#104—22no—
35#419—21no—
36#418—21no—
37#417—22no—
38#416C2214yescomplemented
39#408—22no—
40#408C2214yescomplemented
41#407—22no—
42#344C2214yescomplemented
43#341C2214yescomplemented
44C2×#41—21no—
45C2×#41—21no—
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×#41C2×C412yes—
61C2×#41C2×C412yes—
62C2×#41C2×C412yes—
63C2×#41D412yes—
64C2×#41D412yes—
65C2×#41D412yes—
66C2×#41C2312yes—
67C2×#41C2312yes—
68C22×Q8C2×C411yescharacteristic
69C22×Q8D411yescharacteristic
70C22×D4—21no—
71C22×D4—21no—
72C22×D4C2×C411yescharacteristic
73C22×D4D411yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77#137—24no—
78#137—24no—
79#137—24no—
80#137C2×C418yescomplemented
81AGL(1,ℤ/8)—24no—
82AGL(1,ℤ/8)—24no—
83AGL(1,ℤ/8)—24no—
84AGL(1,ℤ/8)C2×C418yescomplemented
85#135—28no—
86#135—28no—
87C2×Q16—22no—
88C2×SD16—22no—
89C2×SD16—22no—
90C2×D8—22no—
91#131—28no—
92#131—28no—
93#131—28no—
94#131—28no—
95C2×M16—24no—
96C2×M16—28no—
97C2×M16—21no—
98C2×M16—21no—
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—28no—
108C2×M16—21no—
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16C2×C412yes—
115C2×M16C2×C412yes—
116C2×M16D412yes—
117C2×M16D412yes—
118C2×M16D412yes—
119C2×M16C2312yes—
120C2×M16C2312yes—
121C2×M16C2312yes—
122C22×C8—24no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8D412yes—
126C22×C8C2312yes—
127#117—21no—
128#117—21no—
129#117—21no—
130#117C2312yes—
131C2×C4⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C42—21no—
134#104—28no—
135#103—24no—
136#102—24no—
137C22⋊C8—28no—
138C24—21no—
139C24—21no—
140C24#4111yescharacteristic
141#41—22no—
142#41—22no—
143#41—22no—
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—41no—
162#41—41no—
163#41—41no—
164#41—41no—
165#41—41no—
166#41—41no—
167#41—41no—
168#41—41no—
169#41C22⋊C414yes—
170#41C22⋊C414yes—
171#41C22×C414yes—
172#41C2×D414yes—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8C22⋊C412yes—
182C2×Q8C22⋊C412yes—
183C2×Q8C22×C412yes—
184C2×Q8C2×D412yes—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4C22⋊C412yes—
206C2×D4C22⋊C412yes—
207C2×D4C22×C412yes—
208C2×D4C2×D412yes—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4C22×C411yescharacteristic
225C22×C4C22×C411yescharacteristic
226C22×C4C2×D411yescharacteristic
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2×D411yescharacteristic
232C22×C4C2×D411yescharacteristic
233C22×C4C2×D411yescharacteristic
234C22×C4#4111yescharacteristic
235C22×C4#4111yescharacteristic
236C22×C4#4111yescharacteristic
237C22×C4C2411yescharacteristic
238Q16—28no—
239SD16—28no—
240SD16—28no—
241D8—28no—
242M16—216no—
243M16—22no—
244M16—22no—
245M16—22no—
246M16—22no—
247M16—22no—
248M16—22no—
249M16—22no—
250M16—22no—
251M16—22no—
252M16—22no—
253M16—41no—
254M16—41no—
255M16—41no—
256M16—41no—
257M16C22×C414yes—
258M16C2×D414yes—
259C2×C8—216no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
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×C8C2×D414yes—
272C4⋊C4—22no—
273C4⋊C4—41no—
274C22⋊C4—22no—
275C22⋊C4—41no—
276C42—22no—
277C42—41no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—41no—
283C23—41no—
284C23—81no—
285C23—82no—
286C23—82no—
287C23C4×D411yescharacteristic
288C23C4×D411yescharacteristic
289C23#12111yescharacteristic
290C23#12111yescharacteristic
291C23#12311yescharacteristic
292C23#12311yescharacteristic
293C23C2×#4111yescharacteristic
294Q8—22no—
295Q8—22no—
296Q8—22no—
297Q8—22no—
298Q8—22no—
299Q8—41no—
300Q8—41no—
301Q8C2×C22⋊C414yes—
302D4—24no—
303D4—24no—
304D4—24no—
305D4—24no—
306D4—22no—
307D4—22no—
308D4—22no—
309D4—22no—
310D4—22no—
311D4—42no—
312D4—42no—
313D4—42no—
314D4—42no—
315D4—41no—
316D4—41no—
317D4C2×C22⋊C414yes—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4C2×C22⋊C411yescharacteristic
335C2×C4C2×C22⋊C411yescharacteristic
336C2×C4C2×C22⋊C411yescharacteristic
337C2×C4C2×C22⋊C411yescharacteristic
338C2×C4C4×D411yescharacteristic
339C2×C4C4×D411yescharacteristic
340C2×C4C4×D411yescharacteristic
341C2×C4C4×D411yescharacteristic
342C2×C4C4×D411yescharacteristic
343C2×C4C4×D411yescharacteristic
344C2×C4#12011yescharacteristic
345C2×C4#12011yescharacteristic
346C2×C4#12011yescharacteristic
347C2×C4#12011yescharacteristic
348C2×C4#12111yescharacteristic
349C2×C4#12111yescharacteristic
350C2×C4#12111yescharacteristic
351C2×C4#12111yescharacteristic
352C2×C4#12111yescharacteristic
353C2×C4#12111yescharacteristic
354C2×C4#12311yescharacteristic
355C2×C4#12311yescharacteristic
356C2×C4C23×C411yescharacteristic
357C2×C4C22×D411yescharacteristic
358C2×C4C22×D411yescharacteristic
359C2×C4C22×D411yescharacteristic
360C2×C4C22×D411yescharacteristic
361C2×C4C2×#4111yescharacteristic
362C8—24no—
363C8—24no—
364C8—24no—
365C8—42no—
366C22—24no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—44no—
375C22—44no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22#38611yescharacteristic
382C22#38611yescharacteristic
383C22#38611yescharacteristic
384C22#38611yescharacteristic
385C22C2×C4×D411yescharacteristic
386C22C2×#12111yescharacteristic
387C22C2×#12311yescharacteristic
388C4—24no—
389C4—42no—
390C4—82no—
391C4#38611yescharacteristic
392C4#38611yescharacteristic
393C4#38611yescharacteristic
394C4#38611yescharacteristic
395C4C22×C22⋊C411yescharacteristic
396C4C2×C4×D411yescharacteristic
397C4C2×#12011yescharacteristic
398C4C2×#12111yescharacteristic
399C2—21no—
400C2—21no—
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—44no—
407C2—82no—
408C2C2×#38611yescharacteristic
subgroupG/H
centerC4C2×#120
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. #2879C2semidirect
6. #1881C2semidirect
7. C2×#344C2semidirect
8. #1738C2semidirect
13. #575C4semidirect
16. C2×#137C4semidirect
18. C2×AGL(1,ℤ/8)C4semidirect
23. #568C22semidirect
33. #421C22semidirect
38. #416C22semidirect
40. #408C22semidirect
42. #344C22semidirect
43. #341C22semidirect
80. #137C2×C4semidirect
84. AGL(1,ℤ/8)C2×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
#28792
#18812
C2×#3442
#17382