Finite Group Databasebeta

← #17603

Subgroups — #17603

Order 256.

683 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3488C211yesmaximal, characteristic
2#2880C211yesmaximal, characteristic, complemented
3#2339C211yesmaximal, characteristic, complemented
4#1519C211yesmaximal, characteristic, complemented
5#1516C212yesmaximal, complemented
6#1513C211yesmaximal, characteristic, complemented
7#1510C212yesmaximal, complemented
8#1507C211yesmaximal, characteristic, complemented
9#1504C212yesmaximal, complemented
10#1501C211yesmaximal, characteristic, complemented
11C2×#327C212yesmaximal, complemented
12C2×H(5,𝔽2)C411yescharacteristic, complemented
13C22×#41C2211yescharacteristic
14C2×#131—21no—
15#550—21no—
16#548—21no—
17#546—21no—
18#546—22no—
19D42—21no—
20D42—22no—
21#537C412yescomplemented
22#537C411yescharacteristic, complemented
23#535C411yescharacteristic, complemented
24#535C412yescomplemented
25#525C2211yescharacteristic
26#525C2212yes—
27C2×#121C2212yes—
28C2×#121C2211yescharacteristic
29#520C411yescharacteristic, complemented
30C2×C4×D4—21no—
31#408—22no—
32#408—21no—
33#407C2212yescomplemented
34#407C2211yescharacteristic, complemented
35C2×C22⋊C8C2212yescomplemented
36C2×C22⋊C8C2211yescharacteristic, complemented
37#391—21no—
38#386—22no—
39#381C2212yescomplemented
40#380C2211yescharacteristic, complemented
41C4×C22⋊C4C2211yescharacteristic, complemented
42C2×#95C2212yescomplemented
43#328C2211yescharacteristic, complemented
44#328C2211yescharacteristic, complemented
45#328C2212yescomplemented
46#327C2211yescharacteristic, complemented
47#327C2212yescomplemented
48#327C2212yescomplemented
49#327C2212yescomplemented
50#327C2211yescharacteristic, complemented
51#327C2212yescomplemented
52#327C2212yescomplemented
53H(5,𝔽2)—44no—
54C2×#41—21no—
55C2×#41—44no—
56C2×#41C2×C411yescharacteristic
57C2×#41C2×C412yes—
58C2×#41C2×C411yescharacteristic
59C2×#41C2×C412yescomplemented
60C2×#41C2×C411yescharacteristic, complemented
61C2×#41D411yescharacteristic
62C2×#41D412yes—
63C2×#41D411yescharacteristic
64C22×Q8D411yescharacteristic
65C22×D4—22no—
66C22×D4—21no—
67C22×D4—22no—
68C22×D4—21no—
69C22×D4C2×C412yescomplemented
70C22×D4C2×C411yescharacteristic, complemented
71C22×D4D411yescharacteristic
72C22×D4D412yes—
73C23×C4—22no—
74C23×C4C2312yes—
75C23×C4C2311yescharacteristic
76#131—44no—
77C2×M16—22no—
78C2×M16—21no—
79C22×C8—22no—
80C22×C8—21no—
81#127—21no—
82#127—22no—
83#124—21no—
84#124—22no—
85#123—22no—
86#123—21no—
87C22⋊Q8—21no—
88C22⋊Q8—22no—
89C22⋊Q8C2×C411yescharacteristic, complemented
90C22⋊Q8C2×C412yescomplemented
91C22⋊Q8C2311yescharacteristic
92#121—22no—
93#121—22no—
94#121—22no—
95#121—21no—
96#121—22no—
97#121—21no—
98#121—22no—
99#121—21no—
100#121—22no—
101#121C2×C412yescomplemented
102#121C2×C412yescomplemented
103#121C2×C412yescomplemented
104#121C2×C411yescharacteristic, complemented
105#121C2×C412yescomplemented
106#121C2311yescharacteristic
107#121C2312yes—
108#120—22no—
109#120—21no—
110#120—22no—
111#120—22no—
112#120—22no—
113C4×Q8—21no—
114C4×D4—22no—
115C4×D4—21no—
116C4×D4—21no—
117C4×D4—22no—
118C4×D4—21no—
119C4×D4—22no—
120C4×D4—44no—
121#117C2×C411yescharacteristic, complemented
122#117C2×C412yescomplemented
123C2×C4⋊C4—21no—
124C2×C4⋊C4C2×C412yescomplemented
125C2×C4⋊C4C2×C411yescharacteristic, complemented
126C2×C22⋊C4—22no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C42—21no—
132C22⋊C8—21no—
133C22⋊C8—22no—
134C22⋊C8—21no—
135C22⋊C8—22no—
136C22⋊C8C2311yescharacteristic
137C22⋊C8C2312yes—
138C22⋊C8C2311yescharacteristic
139#95—22no—
140#95—22no—
141#95C2311yescharacteristic
142#95C2312yes—
143#95C2311yescharacteristic
144C24—21no—
145C24—22no—
146C24C2×D412yes—
147C24C2×D411yescharacteristic
148#41—24no—
149#41—24no—
150#41—44no—
151#41—44no—
152#41—42no—
153#41—42no—
154#41—44no—
155#41—42no—
156#41—44no—
157C2×Q8—42no—
158C2×Q8C22⋊C411yescharacteristic
159C2×Q8C22⋊C411yescharacteristic
160C2×Q8C22⋊C412yes—
161C2×Q8C22×C411yescharacteristic
162C2×D4—24no—
163C2×D4—21no—
164C2×D4—22no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—21no—
174C2×D4—44no—
175C2×D4—42no—
176C2×D4—44no—
177C2×D4—48no—
178C2×D4—44no—
179C2×D4—82no—
180C2×D4—81no—
181C2×D4—82no—
182C2×D4—81no—
183C2×D4C22⋊C412yes—
184C2×D4C22⋊C412yes—
185C2×D4C22⋊C412yes—
