Finite Group Databasebeta

← #18755

Subgroups — #18755

Order 256.

494 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3481C211yesmaximal, characteristic
2#3210C211yesmaximal, characteristic, complemented
3#3116C211yesmaximal, characteristic, complemented
4#3076C211yesmaximal, characteristic, complemented
5#1644C211yesmaximal, characteristic, complemented
6#1643C211yesmaximal, characteristic, complemented
7#1640C211yesmaximal, characteristic, complemented
8#1639C211yesmaximal, characteristic, complemented
9#1632C211yesmaximal, characteristic, complemented
10#1632C211yesmaximal, characteristic, complemented
11#1630C211yesmaximal, characteristic, complemented
12#1629C211yesmaximal, characteristic, complemented
13#1551C211yesmaximal, characteristic, complemented
14#1488C211yesmaximal, characteristic, complemented
15#1477C211yesmaximal, characteristic, complemented
16#559C2211yescharacteristic
17#554C2211yescharacteristic
18#554C2211yescharacteristic
19#553—21no—
20#553—21no—
21#551—22no—
22#546—21no—
23#546—21no—
24D42C2211yescharacteristic
25#533C2211yescharacteristic, complemented
26#532C2211yescharacteristic
27#529—21no—
28#529—21no—
29#525—22no—
30C4×#41C2211yescharacteristic
31C4×#41C2211yescharacteristic
32C2×C4×D4—21no—
33C2×C4×D4—21no—
34#494C2211yescharacteristic, complemented
35#492C2211yescharacteristic
36#487C2211yescharacteristic, complemented
37#486C2211yescharacteristic
38#485—21no—
39#479—21no—
40#478—21no—
41#477C2211yescharacteristic, complemented
42#477C2211yescharacteristic, complemented
43#476C2211yescharacteristic
44#476C2211yescharacteristic
45#475C2211yescharacteristic, complemented
46#474C2211yescharacteristic
47#464—22no—
48#462C2211yescharacteristic, complemented
49#461C2211yescharacteristic
50#452—21no—
51#452—22no—
52#441—21no—
53#440—21no—
54#432C2211yescharacteristic, complemented
55#431C2211yescharacteristic, complemented
56#427—21no—
57C2×C4⋊C8C2211yescharacteristic, complemented
58#416—21no—
59#333C2211yescharacteristic, complemented
60#333C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62#331C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65Q8⋊C8C2211yescharacteristic, complemented
66Q8⋊C8C2211yescharacteristic, complemented
67D4⋊C8C2211yescharacteristic
68D4⋊C8C2211yescharacteristic
69D4⋊C8C2211yescharacteristic
70D4⋊C8C2211yescharacteristic
71C2×#41D411yescharacteristic
72C2×#41D411yescharacteristic
73C2×#41D412yes—
74C22×D4—21no—
75C22×D4—21no—
76C23×C4—22no—
77C2×Q16—21no—
78C2×Q16—22no—
79C2×SD16—21no—
80C2×SD16—22no—
81C22×C8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127C2311yescharacteristic
86#126—22no—
87#126—22no—
88#125—21no—
89#125—21no—
90#125—21no—
91#124—21no—
92#124—21no—
93#124D412yes—
94#123—22no—
95#123—22no—
96#123—22no—
97#123—22no—
98C22⋊Q8—22no—
99C22⋊Q8—22no—
100C22⋊Q8—21no—
101C22⋊Q8—21no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104#121—22no—
105#121—22no—
106#121—21no—
107#121—21no—
108#121—21no—
109#121—21no—
110#120—22no—
111#120—22no—
112C4×Q8—21no—
113C4×Q8—21no—
114C4×Q8D411yescharacteristic
115C4×Q8D411yescharacteristic
116C4×Q8C2311yescharacteristic
117C4×Q8C2311yescharacteristic
118C4×D4—22no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—21no—
123C4×D4—21no—
124C4×D4—21no—
125C4×D4—21no—
126C4×D4—21no—
127C4×D4—21no—
128C4×D4—21no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4—21no—
132C4×D4D411yescharacteristic
133C4×D4D411yescharacteristic
134C4×D4C2311yescharacteristic
135C4×D4C2311yescharacteristic
136#117—22no—
137#117—21no—
138#117—21no—
139#117—21no—
140#117D411yescharacteristic
141#117D411yescharacteristic
