Finite Group Databasebeta

← #18795

Subgroups — #18795

Order 256.

478 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3475C211yesmaximal, characteristic, complemented
2#3210C211yesmaximal, characteristic, complemented
3#3109C211yesmaximal, characteristic, complemented
4#3106C211yesmaximal, characteristic, complemented
5#1638C211yesmaximal, characteristic, complemented
6#1635C211yesmaximal, characteristic, complemented
7#1634C211yesmaximal, characteristic, complemented
8#1632C211yesmaximal, characteristic, complemented
9#1628C211yesmaximal, characteristic, complemented
10#1625C211yesmaximal, characteristic, complemented
11#1624C211yesmaximal, characteristic, complemented
12#1622C211yesmaximal, characteristic, complemented
13#1554C211yesmaximal, characteristic, complemented
14#1488C211yesmaximal, characteristic, complemented
15#1478C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#555C2211yescharacteristic
18#554C2211yescharacteristic
19#553—21no—
20#552—21no—
21#552—21no—
22#551—22no—
23#547C2211yescharacteristic
24#546—21no—
25#533C2211yescharacteristic
26#532C2211yescharacteristic
27#529—22no—
28#529—21no—
29#525—21no—
30C4×#41—21no—
31C4×#41C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33C2×#117—21no—
34#494C2211yescharacteristic, complemented
35#492C2211yescharacteristic, complemented
36#487C2211yescharacteristic, complemented
37#486C2211yescharacteristic, complemented
38#482—21no—
39#481—21no—
40#477C2211yescharacteristic, complemented
41#476C2211yescharacteristic
42#476C2211yescharacteristic
43#476C2211yescharacteristic, complemented
44#464—22no—
45#460C2211yescharacteristic, complemented
46#459C2211yescharacteristic
47#459C2211yescharacteristic
48#459C2211yescharacteristic, complemented
49#452—22no—
50#449—21no—
51#447—21no—
52#442—21no—
53#442—21no—
54C4×SD16—21no—
55C4×D8—21no—
56#432C2211yescharacteristic
57#431C2211yescharacteristic
58#431C2211yescharacteristic
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—
76C2×Q16—22no—
77C2×SD16—21no—
78C2×SD16—22no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127C2311yescharacteristic
86#126—22no—
87#126—22no—
88#126—22no—
89#125—22no—
90#125—21no—
91#125—21no—
92#124—22no—
93#124—21no—
94#124—21no—
95#123—22no—
96#123—22no—
97#123—22no—
98#123—22no—
99C22⋊Q8—21no—
100C22⋊Q8—21no—
101C22⋊Q8—22no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104C22⋊Q8D412yes—
105#121—22no—
106#121—21no—
107#121—21no—
108#121—21no—
109#121—21no—
110#121—21no—
111#120—22no—
112C4×Q8—21no—
113C4×Q8—21no—
114C4×Q8D411yescharacteristic
115C4×Q8D412yes—
116C4×Q8C2311yescharacteristic
117C4×D4—22no—
118C4×D4—22no—
119C4×D4—22no—
120C4×D4—21no—
121C4×D4—21no—
122C4×D4—21no—
123C4×D4—21no—
124C4×D4—21no—
125C4×D4—21no—
126C4×D4—21no—
127C4×D4D411yescharacteristic
128C4×D4D411yescharacteristic
129C4×D4D411yescharacteristic
130C4×D4C2311yescharacteristic
131C4×D4C2311yescharacteristic
132C4×D4C2311yescharacteristic
133#117—22no—
134#117—22no—
135#117—22no—
136#117—21no—
137#117—21no—
138#117—21no—
139#117—21no—
140#117—21no—
141#117D411yescharacteristic
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4D411yescharacteristic
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C42—22no—
148C2×C42—21no—
149C2×C42C2311yescharacteristic
150#107—21no—
151#106—21no—
152#106—21no—
153#106—21no—
154C4⋊C8—21no—
155C4⋊C8C2311yescharacteristic
156C4⋊C8C2311yescharacteristic
157C4⋊C8C2311yescharacteristic
