Finite Group Databasebeta

← #18522

Subgroups — #18522

Order 256.

500 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3475C211yesmaximal, characteristic, complemented
2#3167C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic, complemented
4#3035C211yesmaximal, characteristic, complemented
5#1654C211yesmaximal, characteristic, complemented
6#1652C211yesmaximal, characteristic, complemented
7#1646C211yesmaximal, characteristic, complemented
8#1644C211yesmaximal, characteristic, complemented
9#1628C211yesmaximal, characteristic, complemented
10#1625C211yesmaximal, characteristic, complemented
11#1624C211yesmaximal, characteristic, complemented
12#1622C211yesmaximal, characteristic, complemented
13#1542C211yesmaximal, characteristic, complemented
14#1476C211yesmaximal, characteristic, complemented
15C2×D4⋊C8C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#556C2211yescharacteristic
18#555C2211yescharacteristic
19#554C2211yescharacteristic
20#553—21no—
21#552—21no—
22#552—21no—
23#551—22no—
24#547C2211yescharacteristic
25#546—21no—
26#533C2211yescharacteristic
27#532C2211yescharacteristic
28#529—22no—
29#529—21no—
30#525—21no—
31C4×#41—21no—
32C4×#41C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C2×#117—21no—
35#498C2211yescharacteristic, complemented
36#492C2211yescharacteristic, complemented
37#487C2211yescharacteristic, complemented
38#486C2211yescharacteristic, complemented
39#480—21no—
40#474C2211yescharacteristic
41#474C2211yescharacteristic
42#465—22no—
43#464—21no—
44#463—21no—
45#462C2211yescharacteristic, complemented
46#461C2211yescharacteristic, complemented
47#460C2211yescharacteristic, complemented
48#459C2211yescharacteristic
49#459C2211yescharacteristic
50#459C2211yescharacteristic, complemented
51#447—21no—
52C4×SD16—22no—
53C4×D8—21no—
54C4×D8—21no—
55#431C2211yescharacteristic
56C2×C4⋊C8C2211yescharacteristic
57#416—21no—
58C2×C4×C8C2211yescharacteristic
59#333C2211yescharacteristic, complemented
60#333C2211yescharacteristic, complemented
61#332C2211yescharacteristic, complemented
62#332C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic
66D4⋊C8C2211yescharacteristic
67D4⋊C8C2211yescharacteristic
68D4⋊C8C2211yescharacteristic
69D4⋊C8C2211yescharacteristic, complemented
70D4⋊C8C2211yescharacteristic, complemented
71C2×#41—21no—
72C2×#41—21no—
73C2×#41D411yescharacteristic
74C22×D4D411yescharacteristic
75C23×C4—21no—
76#135—44no—
77C2×Q16—21no—
78C2×SD16—21no—
79C2×SD16—22no—
80C2×SD16—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88C4⋊Q8C2311yescharacteristic
89#127C2311yescharacteristic
90#126—22no—
91#126—22no—
92#126—22no—
93#125—22no—
94#125—21no—
95#125—21no—
96#124—22no—
97#124—21no—
98#124—21no—
99#123—22no—
100#123—22no—
101#123—22no—
102#123—22no—
103C22⋊Q8—21no—
104C22⋊Q8—21no—
105C22⋊Q8—22no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108C22⋊Q8D412yes—
109#121—22no—
110#121—21no—
111#121—21no—
112#121—21no—
113#121—21no—
114#121—21no—
115#120—22no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8D411yescharacteristic
119C4×Q8D412yes—
120C4×Q8C2311yescharacteristic
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—21no—
125C4×D4—21no—
126C4×D4—21no—
127C4×D4—21no—
128C4×D4—21no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4D411yescharacteristic
132C4×D4D411yescharacteristic
133C4×D4D411yescharacteristic
134C4×D4C2311yescharacteristic
135C4×D4C2311yescharacteristic
136C4×D4C2311yescharacteristic
137#117—22no—
138#117—22no—
139#117—22no—
140#117—21no—
141#117—21no—
142#117—21no—
143#117—21no—
144#117—21no—
145#117D411yescharacteristic
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4D411yescharacteristic
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C42—22no—
152C2×C42—21no—
153C2×C42C2311yescharacteristic
154#107—21no—
155#107—21no—
156#106—22no—
157C4⋊C8—21no—
