Finite Group Databasebeta

← #25172

Subgroups — #25172

Order 256.

655 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#463C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic
4C2×#442C211yesmaximal, characteristic, complemented
5#2427C211yesmaximal, characteristic, complemented
6D4×C22⋊C4C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2033C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1924C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C2×#124C2211yescharacteristic, complemented
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic, complemented
27C22×C22⋊C4—21no—
28#463—24no—
29#450—24no—
30#447—24no—
31#442—24no—
32C2×#107C2211yescharacteristic, complemented
33C2×#106C2211yescharacteristic, complemented
34C2×C4⋊C8C2211yescharacteristic, 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×C22⋊C8C2211yescharacteristic
45C2×#97C2211yescharacteristic, complemented
46#397—21no—
47#394—21no—
48#391C2211yescharacteristic, complemented
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#389C2211yescharacteristic, complemented
52#386—21no—
53#386—21no—
54#386—21no—
55#380—21no—
56#380C2211yescharacteristic
57#379—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×Q8C2311yescharacteristic
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C2311yescharacteristic
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—41no—
86C2×D8—22no—
87C2×D8—22no—
88C2×D8—22no—
89C2×D8—22no—
90C2×D8—41no—
91C2×D8D414yes—
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96#124—22no—
97#124—41no—
98C22⋊Q8—22no—
99C22⋊Q8—41no—
100#121—22no—
101#121—41no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—41no—
106C4×D4—41no—
107C4×D4—41no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
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—44no—
127C2×C22⋊C4D411yescharacteristic
128C2×C22⋊C4D411yescharacteristic
129C2×C42C2311yescharacteristic
130C2×C42C2311yescharacteristic
131#107—41no—
132#106—22no—
133C4⋊C8—22no—
134#103—22no—
135#103—41no—
136#102—22no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102D414yescomplemented
142C22⋊C8—22no—
143C22⋊C8—22no—
144#97—22no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95D411yescharacteristic
150#95D411yescharacteristic
151C24—21no—
152C24—24no—
153C24—21no—
154C24—21no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—42no—
159C24—81no—
160C24—81no—
161C24C2×D411yescharacteristic
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×D4—24no—
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—42no—
183C2×D4—42no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—81no—
191C2×D4C2×D412yes—
192C2×D4C2×D412yes—
193C2×D4C2×D412yes—
194C2×D4#4112yes—
195C2×D4#4112yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4—81no—
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2411yescharacteristic
222SD16—44no—
223SD16—44no—
224D8—28no—
225D8—44no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8C2×D412yes—
237C2×C8C2×D412yes—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C4⋊C4C2×D412yes—
248C22⋊C4—22no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—22no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—22no—
276C23—22no—
277C23—48no—
278C23—48no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—84no—
296C23—84no—
297C23—84no—
298C23—84no—
299C23—84no—
300C23AGL(1,ℤ/8)12yes—
301C23C22×D411yescharacteristic
302Q8—41no—
303Q8—41no—
304Q8—41no—
305Q8—41no—
306D4—22no—
307D4—22no—
308D4—22no—
309D4—22no—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—48no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—84no—
319D4#12114yes—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4C22×D411yescharacteristic
358C2×C4C22×D411yescharacteristic
359C2×C4C2×#4111yescharacteristic
360C2×C4C2×#4111yescharacteristic
361C2×C4H(5,𝔽2)11yescharacteristic
362C2×C4H(5,𝔽2)11yescharacteristic
363C8—22no—
364C8—22no—
365C8—41no—
366C8—41no—
367C22—22no—
368C22—24no—
369C22—22no—
370C22—22no—
371C22—42no—
372C22—41no—
373C22—42no—
374C22—41no—
375C22—42no—
376C22—42no—
377C22—41no—
378C22—42no—
379C22—41no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—816no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#52511yescharacteristic
397C22D4211yescharacteristic
398C22D4211yescharacteristic
399C22C2×AGL(1,ℤ/8)11yescharacteristic
400C22#57511yescharacteristic
401C22#57611yescharacteristic
402C22#57711yescharacteristic
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—42no—
409C4—81no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4C2×#12111yescharacteristic
414C4#53811yescharacteristic
415C4#54611yescharacteristic
416C4#54611yescharacteristic
417C2—42no—
418C2—44no—
419C2—82no—
420C2—82no—
421C2#242411yescharacteristic
422C2#311411yescharacteristic
423C2#311811yescharacteristic
424C2#326411yescharacteristic
425C2#326511yescharacteristic
426C2#326711yescharacteristic
427C2#327311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
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×#463C2semidirect
2. C2×#450C2semidirect
4. C2×#442C2semidirect
5. #2427C2semidirect
6. D4×C22⋊C4C2semidirect
7. #2065C2semidirect
8. #2035C2semidirect
9. #2033C2semidirect
10. #2012C2semidirect
11. #1973C2semidirect
12. #1924C2semidirect
13. #1879C2semidirect
14. #1877C2semidirect
15. #1862C2semidirect
21. C2×#124C22semidirect
26. C2×C4×D4C22semidirect
32. C2×#107C22semidirect
33. C2×#106C22semidirect
34. C2×C4⋊C8C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
41. C2×#102C22semidirect
42. C2×#102C22semidirect
45. C2×#97C22semidirect
48. #391C22semidirect
51. #389C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect
141. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
C2×#4502
C2×#4472
C2×#4422
#24272
D4×C22⋊C42
#20652
#20352
#20332
#20122
#19732
#19242
#18792
#18772
#18622