Finite Group Databasebeta

← #22442

Subgroups — #22442

Order 256.

825 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic, complemented
3C2×C4×D8C211yesmaximal, characteristic, complemented
4#2339C211yesmaximal, characteristic, complemented
5D4×C22⋊C4C211yesmaximal, characteristic, complemented
6#1973C211yesmaximal, characteristic, complemented
7#1968C211yesmaximal, characteristic, complemented
8#1879C211yesmaximal, characteristic, complemented
9#1879C211yesmaximal, characteristic, complemented
10#1877C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, 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×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic, complemented
29C2×C4×D4C2211yescharacteristic, complemented
30C22×C22⋊C4—21no—
31#442—24no—
32C4×D8—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#391C2211yescharacteristic, complemented
47#391C2211yescharacteristic, complemented
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#386—21no—
51#386—21no—
52#386—21no—
53#386—21no—
54#381—21no—
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#379—21no—
58C4×C22⋊C4C2211yescharacteristic
59C2×#95—21no—
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×C414yescomplemented
80C22×D4C2311yescharacteristic
81C22×D4C2311yescharacteristic
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4C2311yescharacteristic
89AGL(1,ℤ/8)—232no—
90C2×SD16—22no—
91C2×SD16—22no—
92C2×SD16—22no—
93C2×SD16—22no—
94C2×SD16C2×C414yescomplemented
95C2×SD16C2×C414yescomplemented
96C2×D8—22no—
97C2×D8—22no—
98C2×D8C2×C414yes—
99C2×D8C2×C414yescomplemented
100C2×D8D414yes—
101C2×D8D414yes—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16C2×C414yescomplemented
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107C22×C8C2311yescharacteristic
108C22×C8C2311yescharacteristic
109C4×D4—22no—
110C4×D4—22no—
111C4×D4—22no—
112C4×D4—22no—
113C4×D4—41no—
114C4×D4—41no—
115C4×D4—41no—
116C4×D4—41no—
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—44no—
132C2×C22⋊C4D411yescharacteristic
133C2×C22⋊C4D411yescharacteristic
134C2×C42C2311yescharacteristic
135C2×C42C2311yescharacteristic
136#107—22no—
137#106—22no—
138#103—41no—
139#103—41no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144#102—41no—
145#102—41no—
146C22⋊C8—22no—
147#97—22no—
148C4×C8—22no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
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×Q8—21no—
176C2×Q8—21no—
177C2×Q8C22×C412yes—
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—44no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4C22×C412yes—
200C2×D4C22×C412yes—
201C2×D4C22×C412yes—
202C2×D4C2×D412yes—
203C2×D4C2×D412yes—
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—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4C22×C412yes—
231C22×C4C2×D411yescharacteristic
232C22×C4C2×D411yescharacteristic
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D411yescharacteristic
235C22×C4C2×D411yescharacteristic
236C22×C4#4112yes—
237C22×C4C2411yescharacteristic
238SD16—28no—
239SD16—28no—
240D8—28no—
241D8C22⋊C4116yescomplemented
242M16—28no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8C22×C412yes—
252C2×C8C22×C412yes—
253C2×C8C2×D412yes—
254C2×C8C2×D412yes—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C22⋊C4—22no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—48no—
293C23—48no—
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—41no—
318Q8—41no—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—22no—
325D4—22no—
326D4—22no—
327D4—48no—
328D4—48no—
329D4—41no—
330D4—41no—
331D4C2×C22⋊C414yes—
332D4C2×C22⋊C414yescomplemented
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—48no—
352C2×C4—41no—
353C2×C4—41no—
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×C4—84no—
369C2×C4—84no—
370C2×C4C4×D412yes—
371C2×C4AGL(1,ℤ/8)12yescomplemented
372C2×C4C23×C411yescharacteristic
373C2×C4C22×D411yescharacteristic
374C2×C4C22×D411yescharacteristic
375C2×C4C2×#4111yescharacteristic
376C2×C4H(5,𝔽2)11yescharacteristic
377C2×C4H(5,𝔽2)11yescharacteristic
378C8—22no—
379C8—41no—
380C8—41no—
381C8C2×C22⋊C414yescomplemented
382C22—22no—
383C22—24no—
384C22—24no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—41no—
399C22—42no—
400C22—42no—
401C22—41no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—816no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22C2×C4×D411yescharacteristic
412C22D4211yescharacteristic
413C22D4211yescharacteristic
414C22C2×AGL(1,ℤ/8)11yescharacteristic
415C22#57511yescharacteristic
416C22#57611yescharacteristic
417C22#57711yescharacteristic
418C4—22no—
419C4—22no—
420C4—42no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—81no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4C22×C22⋊C411yescharacteristic
429C4#51811yescharacteristic
430C4#54611yescharacteristic
431C4#54611yescharacteristic
432C2—42no—
433C2—44no—
434C2—44no—
435C2—82no—
436C2D4×C22⋊C411yescharacteristic
437C2C4×AGL(1,ℤ/8)11yescharacteristic
438C2#294611yescharacteristic
439C2#326411yescharacteristic
440C2#326711yescharacteristic
441C2#327211yescharacteristic
442C2#327311yescharacteristic
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×#442C2semidirect
3. C2×C4×D8C2semidirect
4. #2339C2semidirect
5. D4×C22⋊C4C2semidirect
6. #1973C2semidirect
7. #1968C2semidirect
8. #1879C2semidirect
9. #1879C2semidirect
10. #1877C2semidirect
11. #1877C2semidirect
12. #1809C2semidirect
13. #1788C2semidirect
14. #1786C2semidirect
15. #1753C2semidirect
20. C2×AGL(1,ℤ/8)C4semidirect
28. C2×C4×D4C22semidirect
29. C2×C4×D4C22semidirect
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
46. #391C22semidirect
47. #391C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #336C22semidirect
63. #336C22semidirect
70. C2×#41C2×C4semidirect
79. C22×D4C2×C4semidirect
94. C2×SD16C2×C4semidirect
95. C2×SD16C2×C4semidirect
99. C2×D8C2×C4semidirect
104. C2×M16C2×C4semidirect
241. D8C22⋊C4semidirect
332. D4C2×C22⋊C4semidirect
371. C2×C4AGL(1,ℤ/8)semidirect
381. 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×#4422
C2×C4×D82
#23392
D4×C22⋊C42
#19732
#19682
#18792
#18772
#18092
#17882
#17862
#17532