Finite Group Databasebeta

← #24246

Subgroups — #24246

Order 256.

712 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#548C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic, complemented
4C2×#451C211yesmaximal, characteristic, complemented
5C2×C4×Q16C211yesmaximal, characteristic, complemented
6#2536C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1958C211yesmaximal, characteristic, complemented
12#1954C211yesmaximal, characteristic, complemented
13#1805C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21#548—24no—
22#548—24no—
23C2×#124—21no—
24C2×#124C2211yescharacteristic
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×#121—21no—
30C2×#121C2211yescharacteristic
31C2×C4×Q8C2211yescharacteristic
32C2×C4×Q8C2211yescharacteristic
33C2×C4×D4—21no—
34C2×C4×D4C2211yescharacteristic
35#464—24no—
36#452—24no—
37#451—24no—
38C4×Q16—24no—
39C2×#107C2211yescharacteristic
40C2×C4⋊C8C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#103C2211yescharacteristic
47C2×#103C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic
54C2×C22⋊C8C2211yescharacteristic
55C2×C4×C8C2211yescharacteristic
56#398—21no—
57#394—21no—
58#388—21no—
59#388—21no—
60#382C2211yescharacteristic
61#380C2211yescharacteristic
62#380C2211yescharacteristic
63#340C2211yescharacteristic
64#340C2211yescharacteristic
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—41no—
70C2×#41—41no—
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—21no—
74C22×Q8—22no—
75C22×Q8—22no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×Q8C2311yescharacteristic
81C22×D4—21no—
82C22×D4—21no—
83C22×D4D411yescharacteristic
84C22×D4C2311yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—22no—
93C2×Q16—41no—
94C2×Q16D414yescomplemented
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—41no—
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C22×C8C2311yescharacteristic
102#124—28no—
103#124—22no—
104#124D414yes—
105C22⋊Q8—28no—
106C22⋊Q8—22no—
107C22⋊Q8—22no—
108C22⋊Q8—22no—
109C22⋊Q8—22no—
110C22⋊Q8—41no—
111C22⋊Q8—41no—
112#121—28no—
113#121—22no—
114#121—22no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8D414yes—
119C4×D4—28no—
120C4×D4—22no—
121C4×D4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4D411yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4D411yescharacteristic
141C2×C22⋊C4D411yescharacteristic
142C2×C42—21no—
143C2×C42—21no—
144C2×C42C2311yescharacteristic
145#107—22no—
146C4⋊C8—22no—
147#103—22no—
148#103—22no—
149#103—22no—
150#103—22no—
151#103—22no—
152#103—22no—
153#103—41no—
154#102—22no—
155#102—22no—
156#102—41no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C4×C8—22no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166C24—21no—
167C24C2×D411yescharacteristic
168C24C2×D411yescharacteristic
169#41—44no—
170#41—44no—
171#41—44no—
172#41—44no—
173C2×Q8—24no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—44no—
190C2×Q8—44no—
191C2×Q8—44no—
192C2×Q8—44no—
193C2×Q8C2×D412yes—
194C2×Q8C2×D412yes—
195C2×Q8C2×D412yes—
196C2×Q8C2×D412yes—
197C2×Q8#4112yes—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—41no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4C2×D412yes—
213C22×C4—21no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4C2×D411yescharacteristic
239C22×C4C2×D411yescharacteristic
240C22×C4C2×D411yescharacteristic
241C22×C4C2×D411yescharacteristic
242C22×C4C2411yescharacteristic
243Q16—28no—
244Q16—44no—
245SD16—44no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8C2×D412yes—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4C2×D412yes—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—24no—
283C42—42no—
284C42C2×D412yes—
285C42C2×D412yes—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23#13512yes—
307C23C22×D411yescharacteristic
308Q8—22no—
309Q8—22no—
310Q8—22no—
311Q8—22no—
312Q8—22no—
313Q8—22no—
314Q8—22no—
315Q8—48no—
316Q8—41no—
317Q8—41no—
318Q8—41no—
319Q8—41no—
320Q8#12114yes—
321D4—22no—
322D4—22no—
323D4—44no—
324D4—48no—
325D4—48no—
326D4—41no—
327D4—41no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4#12012yes—
370C2×C4#12012yes—
371C2×C4C22×D411yescharacteristic
372C2×C4C22×D411yescharacteristic
373C2×C4C22×D411yescharacteristic
374C2×C4C2×#4111yescharacteristic
375C2×C4H(5,𝔽2)11yescharacteristic
376C2×C4H(5,𝔽2)11yescharacteristic
377C8—22no—
378C8—41no—
379C8—41no—
380C8—41no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—42no—
392C22—41no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22C2×#12011yescharacteristic
403C22D4211yescharacteristic
404C22#54611yescharacteristic
405C22C2×#13511yescharacteristic
406C22#57511yescharacteristic
407C22#57811yescharacteristic
408C22#57811yescharacteristic
409C4—24no—
410C4—22no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4C2×#12111yescharacteristic
421C4#53511yescharacteristic
422C4D4211yescharacteristic
423C4#54611yescharacteristic
424C2—42no—
425C2—81no—
426C2—82no—
427C2#240811yescharacteristic
428C2#300211yescharacteristic
429C2#301111yescharacteristic
430C2#327711yescharacteristic
431C2#328511yescharacteristic
432C2#330011yescharacteristic
433C2#332711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#548C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#464C2semidirect
3. C2×#452C2semidirect
4. C2×#451C2semidirect
5. C2×C4×Q16C2semidirect
6. #2536C2semidirect
7. #2036C2semidirect
8. #2033C2semidirect
9. #2001C2semidirect
10. #2000C2semidirect
11. #1958C2semidirect
12. #1954C2semidirect
13. #1805C2semidirect
14. #1788C2semidirect
15. #1787C2semidirect
94. C2×Q16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5482
C2×#4642
C2×#4522
C2×#4512
C2×C4×Q162
#25362
#20362
#20332
#20012
#20002
#19582
#19542
#18052
#17882
#17872