Finite Group Databasebeta

← #17559

Subgroups — #17559

Order 256.

719 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
1C2×#525C211yesmaximal, characteristic
2C2×#407C211yesmaximal, characteristic, complemented
3#2280C211yesmaximal, characteristic, complemented
4#1513C214yesmaximal, complemented
5#1511C212yesmaximal, complemented
6#1510C212yesmaximal, 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×#117—21no—
28C2×#117C411yescharacteristic, complemented
29#407—21no—
30#407—21no—
31#407C2212yes—
32#407C2212yescomplemented
33C2×C22⋊C8—21no—
34C2×C22⋊C8C2212yescomplemented
35#381C2212yes—
36#380—21no—
37#380C2212yescomplemented
38#379—21no—
39C4×C22⋊C4—21no—
40C4×C22⋊C4C2212yescomplemented
41#328C2218yescomplemented
42#327C2218yescomplemented
43C25D411yescharacteristic
44C2×#41—21no—
45C2×#41—21no—
46C2×#41—44no—
47C2×#41C2×C412yes—
48C2×#41C2×C412yes—
49C22×Q8C2×C411yescharacteristic
50C22×D4—21no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4C2×C411yescharacteristic
54C23×C4—21no—
55C23×C4—44no—
56C23×C4C2×C411yescharacteristic
57C23×C4C2×C411yescharacteristic
58C23×C4C2×C411yescharacteristic
59C23×C4C2×C411yescharacteristic
60C23×C4D411yescharacteristic
61C23×C4D411yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4D411yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—44no—
73C22×C8—22no—
74#123—22no—
75#123—22no—
76#123—22no—
77#123—22no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8C2×C412yes—
83C22⋊Q8C2×C412yes—
84C22⋊Q8C2×C412yes—
85C22⋊Q8C2312yes—
86#121—21no—
87#121—21no—
88#121—21no—
89#121—21no—
90#121C2×C412yes—
91#121C2×C412yes—
92#121C2×C412yes—
93#121C2312yes—
94#120—22no—
95#120—22no—
96#120—22no—
97#120—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106#117—21no—
107#117—21no—
108#117—44no—
109#117C2×C412yes—
110#117C2×C412yes—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4C2×C411yescharacteristic
115C2×C4⋊C4C2×C411yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C42—21no—
125C2×C42—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8C2314yes—
130#95—22no—
131#95C2314yes—
132C24—21no—
133C24—42no—
134C24—42no—
135C24—42no—
136C24C22⋊C411yescharacteristic
137C24C22⋊C411yescharacteristic
138C24C22⋊C411yescharacteristic
139C24C22⋊C411yescharacteristic
140C24C2×D411yescharacteristic
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143#41—44no—
144#41—44no—
145#41—44no—
146#41—44no—
147C2×Q8—21no—
148C2×Q8—42no—
149C2×Q8C22×C412yes—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—42no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—44no—
161C2×D4C22×C412yes—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—42no—
172C22×C4—42no—
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—44no—
183C22×C4C22⋊C411yescharacteristic
184C22×C4C22⋊C411yescharacteristic
185C22×C4C22⋊C411yescharacteristic
186C22×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2411yescharacteristic
211M16—44no—
212M16—44no—
213C2×C8—24no—
214C2×C8—44no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—22no—
221C4⋊C4—44no—
222C4⋊C4C22×C412yes—
223C4⋊C4C22×C412yes—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C42—22no—
238C42—22no—
239C42—44no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—42no—
263C23—42no—
264C23—416no—
265C23—42no—
266C23C2×C22⋊C411yescharacteristic
267C23C2×C22⋊C411yescharacteristic
268C23C2×C22⋊C411yescharacteristic
269C23C2×C22⋊C411yescharacteristic
270C23C2×C22⋊C411yescharacteristic
271C23C2×C22⋊C411yescharacteristic
272C23C22×D411yescharacteristic
273Q8—42no—
274Q8—42no—
275D4—42no—
276D4—42no—
277D4—44no—
278D4—44no—
279D4—44no—
280D4—44no—
281D4—44no—
282D4—44no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
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—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
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—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4#9914yes—
322C2×C4C2×C22⋊C411yescharacteristic
323C2×C4C2×C22⋊C411yescharacteristic
324C2×C4C2×C22⋊C411yescharacteristic
325C2×C4C2×C22⋊C411yescharacteristic
326C2×C4C2×C22⋊C411yescharacteristic
327C2×C4C2×C22⋊C411yescharacteristic
328C2×C4C23×C411yescharacteristic
329C2×C4C22×D411yescharacteristic
330C8—44no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
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—44no—
350C22—44no—
351C22—44no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—84no—
360C22C2×#9912yes—
361C22#42112yes—
362C22C22×C22⋊C411yescharacteristic
363C22#57512yes—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—84no—
374C4C2×#9912yes—
375C4C2×#9912yes—
376C2—22no—
377C2—22no—
378C2—22no—
379C2—41no—
380C2—41no—
381C2—41no—
382C2—41no—
383C2—82no—
384C2C22×#9911yescharacteristic
385C2#289611yescharacteristic
386C2C2×#42111yescharacteristic
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. #2280C2semidirect
4. #1513C2semidirect
5. #1511C2semidirect
6. #1510C2semidirect
7. #1501C2semidirect
8. C22×#41C4semidirect
15. #525C4semidirect
16. #525C4semidirect
20. C2×C22⋊Q8C4semidirect
22. C2×#121C4semidirect
28. C2×#117C4semidirect
32. #407C22semidirect
34. C2×C22⋊C8C22semidirect
37. #380C22semidirect
40. C4×C22⋊C4C22semidirect
41. #328C22semidirect
42. #327C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4072
#22802
#15132
#15112
#15102
#15012