Finite Group Databasebeta

← #23972

Subgroups — #23972

Order 256.

758 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547C211yesmaximal, characteristic, complemented
2C2×#449C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic, complemented
4#2407C211yesmaximal, characteristic, complemented
5#2024C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#2000C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1872C211yesmaximal, characteristic, complemented
14#1870C211yesmaximal, characteristic, complemented
15#1824C211yesmaximal, characteristic, complemented
16C22×#41—21no—
17C22×#41C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×M16C2211yescharacteristic
21#547—24no—
22#547—24no—
23C2×C4⋊Q8C2211yescharacteristic
24C2×#123—21no—
25C2×#123—21no—
26C2×C22⋊Q8—21no—
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34C22×C4⋊C4C2211yescharacteristic
35#449—24no—
36#417—24no—
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#394—21no—
49#392—21no—
50#391C2211yescharacteristic
51#390C2211yescharacteristic
52#386—21no—
53#385—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—28no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—41no—
69C2×#41—41no—
70C2×#41—44no—
71C2×#41—44no—
72C22×Q8—21no—
73C22×Q8C2311yescharacteristic
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4D411yescharacteristic
79C22×D4C2311yescharacteristic
80C22×D4C2311yescharacteristic
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C2×SD16—22no—
88C2×SD16—22no—
89C2×SD16—41no—
90C2×D8—22no—
91C2×D8—22no—
92C2×D8—41no—
93C2×M16—22no—
94C2×M16—41no—
95C2×M16—41no—
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102#123—28no—
103#123—28no—
104C22⋊Q8—28no—
105C22⋊Q8—22no—
106C22⋊Q8—41no—
107#121—22no—
108#121—22no—
109#121—22no—
110#121—22no—
111#121—41no—
112#121D414yescomplemented
113C4×D4—22no—
114C4×D4—22no—
115C4×D4—22no—
116C4×D4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4D411yescharacteristic
125C2×C4⋊C4D414yescomplemented
126C2×C4⋊C4C2311yescharacteristic
127C2×C4⋊C4C2311yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
140C2×C22⋊C4D411yescharacteristic
141C2×C42C2311yescharacteristic
142#103—22no—
143#103—22no—
144#103—22no—
145#102—22no—
146#102—22no—
147#102—22no—
148#102—41no—
149#102—41no—
150C22⋊C8—22no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153#95—21no—
154#95—21no—
155#95—21no—
156C24—21no—
157C24—21no—
158C24C2×D411yescharacteristic
159C24C2×D411yescharacteristic
160#41—28no—
161#41—44no—
162#41—44no—
163#41—44no—
164#41—432no—
165C2×Q8—24no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8#4112yescomplemented
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—24no—
177C2×D4—24no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—81no—
199C2×D4C2×D412yes—
200C2×D4#4112yescomplemented
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2×D411yescharacteristic
232C22×C4C2×D412yes—
233C22×C4C2×D412yes—
234C22×C4C2×D411yescharacteristic
235C22×C4C2411yescharacteristic
236SD16—44no—
237D8—44no—
238M16—44no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—42no—
263C4⋊C4C2×D412yes—
264C4⋊C4C2×D412yes—
265C4⋊C4C2×D412yes—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—22no—
269C22⋊C4—24no—
270C22⋊C4—22no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C42—21no—
277C42—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—22no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#12012yes—
305C23AGL(1,ℤ/8)12yes—
306C23C22×D411yescharacteristic
307Q8—22no—
308Q8—22no—
309Q8—48no—
310Q8—41no—
311Q8—41no—
312D4—28no—
313D4—22no—
314D4—22no—
315D4—22no—
316D4—22no—
317D4—48no—
318D4—48no—
319D4—48no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—84no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—48no—
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#12012yes—
366C2×C4AGL(1,ℤ/8)12yes—
367C2×C4C22×D411yescharacteristic
368C2×C4C22×D411yescharacteristic
369C2×C4C2×#4111yescharacteristic
370C2×C4C2×#4111yescharacteristic
371C2×C4H(5,𝔽2)11yescharacteristic
372C2×C4H(5,𝔽2)11yescharacteristic
373C8—41no—
374C8—41no—
375C8—41no—
376C8—41no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22C2×#12011yescharacteristic
406C22#54611yescharacteristic
407C22#54611yescharacteristic
408C22C2×AGL(1,ℤ/8)11yescharacteristic
409C22C2×AGL(1,ℤ/8)11yescharacteristic
410C22#57711yescharacteristic
411C22#57811yescharacteristic
412C4—22no—
413C4—22no—
414C4—41no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—41no—
420C4—81no—
421C4—81no—
422C4#52511yescharacteristic
423C4#53511yescharacteristic
424C4#54611yescharacteristic
425C4#54611yescharacteristic
426C2—42no—
427C2—42no—
428C2—82no—
429C2—82no—
430C2#240411yescharacteristic
431C2#300311yescharacteristic
432C2#301311yescharacteristic
433C2#330611yescharacteristic
434C2#330811yescharacteristic
435C2#333011yescharacteristic
436C2#333211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#547C2semidirect
2. C2×#449C2semidirect
3. C2×#417C2semidirect
4. #2407C2semidirect
5. #2024C2semidirect
6. #2023C2semidirect
7. #2000C2semidirect
8. #1999C2semidirect
9. #1895C2semidirect
10. #1879C2semidirect
11. #1877C2semidirect
12. #1874C2semidirect
13. #1872C2semidirect
14. #1870C2semidirect
15. #1824C2semidirect
112. #121D4semidirect
125. C2×C4⋊C4D4semidirect
173. C2×Q8#41semidirect
200. C2×D4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4492
C2×#4172
#24072
#20242
#20232
#20002
#19992
#18952
#18792
#18772
#18742
#18722
#18702
#18242