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42C2311yescharacteristic
149#107—21no—
150#107—21no—
151#107—21no—
152#106—21no—
153#106—21no—
154#106—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8C2311yescharacteristic
158C4⋊C8C2311yescharacteristic
159C4⋊C8C2311yescharacteristic
160C4⋊C8C2311yescharacteristic
161#103—21no—
162#103—21no—
163#103—21no—
164#103—22no—
165#103—21no—
166#103—21no—
167#102—21no—
168#102—21no—
169#102—21no—
170#102—22no—
171C22⋊C8—21no—
172C22⋊C8—21no—
173#97—21no—
174C4×C8C2311yescharacteristic
175C4×C8C2311yescharacteristic
176C24—22no—
177#41—22no—
178#41—22no—
179#41—41no—
180#41—41no—
181#41—82no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8C2×D411yescharacteristic
185C2×Q8C2×D411yescharacteristic
186C2×Q8C2×D412yes—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—22no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4C2×D411yescharacteristic
199C2×D4C2×D411yescharacteristic
200C2×D4C2×D412yes—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—84no—
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215Q16—42no—
216Q16—82no—
217SD16—42no—
218SD16—82no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—42no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4C2×D411yescharacteristic
243C4⋊C4C2×D411yescharacteristic
244C4⋊C4C2×D411yescharacteristic
245C4⋊C4C2×D411yescharacteristic
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—41no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—41no—
268C42C2×D411yescharacteristic
269C42C2×D411yescharacteristic
270C42C2×D411yescharacteristic
271C42C2411yescharacteristic
272C23—21no—
273C23—21no—
274C23—22no—
275C23—44no—
276C23—44no—
277C23—82no—
278C23#12011yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—42no—
284Q8—42no—
285Q8—41no—
286Q8—41no—
287D4—21no—
288D4—21no—
289D4—42no—
290D4—42no—
291D4—42no—
292D4—44no—
293D4—44no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—82no—
299D4#13512yes—
300D4#13712yes—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4#12011yescharacteristic
328C2×C4#12011yescharacteristic
329C2×C4#12011yescharacteristic
330C2×C4C22×D411yescharacteristic
331C2×C4C22×D411yescharacteristic
332C2×C4C22×D411yescharacteristic
333C8—41no—
334C8—41no—
335C8—41no—
336C8—41no—
337C8—41no—
338C8—41no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—22no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—88no—
352C22C2×#12011yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—42no—
364C4—81no—
365C4—81no—
366C4C2×#13511yescharacteristic
367C4C2×#13711yescharacteristic
368C4#57511yescharacteristic
369C4#57511yescharacteristic
370C2—41no—
371C2—42no—
372C2—42no—
373C2—82no—
374C2#300211yescharacteristic
375C2#300711yescharacteristic
376C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3481C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3210C2semidirect
3. #3116C2semidirect
4. #3076C2semidirect
5. #1644C2semidirect
6. #1643C2semidirect
7. #1640C2semidirect
8. #1639C2semidirect
9. #1632C2semidirect
10. #1632C2semidirect
11. #1630C2semidirect
12. #1629C2semidirect
13. #1551C2semidirect
14. #1488C2semidirect
15. #1477C2semidirect
25. #533C22semidirect
34. #494C22semidirect
36. #487C22semidirect
41. #477C22semidirect
42. #477C22semidirect
45. #475C22semidirect
48. #462C22semidirect
54. #432C22semidirect
55. #431C22semidirect
57. C2×C4⋊C8C22semidirect
59. #333C22semidirect
60. #333C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
65. Q8⋊C8C22semidirect
66. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34812
#32102
#31162
#30762
#16442
#16432
#16402
#16392
#16322
#16302
#16292
#15512
#14882
#14772