Finite Group Databasebeta

← #17592

Subgroups — #17592

Order 256.

755 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2C2×#407C211yesmaximal, characteristic, complemented
3C2×#380C211yesmaximal, characteristic, complemented
4#1520C212yesmaximal, complemented
5#1519C212yesmaximal, complemented
6#1513C214yesmaximal, complemented
7#1504C214yesmaximal, complemented
8C22×#41C411yescharacteristic, complemented
9C24×C4C2211yescharacteristic
10C22×M16—21no—
11#525—21no—
12#525—21no—
13#525—21no—
14#525—21no—
15#525C412yescomplemented
16#525C412yescomplemented
17#525C2212yes—
18#525C2212yes—
19C2×#123—21no—
20C2×C22⋊Q8C411yescharacteristic, complemented
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C411yescharacteristic, complemented
23C2×#121C2211yescharacteristic
24C2×#120—21no—
25C2×C4×D4—21no—
26C2×C4×D4—21no—
27C2×#117C411yescharacteristic, complemented
28C22×C4⋊C4—21no—
29C22×C22⋊C4—21no—
30#407—21no—
31#407—21no—
32#407C2212yescomplemented
33#407C2212yescomplemented
34C2×C22⋊C8—21no—
35C2×C22⋊C8C2212yes—
36#380—21no—
37#380—21no—
38#380C2212yes—
39#380C2212yescomplemented
40C2×#95C2212yescomplemented
41#328C2218yescomplemented
42#327C2218yescomplemented
43C25D411yescharacteristic
44C2×#41—21no—
45C2×#41—21no—
46C2×#41—44no—
47C2×#41C2×C412yes—
48C2×#41C2×C412yescomplemented
49C22×Q8C2×C411yescharacteristic, complemented
50C22×D4—21no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4C2×C411yescharacteristic, complemented
54C23×C4—21no—
55C23×C4—22no—
56C23×C4—44no—
57C23×C4C2×C411yescharacteristic
58C23×C4C2×C411yescharacteristic
59C23×C4C2×C411yescharacteristic
60C23×C4C2×C411yescharacteristic
61C23×C4D411yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4D411yescharacteristic
67C23×C4D411yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—44no—
74C22×C8—22no—
75#123—22no—
76#123—22no—
77#123—22no—
78#123—22no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8—21no—
83C22⋊Q8C2×C412yes—
84C22⋊Q8C2×C412yescomplemented
85C22⋊Q8C2×C412yescomplemented
86C22⋊Q8C2312yes—
87#121—21no—
88#121—21no—
89#121—21no—
90#121—21no—
91#121C2×C412yes—
92#121C2×C412yescomplemented
93#121C2×C412yescomplemented
94#121C2312yes—
95#120—22no—
96#120—22no—
97#120—22no—
98#120—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4—22no—
107#117—21no—
108#117—21no—
109#117C2×C412yes—
110#117C2×C412yescomplemented
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4C2×C411yescharacteristic, complemented
117C2×C4⋊C4C2×C411yescharacteristic, complemented
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—44no—
126C2×C42—21no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8C2314yes—
131#95—22no—
132#95C2314yes—
133C24—22no—
134C24—22no—
135C24—21no—
136C24—42no—
137C24—42no—
138C24C22⋊C411yescharacteristic
139C24C22⋊C411yescharacteristic
140C24C22⋊C411yescharacteristic
141C24C22⋊C411yescharacteristic
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144C24C2×D411yescharacteristic
145#41—44no—
146#41—44no—
147#41—44no—
148#41—44no—
149C2×Q8—21no—
150C2×Q8—42no—
151C2×Q8C22×C412yes—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—21no—
159C2×D4—42no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4C22×C412yes—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4C22⋊C411yescharacteristic
190C22×C4C22⋊C411yescharacteristic
