Finite Group Databasebeta

← #17542

Subgroups — #17542

Order 256.

787 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2C2×#407C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C4C211yesmaximal, characteristic, complemented
4#1508C212yesmaximal, complemented
5#1507C212yesmaximal, complemented
6#1504C214yesmaximal, complemented
7#1501C214yesmaximal, 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×C22⋊C4—21no—
29C22×C42—21no—
30#407—21no—
31#407—21no—
32#407C2212yes—
33#407C2212yescomplemented
34C2×C22⋊C8—21no—
35C2×C22⋊C8C2212yescomplemented
36C4×C22⋊C4—21no—
37C4×C22⋊C4—21no—
38C4×C22⋊C4C2212yes—
39C4×C22⋊C4C2212yescomplemented
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—22no—
55C23×C4—21no—
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⋊C4C2×C411yescharacteristic, complemented
114C2×C4⋊C4C2×C411yescharacteristic, complemented
115C2×C22⋊C4—28no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C42—21no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—44no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8C2314yes—
131#95—22no—
132#95C2314yes—
133C24—24no—
134C24—21no—
135C24—42no—
136C24—42no—
137C24C22⋊C411yescharacteristic
138C24C22⋊C411yescharacteristic
139C24C22⋊C411yescharacteristic
140C24C22⋊C411yescharacteristic
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144#41—44no—
145#41—44no—
146#41—44no—
147#41—44no—
148C2×Q8—21no—
149C2×Q8—42no—
150C2×Q8C22×C412yes—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—21no—
158C2×D4—42no—
159C2×D4—44no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4C22×C412yes—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—28no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—42no—
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—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4C22⋊C411yescharacteristic
188C22×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2411yescharacteristic
215M16—44no—
216M16—44no—
217C2×C8—24no—
218C2×C8—44no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—22no—
225C4⋊C4C22×C412yes—
226C4⋊C4C22×C412yes—
227C22⋊C4—28no—
228C22⋊C4—28no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C42—22no—
240C42—22no—
241C42—44no—
242C42—44no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—416no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23#10414yes—
269C23C2×C22⋊C411yescharacteristic
270C23C2×C22⋊C411yescharacteristic
271C23C2×C22⋊C411yescharacteristic
272C23C2×C22⋊C411yescharacteristic
273C23C2×C22⋊C411yescharacteristic
274C23C2×C22⋊C411yescharacteristic
275C23C22×D411yescharacteristic
276Q8—42no—
277Q8—42no—
278D4—42no—
279D4—44no—
280D4—44no—
281D4—44no—
282D4—44no—
283D4—44no—
284D4—44no—
285D4—42no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
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—28no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—416no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4#9914yes—
326C2×C4C2×C22⋊C411yescharacteristic
327C2×C4C2×C22⋊C411yescharacteristic
328C2×C4C2×C22⋊C411yescharacteristic
329C2×C4C2×C22⋊C411yescharacteristic
330C2×C4C2×C22⋊C411yescharacteristic
331C2×C4C2×C22⋊C411yescharacteristic
332C2×C4C23×C411yescharacteristic
333C2×C4C22×D411yescharacteristic
334C8—44no—
335C22—28no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—84no—
362C22C2×#9912yes—
363C22C2×#10412yes—
364C22C2×#10412yes—
365C22C2×#10412yes—
366C22C22×C22⋊C411yescharacteristic
367C22#57512yes—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—48no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4C2×#9912yes—
379C4C2×#9912yes—
380C2—22no—
381C2—22no—
382C2—22no—
383C2—22no—
384C2—22no—
385C2—41no—
386C2—41no—
387C2—82no—
388C2C22×#9911yescharacteristic
389C2#289611yescharacteristic
390C2C22×#10411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C22×C22⋊C4
prime radicalC2×#525C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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×C4×C22⋊C4C2semidirect
4. #1508C2semidirect
5. #1507C2semidirect
6. #1504C2semidirect
7. #1501C2semidirect
8. C22×#41C4semidirect
15. #525C4semidirect
16. #525C4semidirect
20. C2×C22⋊Q8C4semidirect
22. C2×#121C4semidirect
27. C2×#117C4semidirect
33. #407C22semidirect
35. C2×C22⋊C8C22semidirect
39. C4×C22⋊C4C22semidirect
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
113. C2×C4⋊C4C2×C4semidirect
114. 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×C4×C22⋊C42
#15082
#15072
#15042
#15012