158C4⋊C8C2311yescharacteristic
159C4⋊C8C2311yescharacteristic
160C4⋊C8C2311yescharacteristic
161C4⋊C8C2311yescharacteristic
162#103—22no—
163#103—21no—
164#103—21no—
165#102—21no—
166#102—21no—
167#102—21no—
168#102—21no—
169#102—21no—
170#102—21no—
171#102—22no—
172C22⋊C8—21no—
173C4×C8—21no—
174C4×C8—21no—
175C4×C8C2311yescharacteristic
176C4×C8C2311yescharacteristic
177C24—21no—
178#41—22no—
179#41—22no—
180#41—42no—
181#41—42no—
182#41—42no—
183#41—42no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8C2×D411yescharacteristic
187C2×Q8C2×D412yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—22no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4C2×D411yescharacteristic
199C2×D4C2×D411yescharacteristic
200C2×D4C2×D411yescharacteristic
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217Q16—42no—
218SD16—44no—
219SD16—42no—
220SD16—42no—
221D8—42no—
222D8—42no—
223D8—42no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—42no—
231C2×C8—42no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—21no—
237C4⋊C4—22no—
238C4⋊C4—21no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4C2×D411yescharacteristic
247C4⋊C4C2×D411yescharacteristic
248C4⋊C4C2×D412yes—
249C4⋊C4C2×D411yescharacteristic
250C4⋊C4C2×D411yescharacteristic
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—21no—
258C22⋊C4—22no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C42—22no—
268C42—22no—
269C42—21no—
270C42—21no—
271C42—22no—
272C42—21no—
273C42C2×D411yescharacteristic
274C42C2×D411yescharacteristic
275C42C2×D411yescharacteristic
276C42C2411yescharacteristic
277C23—21no—
278C23—21no—
279C23—21no—
280C23—42no—
281C23—42no—
282C23—41no—
283C23—81no—
284C23#12011yescharacteristic
285Q8—21no—
286Q8—21no—
287Q8—42no—
288Q8—41no—
289Q8—41no—
290Q8—41no—
291Q8—41no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—42no—
299D4—44no—
300D4—42no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—82no—
306D4—82no—
307D4#13512yescomplemented
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4#12011yescharacteristic
334C2×C4#12011yescharacteristic
335C2×C4#12011yescharacteristic
336C2×C4#13512yes—
337C2×C4AGL(1,ℤ/8)12yes—
338C2×C4C22×D411yescharacteristic
339C2×C4C22×D411yescharacteristic
340C2×C4C22×D411yescharacteristic
341C8—22no—
342C8—41no—
343C8—41no—
344C8—41no—
345C8—41no—
346C8—41no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22#44912yes—
363C22C2×#12011yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—42no—
372C4—41no—
373C4—42no—
374C4—81no—
375C4#44912yes—
376C4C2×#13511yescharacteristic
377C4C2×#13511yescharacteristic
378C4C2×AGL(1,ℤ/8)11yescharacteristic
379C4#57511yescharacteristic
380C2—22no—
381C2—42no—
382C2—81no—
383C2—81no—
384C2—161no—
385C2C2×#44911yescharacteristic
386C2#300211yescharacteristic
387C2#301811yescharacteristic
subgroupG/H
centerC2×C4#120
commutator, Frattini, cosocleC42C24
socleC22C2×#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3475C2semidirect
2. #3167C2semidirect
3. #3100C2semidirect
4. #3035C2semidirect
5. #1654C2semidirect
6. #1652C2semidirect
7. #1646C2semidirect
8. #1644C2semidirect
9. #1628C2semidirect
10. #1625C2semidirect
11. #1624C2semidirect
12. #1622C2semidirect
13. #1542C2semidirect
14. #1476C2semidirect
15. C2×D4⋊C8C2semidirect
35. #498C22semidirect
36. #492C22semidirect
37. #487C22semidirect
38. #486C22semidirect
45. #462C22semidirect
46. #461C22semidirect
47. #460C22semidirect
50. #459C22semidirect
59. #333C22semidirect
60. #333C22semidirect
61. #332C22semidirect
62. #332C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
69. D4⋊C8C22semidirect
70. D4⋊C8C22semidirect
307. D4#135semidirect

Maximal subgroups

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

subgroupindex
#34752
#31672
#31002
#30352
#16542
#16522
#16462
#16442
#16282
#16252
#16242
#16222
#15422
#14762
C2×D4⋊C82