Finite Group Databasebeta

← #24274

Subgroups — #24274

Order 256.

709 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#137C211yesmaximal, characteristic, complemented
2C2×#467C211yesmaximal, characteristic, complemented
3C2×#465C211yesmaximal, characteristic, complemented
4C2×#435C211yesmaximal, characteristic, complemented
5#2409C211yesmaximal, characteristic, complemented
6#2407C211yesmaximal, characteristic
7#2056C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1971C211yesmaximal, characteristic, complemented
13#1809C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C2×#137—24no—
19C2×#137—24no—
20C2×#137—24no—
21C22×Q16C2211yescharacteristic
22C22×Q16C2211yescharacteristic
23C22×SD16C2211yescharacteristic
24C22×SD16C2211yescharacteristic
25C22×M16C2211yescharacteristic
26C23×C8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×C22⋊Q8C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32#467—24no—
33#465—24no—
34#435—24no—
35C2×#107C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×#97C2211yescharacteristic
49#395—21no—
50#394—21no—
51#393—21no—
52#392—21no—
53#391C2211yescharacteristic
54#391C2211yescharacteristic
55#391C2211yescharacteristic
56#390C2211yescharacteristic
57#386—21no—
58#386—21no—
59#385—21no—
60#385—21no—
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—21no—
72C22×Q8—22no—
73C22×Q8—22no—
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
89#137—232no—
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—22no—
93C2×Q16—22no—
94C2×Q16—22no—
95C2×Q16D414yescomplemented
96C2×SD16—22no—
97C2×SD16—22no—
98C2×SD16—22no—
99C2×SD16—22no—
100C2×SD16—22no—
101C2×SD16D414yescomplemented
102C2×M16—22no—
103C2×M16—22no—
104C2×M16D414yescomplemented
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8C2311yescharacteristic
109C22×C8C2311yescharacteristic
110C22×C8C2311yescharacteristic
111C22×C8C2311yescharacteristic
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C22⋊Q8—22no—
115C22⋊Q8—41no—
116C22⋊Q8—41no—
117C22⋊Q8—41no—
118#121—22no—
119#121—41no—
120C4×D4—22no—
121C4×D4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4D411yescharacteristic
126C2×C4⋊C4C2311yescharacteristic
127C2×C4⋊C4C2311yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
139C2×C22⋊C4D411yescharacteristic
140C2×C42C2311yescharacteristic
141#107—22no—
142#106—22no—
143C4⋊C8—22no—
144#103—22no—
145#103—22no—
146#103—22no—
147#103—41no—
148#103—41no—
149#103—41no—
150#102—22no—
151#102—41no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154#97—22no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—21no—
163C24—21no—
164C24C2×D411yescharacteristic
165C24C2×D411yescharacteristic
166#41—28no—
167#41—44no—
168#41—44no—
169#41—44no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—28no—
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—42no—
187C2×Q8—42no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—44no—
191C2×Q8—42no—
192C2×Q8C2×D412yes—
193C2×Q8C2×D412yes—
194C2×Q8C2×D412yes—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—41no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—81no—
209C2×D4C2×D412yes—
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—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D411yescharacteristic
235C22×C4C2×D411yescharacteristic
236C22×C4C2×D411yescharacteristic
237C22×C4C2×D412yes—
238C22×C4#4112yes—
239C22×C4C2411yescharacteristic
240Q16—28no—
241Q16—28no—
242SD16—28no—
243SD16—28no—
244M16—28no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—44no—
254C2×C8C2×D412yes—
255C2×C8C2×D412yes—
256C2×C8C2×D412yes—
257C2×C8C2×D412yes—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C22⋊C4—22no—
271C22⋊C4—22no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C42—21no—
281C42—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—22no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23#12112yes—
302C23#13712yescomplemented
303C23C22×D411yescharacteristic
304Q8—22no—
305Q8—22no—
306Q8—22no—
307Q8—22no—
308Q8—22no—
309Q8—22no—
310Q8—48no—
311Q8—41no—
312Q8—41no—
313Q8—41no—
314Q8—41no—
315Q8—41no—
316Q8—41no—
317D4—22no—
318D4—22no—
319D4—44no—
320D4—48no—
321D4—41no—
322D4—41no—
323D4—84no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—41no—
345C2×C4—41no—
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#12112yes—
367C2×C4C22×D411yescharacteristic
368C2×C4C22×D411yescharacteristic
369C2×C4C22×D411yescharacteristic
370C2×C4C2×#4111yescharacteristic
371C2×C4H(5,𝔽2)11yescharacteristic
372C2×C4H(5,𝔽2)11yescharacteristic
373C8—22no—
374C8—41no—
375C8—41no—
376C8#12014yes—
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—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22C2×#12111yescharacteristic
399C22D4211yescharacteristic
400C22D4211yescharacteristic
401C22C2×#13711yescharacteristic
402C22#57511yescharacteristic
403C22#57711yescharacteristic
404C22#57811yescharacteristic
405C4—22no—
406C4—41no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C4—81no—
415C4—81no—
416C4C2×#12011yescharacteristic
417C4#53511yescharacteristic
418C4#54611yescharacteristic
419C4#54611yescharacteristic
420C2—42no—
421C2—42no—
422C2—82no—
423C2#240811yescharacteristic
424C2#305611yescharacteristic
425C2#306311yescharacteristic
426C2#326611yescharacteristic
427C2#326911yescharacteristic
428C2#327011yescharacteristic
429C2#327111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2407C2
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. C22×#137C2semidirect
2. C2×#467C2semidirect
3. C2×#465C2semidirect
4. C2×#435C2semidirect
5. #2409C2semidirect
7. #2056C2semidirect
8. #2055C2semidirect
9. #2001C2semidirect
10. #2000C2semidirect
11. #1973C2semidirect
12. #1971C2semidirect
13. #1809C2semidirect
14. #1788C2semidirect
15. #1787C2semidirect
95. C2×Q16D4semidirect
101. C2×SD16D4semidirect
104. C2×M16D4semidirect
302. C23#137semidirect

Maximal subgroups

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

subgroupindex
C22×#1372
C2×#4672
C2×#4652
C2×#4352
#24092
#24072
#20562
#20552
#20012
#20002
#19732
#19712
#18092
#17882
#17872