158C4⋊C8C2311yescharacteristic
159#103—21no—
160#103—22no—
161#102—21no—
162#102—21no—
163#102—21no—
164#102—21no—
165#102—21no—
166#102—21no—
167#102—21no—
168#102—22no—
169C22⋊C8—21no—
170C22⋊C8—21no—
171C22⋊C8—21no—
172#97—21no—
173C4×C8—21no—
174C4×C8C2311yescharacteristic
175C4×C8C2311yescharacteristic
176C24—21no—
177#41—41no—
178#41—41no—
179#41—42no—
180#41—42no—
181#41—82no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8C2×D411yescharacteristic
185C2×Q8C2×D412yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4C2×D411yescharacteristic
197C2×D4C2×D411yescharacteristic
198C2×D4C2×D411yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215Q16—82no—
216SD16—42no—
217SD16—82no—
218D8—42no—
219D8—42no—
220D8—42no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—21no—
232C4⋊C4—22no—
233C4⋊C4—21no—
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4C2×D412yes—
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—22no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C42—22no—
263C42—22no—
264C42—21no—
265C42—22no—
266C42—21no—
267C42—21no—
268C42C2×D411yescharacteristic
269C42C2×D411yescharacteristic
270C42C2×D411yescharacteristic
271C42C2411yescharacteristic
272C23—21no—
273C23—21no—
274C23—21no—
275C23—42no—
276C23—42no—
277C23—41no—
278C23—81no—
279C23#12011yescharacteristic
280Q8—21no—
281Q8—21no—
282Q8—42no—
283Q8—42no—
284Q8—41no—
285Q8—41no—
286D4—21no—
287D4—21no—
288D4—21no—
289D4—21no—
290D4—21no—
291D4—42no—
292D4—42no—
293D4—44no—
294D4—42no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—82no—
300D4—82no—
301D4#13512yes—
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—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#12011yescharacteristic
330C2×C4#12011yescharacteristic
331C2×C4#12011yescharacteristic
332C2×C4C22×D411yescharacteristic
333C2×C4C22×D411yescharacteristic
334C2×C4C22×D411yescharacteristic
335C8—41no—
336C8—41no—
337C8—41no—
338C8—41no—
339C8—41no—
340C8—41no—
341C22—21no—
342C22—22no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—84no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22C2×#12011yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—41no—
363C4—42no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—42no—
369C4—81no—
370C4C2×#13511yescharacteristic
371C4#57511yescharacteristic
372C4#57511yescharacteristic
373C4#57511yescharacteristic
374C2—41no—
375C2—42no—
376C2—81no—
377C2—81no—
378C2—161no—
379C2#300211yescharacteristic
380C2#300511yescharacteristic
381C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
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
1. #3475C2semidirect
2. #3210C2semidirect
3. #3109C2semidirect
4. #3106C2semidirect
5. #1638C2semidirect
6. #1635C2semidirect
7. #1634C2semidirect
8. #1632C2semidirect
9. #1628C2semidirect
10. #1625C2semidirect
11. #1624C2semidirect
12. #1622C2semidirect
13. #1554C2semidirect
14. #1488C2semidirect
15. #1478C2semidirect
34. #494C22semidirect
35. #492C22semidirect
36. #487C22semidirect
37. #486C22semidirect
40. #477C22semidirect
43. #476C22semidirect
45. #460C22semidirect
48. #459C22semidirect
59. #333C22semidirect
60. #333C22semidirect
61. #332C22semidirect
62. #332C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
69. D4⋊C8C22semidirect
70. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34752
#32102
#31092
#31062
#16382
#16352
#16342
#16322
#16282
#16252
#16242
#16222
#15542
#14882
#14782