191C22×C4C22⋊C411yescharacteristic
192C22×C4C22⋊C411yescharacteristic
193C22×C4C22⋊C411yescharacteristic
194C22×C4C22⋊C411yescharacteristic
195C22×C4C22⋊C411yescharacteristic
196C22×C4C22⋊C411yescharacteristic
197C22×C4C22⋊C411yescharacteristic
198C22×C4C22⋊C411yescharacteristic
199C22×C4C22⋊C411yescharacteristic
200C22×C4C22⋊C411yescharacteristic
201C22×C4C22×C411yescharacteristic
202C22×C4C22×C411yescharacteristic
203C22×C4C22×C411yescharacteristic
204C22×C4C22×C411yescharacteristic
205C22×C4C22×C411yescharacteristic
206C22×C4C22×C411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2411yescharacteristic
217M16—44no—
218M16—44no—
219C2×C8—24no—
220C2×C8—44no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—22no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4C22×C412yes—
230C4⋊C4C22×C412yes—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C42—22no—
244C42—22no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—48no—
269C23—48no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23C2×C22⋊C411yescharacteristic
274C23C2×C22⋊C411yescharacteristic
275C23C2×C22⋊C411yescharacteristic
276C23C2×C22⋊C411yescharacteristic
277C23C2×C22⋊C411yescharacteristic
278C23C2×C22⋊C411yescharacteristic
279C23AGL(1,ℤ/8)12yes—
280C23#13712yes—
281C23C22×D411yescharacteristic
282Q8—42no—
283Q8—42no—
284D4—42no—
285D4—44no—
286D4—44no—
287D4—42no—
288D4—42no—
289D4—44no—
290D4—44no—
291D4—44no—
292D4—42no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—88no—
332C2×C4#9914yes—
333C2×C4C2×C22⋊C411yescharacteristic
334C2×C4C2×C22⋊C411yescharacteristic
335C2×C4C2×C22⋊C411yescharacteristic
336C2×C4C2×C22⋊C411yescharacteristic
337C2×C4C2×C22⋊C411yescharacteristic
338C2×C4C2×C22⋊C411yescharacteristic
339C2×C4C23×C411yescharacteristic
340C2×C4C22×D411yescharacteristic
341C8—44no—
342C22—24no—
343C22—24no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—42no—
361C22—42no—
362C22—44no—
363C22—44no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—84no—
372C22C2×#9912yes—
373C22#41712yes—
374C22#41712yes—
375C22#42112yes—
376C22C22×C22⋊C411yescharacteristic
377C22C2×AGL(1,ℤ/8)11yescharacteristic
378C22C2×#13711yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—42no—
386C4—42no—
387C4—42no—
388C4—84no—
389C4C2×#9912yes—
390C4C2×#9912yes—
391C2—21no—
392C2—21no—
393C2—22no—
394C2—22no—
395C2—22no—
396C2—22no—
397C2—41no—
398C2—41no—
399C2—82no—
400C2C22×#9911yescharacteristic
401C2C2×#41711yescharacteristic
402C2C2×#42111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#525C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#407C2semidirect
3. C2×#380C2semidirect
4. #1520C2semidirect
5. #1519C2semidirect
6. #1513C2semidirect
7. #1504C2semidirect
8. C22×#41C4semidirect
15. #525C4semidirect
16. #525C4semidirect
20. C2×C22⋊Q8C4semidirect
22. C2×#121C4semidirect
27. C2×#117C4semidirect
32. #407C22semidirect
33. #407C22semidirect
39. #380C22semidirect
40. C2×#95C22semidirect
41. #328C22semidirect
42. #327C22semidirect
48. C2×#41C2×C4semidirect
49. C22×Q8C2×C4semidirect
53. C22×D4C2×C4semidirect
84. C22⋊Q8C2×C4semidirect
85. C22⋊Q8C2×C4semidirect
92. #121C2×C4semidirect
93. #121C2×C4semidirect
110. #117C2×C4semidirect
116. C2×C4⋊C4C2×C4semidirect
117. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4072
C2×#3802
#15202
#15192
#15132
#15042