Finite Group Databasebeta

← #22443

Subgroups — #22443

Order 256.

811 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic, complemented
3C2×C4×SD16C211yesmaximal, characteristic, complemented
4#2341C211yesmaximal, characteristic, complemented
5D4×C22⋊C4C211yesmaximal, characteristic, complemented
6#1972C211yesmaximal, characteristic, complemented
7#1970C211yesmaximal, characteristic, complemented
8#1880C211yesmaximal, characteristic, complemented
9#1880C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1809C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C2×AGL(1,ℤ/8)—24no—
19C2×AGL(1,ℤ/8)—24no—
20C2×AGL(1,ℤ/8)C418yescomplemented
21C22×SD16C2211yescharacteristic
22C22×SD16C2211yescharacteristic
23C22×D8C2211yescharacteristic
24C22×D8C2211yescharacteristic
25C22×M16C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic, complemented
27C2×C4×Q8C2211yescharacteristic, complemented
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C22×C22⋊C4—21no—
31#440—24no—
32C4×SD16—24no—
33C2×#107C2211yescharacteristic, complemented
34C2×#106C2211yescharacteristic, 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×#97C2211yescharacteristic, complemented
45C2×C4×C8C2211yescharacteristic, complemented
46#390—21no—
47#390—21no—
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#390C2211yescharacteristic, complemented
51#390C2211yescharacteristic, complemented
52#386—21no—
53#386—21no—
54#380—21no—
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#379—21no—
58C4×C22⋊C4—21no—
59C4×C22⋊C4C2211yescharacteristic
60#340C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63#336C2211yescharacteristic, complemented
64C25—21no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—41no—
69C2×#41—41no—
70C2×#41C2×C414yescomplemented
71C22×Q8C2311yescharacteristic
72C22×D4—22no—
73C22×D4—22no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—22no—
77C22×D4—41no—
78C22×D4—41no—
79C22×D4C2×C414yes—
80C22×D4C2311yescharacteristic
81C22×D4C2311yescharacteristic
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C2311yescharacteristic
87AGL(1,ℤ/8)—232no—
88C2×SD16—22no—
89C2×SD16—22no—
90C2×SD16C2×C414yes—
91C2×SD16C2×C414yescomplemented
92C2×SD16D414yes—
93C2×SD16D414yes—
94C2×D8—22no—
95C2×D8—22no—
96C2×D8—22no—
97C2×D8—22no—
98C2×D8C2×C414yes—
99C2×D8C2×C414yescomplemented
100C2×M16—22no—
101C2×M16—22no—
102C2×M16C2×C414yescomplemented
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107C4×Q8—22no—
108C4×Q8—22no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×D4—22no—
112C4×D4—22no—
113C4×D4—41no—
114C4×D4—41no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
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—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—44no—
134C2×C22⋊C4D411yescharacteristic
135C2×C22⋊C4D411yescharacteristic
136C2×C42—21no—
137C2×C42—21no—
138C2×C42C2311yescharacteristic
139C2×C42C2311yescharacteristic
140#107—22no—
141#106—22no—
142#103—22no—
143#103—22no—
144#102—22no—
145#102—22no—
146#102—41no—
147#102—41no—
148#102—41no—
149#102—41no—
150C22⋊C8—22no—
151#97—22no—
152C4×C8—22no—
153#95—21no—
154#95—21no—
155#95D411yescharacteristic
156#95D411yescharacteristic
157C24—21no—
158C24—21no—
159C24—21no—
160C24—24no—
161C24—42no—
162C24—42no—
163C24—42no—
164C24—42no—
165C24—81no—
166C24—81no—
167C24C2×D411yescharacteristic
168#41—28no—
169#41—44no—
170#41—44no—
171#41—44no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8C22×C412yes—
176C2×Q8C2×D412yes—
177C2×Q8C2×D412yes—
178C2×D4—28no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4C22×C412yes—
202C2×D4C22×C412yes—
203C2×D4C22×C412yes—
