Finite Group Databasebeta

← #24446

Subgroups — #24446

Order 256.

729 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547C211yesmaximal, characteristic
2C2×#470C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic, complemented
4C2×#452C211yesmaximal, characteristic, complemented
5C2×#417C211yesmaximal, characteristic, complemented
6#2417C211yesmaximal, characteristic, complemented
7#2043C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2016C211yesmaximal, characteristic, complemented
10#2001C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1872C211yesmaximal, characteristic, complemented
13#1870C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×#41C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×M16C2211yescharacteristic
22#547—24no—
23#547—24no—
24C2×C4⋊Q8C2211yescharacteristic
25C2×#123—21no—
26C2×#123—21no—
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×C22⋊Q8C2211yescharacteristic
30C2×C22⋊Q8C2211yescharacteristic
31C2×#121—21no—
32C2×C4×D4C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C22×C4⋊C4—21no—
35C22×C4⋊C4C2211yescharacteristic
36#470—24no—
37#452—24no—
38#452—24no—
39C2×#106C2211yescharacteristic
40#417—24no—
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
54#398—21no—
55#396—21no—
56#391C2211yescharacteristic
57#389C2211yescharacteristic
58#386—21no—
59#380—21no—
60#380C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63#336C2211yescharacteristic
64#336C2211yescharacteristic
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—22no—
71C2×#41—22no—
72C2×#41—41no—
73C2×#41—41no—
74C2×#41—41no—
75C2×#41—41no—
76C2×#41D414yes—
77C22×Q8C2311yescharacteristic
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×D4—21no—
81C22×D4—21no—
82C22×D4D411yescharacteristic
83C22×D4C2311yescharacteristic
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—22no—
93C2×Q16—41no—
94C2×Q16—41no—
95C2×Q16—41no—
96C2×SD16—22no—
97C2×SD16—22no—
98C2×SD16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—41no—
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108#123—28no—
109#123—28no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C22⋊Q8—22no—
115C22⋊Q8—22no—
116C22⋊Q8—41no—
117C22⋊Q8D414yes—
118#121—28no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—22no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4D411yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4D411yescharacteristic
146C2×C22⋊C4D411yescharacteristic
147C2×C42C2311yescharacteristic
148#106—22no—
149#103—22no—
150#103—22no—
151#103—22no—
152#103—22no—
153#103—22no—
154#103—22no—
155#103—41no—
156#102—22no—
157#102—22no—
158#102D414yescomplemented
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—41no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166C24—21no—
167C24C2×D411yescharacteristic
168C24C2×D411yescharacteristic
169#41—28no—
170#41—44no—
171#41—44no—
172#41—44no—
173#41—44no—
174#41—44no—
175#41—44no—
176#41—44no—
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—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8C2×D412yes—
194C2×Q8C2×D412yes—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—24no—
198C2×D4—22no—
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×D4C2×D412yes—
208C2×D4C2×D412yes—
209C2×D4#4112yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—24no—
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—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—81no—
239C22×C4C2×D411yescharacteristic
240C22×C4C2×D411yescharacteristic
241C22×C4C2×D411yescharacteristic
242C22×C4C2×D412yes—
243C22×C4C2×D412yes—
244C22×C4C2×D411yescharacteristic
245C22×C4C2411yescharacteristic
246Q16—44no—
247Q16—44no—
248SD16—44no—
249M16—44no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8C2×D412yes—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—81no—
277C4⋊C4C2×D412yes—
278C4⋊C4C2×D412yes—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—81no—
287C42—21no—
288C42—21no—
289C23—22no—
290C23—21no—
291C23—22no—
292C23—21no—
293C23—22no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#12012yes—
310C23C22×D411yescharacteristic
311Q8—22no—
312Q8—22no—
313Q8—22no—
314Q8—22no—
315Q8—41no—
316Q8—41no—
317Q8—41no—
318Q8—41no—
319Q8—41no—
320Q8—41no—
321Q8—41no—
322Q8—41no—
323D4—22no—
324D4—22no—
325D4—22no—
326D4—44no—
327D4—48no—
328D4—48no—
329D4#12114yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—48no—
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—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—84no—
373C2×C4#12012yes—
374C2×C4#13712yes—
375C2×C4C22×D411yescharacteristic
376C2×C4C22×D411yescharacteristic
377C2×C4C22×D411yescharacteristic
378C2×C4C2×#4111yescharacteristic
379C2×C4H(5,𝔽2)11yescharacteristic
380C2×C4H(5,𝔽2)11yescharacteristic
381C8—22no—
382C8—41no—
383C8—41no—
384C8—41no—
385C22—22no—
386C22—24no—
387C22—22no—
388C22—22no—
389C22—41no—
390C22—42no—
391C22—42no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—42no—
397C22—41no—
398C22—42no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22C2×#12011yescharacteristic
407C22D4211yescharacteristic
408C22#54611yescharacteristic
409C22C2×#13711yescharacteristic
410C22#57511yescharacteristic
411C22#57711yescharacteristic
412C22#57811yescharacteristic
413C4—22no—
414C4—22no—
415C4—41no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—41no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4C2×#12111yescharacteristic
425C4#53411yescharacteristic
426C4D4211yescharacteristic
427C4#54611yescharacteristic
428C2—42no—
429C2—44no—
430C2—81no—
431C2#241011yescharacteristic
432C2#300711yescharacteristic
433C2#301311yescharacteristic
434C2#326711yescharacteristic
435C2#327611yescharacteristic
436C2#331511yescharacteristic
437C2#333111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
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×#470C2semidirect
3. C2×#452C2semidirect
4. C2×#452C2semidirect
5. C2×#417C2semidirect
6. #2417C2semidirect
7. #2043C2semidirect
8. #2035C2semidirect
9. #2016C2semidirect
10. #2001C2semidirect
11. #1877C2semidirect
12. #1872C2semidirect
13. #1870C2semidirect
14. #1864C2semidirect
15. #1788C2semidirect
158. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4702
C2×#4522
C2×#4172
#24172
#20432
#20352
#20162
#20012
#18772
#18722
#18702
#18642
#17882