186C2×D4C22⋊C411yescharacteristic
187C2×D4C22⋊C412yes—
188C2×D4C22⋊C412yes—
189C2×D4C22⋊C411yescharacteristic
190C2×D4C22×C411yescharacteristic
191C2×D4C22×C412yes—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—82no—
204C22×C4C22×C412yes—
205C22×C4C22×C411yescharacteristic
206C22×C4C22×C412yes—
207C22×C4C22×C411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D412yes—
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D412yes—
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D412yes—
214C22×C4C2411yescharacteristic
215M16—44no—
216M16—42no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—22no—
220C2×C8—44no—
221C2×C8—42no—
222C4⋊C4—22no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—22no—
226C4⋊C4—21no—
227C4⋊C4—42no—
228C4⋊C4C22×C411yescharacteristic
229C4⋊C4C22×C412yes—
230C4⋊C4C22×C411yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—22no—
233C22⋊C4—21no—
234C22⋊C4—22no—
235C22⋊C4—21no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—21no—
240C22⋊C4—22no—
241C22⋊C4—21no—
242C22⋊C4—22no—
243C22⋊C4—42no—
244C22⋊C4—44no—
245C22⋊C4—81no—
246C42—22no—
247C42—21no—
248C42—42no—
249C23—22no—
250C23—22no—
251C23—24no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—21no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23—42no—
260C23—44no—
261C23—82no—
262C23—81no—
263C23—82no—
264C23—81no—
265C23C2×C22⋊C411yescharacteristic
266C23C2×C22⋊C411yescharacteristic
267C23C2×C22⋊C412yes—
268C23C2×C22⋊C412yes—
269C23C22×D411yescharacteristic
270Q8—22no—
271Q8—22no—
272Q8—42no—
273Q8—41no—
274D4—24no—
275D4—24no—
276D4—24no—
277D4—22no—
278D4—42no—
279D4—42no—
280D4—41no—
281D4—42no—
282D4—44no—
283D4—42no—
284D4—41no—
285D4—42no—
286D4—41no—
287D4—42no—
288D4—44no—
289D4—42no—
290D4—82no—
291D4—81no—
292D4—82no—
293D4—82no—
294D4—82no—
295D4#9914yes—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—44no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—88no—
325C2×C4C2×C22⋊C412yes—
326C2×C4C2×C22⋊C411yescharacteristic
327C2×C4C2×C22⋊C412yes—
328C2×C4C2×C22⋊C411yescharacteristic
329C2×C4C23×C411yescharacteristic
330C2×C4C22×D411yescharacteristic
331C8—41no—
332C8—42no—
333C8—41no—
334C22—21no—
335C22—22no—
336C22—21no—
337C22—22no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—22no—
343C22—21no—
344C22—22no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—44no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22—42no—
353C22—41no—
354C22—42no—
355C22—81no—
356C22—81no—
357C22—164no—
358C22—162no—
359C22C2×#9914yes—
360C22C22×C22⋊C411yescharacteristic
361C22#57611yescharacteristic
362C22#57711yescharacteristic
363C4—21no—
364C4—21no—
365C4—22no—
366C4—21no—
367C4—22no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—42no—
372C4—41no—
373C4—81no—
374C4—82no—
375C4C2×#9912yes—
376C2—21no—
377C2—21no—
378C2—24no—
379C2—42no—
380C2—41no—
381C2—41no—
382C2—81no—
383C2—82no—
384C2C22×#9911yescharacteristic
385C2#289711yescharacteristic
386C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3488C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2880C2semidirect
3. #2339C2semidirect
4. #1519C2semidirect
5. #1516C2semidirect
6. #1513C2semidirect
7. #1510C2semidirect
8. #1507C2semidirect
9. #1504C2semidirect
10. #1501C2semidirect
11. C2×#327C2semidirect
12. C2×H(5,𝔽2)C4semidirect
21. #537C4semidirect
22. #537C4semidirect
23. #535C4semidirect
24. #535C4semidirect
29. #520C4semidirect
33. #407C22semidirect
34. #407C22semidirect
35. C2×C22⋊C8C22semidirect
36. C2×C22⋊C8C22semidirect
39. #381C22semidirect
40. #380C22semidirect
41. C4×C22⋊C4C22semidirect
42. C2×#95C22semidirect
43. #328C22semidirect
44. #328C22semidirect
45. #328C22semidirect
46. #327C22semidirect
47. #327C22semidirect
48. #327C22semidirect
49. #327C22semidirect
50. #327C22semidirect
51. #327C22semidirect
52. #327C22semidirect
59. C2×#41C2×C4semidirect
60. C2×#41C2×C4semidirect
69. C22×D4C2×C4semidirect
70. C22×D4C2×C4semidirect
89. C22⋊Q8C2×C4semidirect
90. C22⋊Q8C2×C4semidirect
101. #121C2×C4semidirect
102. #121C2×C4semidirect
103. #121C2×C4semidirect
104. #121C2×C4semidirect
105. #121C2×C4semidirect
121. #117C2×C4semidirect
122. #117C2×C4semidirect
124. C2×C4⋊C4C2×C4semidirect
125. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#34882
#28802
#23392
#15192
#15162
#15132
#15102
#15072
#15042
#15012
C2×#3272