Finite Group Databasebeta

← #25168

Subgroups — #25168

Order 256.

698 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#459C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic, complemented
4C2×#442C211yesmaximal, characteristic, complemented
5#2428C211yesmaximal, characteristic, complemented
6D4×C22⋊C4C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#2011C211yesmaximal, characteristic, complemented
11#1972C211yesmaximal, characteristic, complemented
12#1924C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1879C211yesmaximal, characteristic, complemented
15#1864C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C2×#127C2211yescharacteristic, complemented
22C2×#121—21no—
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic, complemented
28C22×C22⋊C4—21no—
29#459—24no—
30#447—24no—
31#447—24no—
32#442—24no—
33C2×#106C2211yescharacteristic, complemented
34C2×#106C2211yescharacteristic, complemented
35C2×C4⋊C8C2211yescharacteristic, complemented
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic, complemented
41C2×#102C2211yescharacteristic, complemented
42C2×#102C2211yescharacteristic, complemented
43C2×#102C2211yescharacteristic, complemented
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×#97C2211yescharacteristic, complemented
47#394—21no—
48#392—21no—
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#390C2211yescharacteristic, complemented
52#389C2211yescharacteristic, complemented
53#386—21no—
54#386—21no—
55#386—21no—
56#380C2211yescharacteristic
57#379—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—21no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C22×D4C2311yescharacteristic
78C22×D4C2311yescharacteristic
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4C2311yescharacteristic
83C2×D8—22no—
84C2×D8—22no—
85C2×D8—22no—
86C2×D8—22no—
87C2×D8—22no—
88C2×D8—22no—
89C2×D8—22no—
90C2×D8—22no—
91C2×D8—22no—
92C2×D8—41no—
93C2×D8—41no—
94C2×D8D414yescomplemented
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99#127—22no—
100#127—41no—
101#121—22no—
102#121—22no—
103#121—44no—
104#121—41no—
105#121—41no—
106C4×D4—22no—
107C4×D4—22no—
108C4×D4—22no—
109C4×D4—41no—
110C4×D4—41no—
111C4×D4—41no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
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—44no—
130C2×C22⋊C4D411yescharacteristic
131C2×C22⋊C4D411yescharacteristic
132C2×C42C2311yescharacteristic
133C2×C42C2311yescharacteristic
134#106—22no—
135#106—41no—
136C4⋊C8—22no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—41no—
144#102D414yescomplemented
145C22⋊C8—22no—
146C22⋊C8—22no—
147#97—22no—
148#95—21no—
149#95—21no—
150#95D411yescharacteristic
151#95D411yescharacteristic
152C24—21no—
153C24—24no—
154C24—21no—
155C24—21no—
156C24—21no—
157C24—42no—
158C24—42no—
159C24—42no—
160C24—42no—
161C24—81no—
162C24—81no—
163C24C2×D411yescharacteristic
164C2×D4—24no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—81no—
199C2×D4—81no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4C2×D412yes—
203C2×D4C2×D412yes—
204C2×D4C2×D412yes—
205C2×D4#4112yes—
206C2×D4#4112yes—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4C2×D411yescharacteristic
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2411yescharacteristic
232D8—28no—
233D8—44no—
234D8—44no—
235D8—44no—
236C2×C8—21no—
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×C8—21no—
246C2×C8C2×D412yes—
247C2×C8C2×D412yes—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—42no—
256C4⋊C4C2×D412yes—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—42no—
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—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—48no—
287C23—48no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
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—41no—
309C23—41no—
310C23—41no—
311C23—84no—
312C23—84no—
313C23—84no—
314C23—84no—
315C23—84no—
316C23AGL(1,ℤ/8)12yes—
317C23C22×D411yescharacteristic
318D4—22no—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—22no—
325D4—48no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—41no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—41no—
334D4—84no—
335D4—84no—
336D4—84no—
337D4—84no—
338D4#12114yescomplemented
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
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—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4AGL(1,ℤ/8)12yes—
373C2×C4C22×D411yescharacteristic
374C2×C4C22×D411yescharacteristic
375C2×C4C2×#4111yescharacteristic
376C2×C4C2×#4111yescharacteristic
377C2×C4H(5,𝔽2)11yescharacteristic
378C2×C4H(5,𝔽2)11yescharacteristic
379C8—22no—
380C8—22no—
381C8—41no—
382C8—41no—
383C22—22no—
384C22—24no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—41no—
389C22—42no—
390C22—41no—
391C22—42no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—816no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22#52511yescharacteristic
420C22D4211yescharacteristic
421C22D4211yescharacteristic
422C22C2×AGL(1,ℤ/8)11yescharacteristic
423C22C2×AGL(1,ℤ/8)11yescharacteristic
424C22#57611yescharacteristic
425C22#57611yescharacteristic
426C4—22no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—42no—
431C4—41no—
432C4—81no—
433C4—81no—
434C4—81no—
435C4C2×#12111yescharacteristic
436C4#53811yescharacteristic
437C4#54611yescharacteristic
438C4#54611yescharacteristic
439C2—42no—
440C2—44no—
441C2—82no—
442C2—82no—
443C2—161no—
444C2#242411yescharacteristic
445C2#310911yescharacteristic
446C2#312211yescharacteristic
447C2#326411yescharacteristic
448C2#326411yescharacteristic
449C2#326411yescharacteristic
450C2#327411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×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
1. C2×#459C2semidirect
2. C2×#447C2semidirect
3. C2×#447C2semidirect
4. C2×#442C2semidirect
5. #2428C2semidirect
6. D4×C22⋊C4C2semidirect
7. #2065C2semidirect
8. #2034C2semidirect
9. #2034C2semidirect
10. #2011C2semidirect
11. #1972C2semidirect
12. #1924C2semidirect
13. #1879C2semidirect
14. #1879C2semidirect
15. #1864C2semidirect
21. C2×#127C22semidirect
27. C2×C4×D4C22semidirect
33. C2×#106C22semidirect
34. C2×#106C22semidirect
35. C2×C4⋊C8C22semidirect
40. C2×#102C22semidirect
41. C2×#102C22semidirect
42. C2×#102C22semidirect
43. C2×#102C22semidirect
46. C2×#97C22semidirect
51. #390C22semidirect
52. #389C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect
94. C2×D8D4semidirect
144. #102D4semidirect
338. D4#121semidirect

Maximal subgroups

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

subgroupindex
C2×#4592
C2×#4472
C2×#4422
#24282
D4×C22⋊C42
#20652
#20342
#20112
#19722
#19242
#18792
#18642