204C2×D4C2×D412yes—
205C2×D4C2×D412yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4C22×C412yes—
225C22×C4C2×D411yescharacteristic
226C22×C4C2×D411yescharacteristic
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4#4112yes—
231C22×C4C2411yescharacteristic
232SD16—28no—
233SD16C22⋊C4116yescomplemented
234D8—28no—
235D8—28no—
236M16—28no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8C22×C412yes—
246C2×C8C22×C412yes—
247C2×C8C2×D412yes—
248C2×C8C2×D412yes—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C22⋊C4—22no—
260C22⋊C4—41no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—42no—
278C42—42no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—22no—
290C23—48no—
291C23—48no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—84no—
309C23—84no—
310C23—84no—
311C23—84no—
312C23—84no—
313C23C4×D412yes—
314C23C22×D411yescharacteristic
315Q8—22no—
316Q8—22no—
317Q8—22no—
318Q8C2×C22⋊C414yescomplemented
319D4—22no—
320D4—22no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—22no—
325D4—22no—
326D4—48no—
327D4—48no—
328D4—41no—
329D4—41no—
330D4—41no—
331D4—41no—
332D4C2×C22⋊C414yes—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—48no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4C4×D412yes—
369C2×C4AGL(1,ℤ/8)12yescomplemented
370C2×C4C23×C411yescharacteristic
371C2×C4C22×D411yescharacteristic
372C2×C4C22×D411yescharacteristic
373C2×C4C2×#4111yescharacteristic
374C2×C4H(5,𝔽2)11yescharacteristic
375C2×C4H(5,𝔽2)11yescharacteristic
376C8—22no—
377C8—41no—
378C8—41no—
379C8C2×C22⋊C414yescomplemented
380C22—22no—
381C22—24no—
382C22—22no—
383C22—22no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—816no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22C2×C4×D411yescharacteristic
410C22D4211yescharacteristic
411C22D4211yescharacteristic
412C22C2×AGL(1,ℤ/8)11yescharacteristic
413C22#57511yescharacteristic
414C22#57611yescharacteristic
415C22#57711yescharacteristic
416C4—22no—
417C4—24no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4—81no—
426C4C22×C22⋊C411yescharacteristic
427C4#51811yescharacteristic
428C4#54611yescharacteristic
429C4#54611yescharacteristic
430C2—42no—
431C2—44no—
432C2—82no—
433C2—82no—
434C2D4×C22⋊C411yescharacteristic
435C2C4×AGL(1,ℤ/8)11yescharacteristic
436C2#294611yescharacteristic
437C2#326511yescharacteristic
438C2#326811yescharacteristic
439C2#327411yescharacteristic
440C2#327511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×AGL(1,ℤ/8)C2
LCSG ▷ C2×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
2. C2×#440C2semidirect
3. C2×C4×SD16C2semidirect
4. #2341C2semidirect
5. D4×C22⋊C4C2semidirect
6. #1972C2semidirect
7. #1970C2semidirect
8. #1880C2semidirect
9. #1880C2semidirect
10. #1879C2semidirect
11. #1879C2semidirect
12. #1809C2semidirect
13. #1788C2semidirect
14. #1786C2semidirect
15. #1753C2semidirect
20. C2×AGL(1,ℤ/8)C4semidirect
26. C2×C4×Q8C22semidirect
27. C2×C4×Q8C22semidirect
33. C2×#107C22semidirect
34. C2×#106C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
41. C2×#102C22semidirect
42. C2×#102C22semidirect
44. C2×#97C22semidirect
45. C2×C4×C8C22semidirect
50. #390C22semidirect
51. #390C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #336C22semidirect
63. #336C22semidirect
70. C2×#41C2×C4semidirect
91. C2×SD16C2×C4semidirect
99. C2×D8C2×C4semidirect
102. C2×M16C2×C4semidirect
233. SD16C22⋊C4semidirect
318. Q8C2×C22⋊C4semidirect
369. C2×C4AGL(1,ℤ/8)semidirect
379. C8C2×C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×AGL(1,ℤ/8)2
C2×#4402
C2×C4×SD162
#23412
D4×C22⋊C42
#19722
#19702
#18802
#18792
#18092
#17882
#17862
#17532