Finite Group Databasebeta

← #22421

Subgroups — #22421

Order 256.

765 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#548C211yesmaximal, characteristic
2C2×#435C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic, complemented
4C22×#103C211yesmaximal, characteristic, complemented
5D4×C4⋊C4C211yesmaximal, characteristic, complemented
6#1983C211yesmaximal, characteristic, complemented
7#1959C211yesmaximal, characteristic, complemented
8#1894C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1874C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1803C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic, complemented
14#1787C211yesmaximal, characteristic, complemented
15#1763C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#548—24no—
21#548C418yescomplemented
22C2×#124—21no—
23C2×#124C2211yescharacteristic
24C2×C22⋊Q8—21no—
25C2×C22⋊Q8C2211yescharacteristic
26C2×#121—21no—
27C2×#121C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4—21no—
31C2×C4×D4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34#435—24no—
35C2×C4⋊C8C2211yescharacteristic
36#417—24no—
37C2×#103—24no—
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×#97C2211yescharacteristic
53#389—21no—
54#385—21no—
55#385—21no—
56#384C2211yescharacteristic
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59C4×C4⋊C4C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63#340C2211yescharacteristic
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—41no—
68C2×#41—41no—
69C2×#41C2×C414yescomplemented
70C22×Q8—21no—
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8C2×C414yescomplemented
77C22×Q8C2311yescharacteristic
78C22×Q8C2311yescharacteristic
79C22×D4—21no—
80C22×D4—21no—
81C22×D4D411yescharacteristic
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—41no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—41no—
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99#124—28no—
100#124—22no—
101#124C2×C414yescomplemented
102C22⋊Q8—22no—
103C22⋊Q8—28no—
104C22⋊Q8C2×C414yescomplemented
105#121—28no—
106#121—22no—
107#121C2×C414yescomplemented
108C4×Q8—22no—
109C4×Q8C2×C414yescomplemented
110C4×D4—28no—
111C4×D4—44no—
112C4×D4C2×C414yes—
113C4×D4D414yes—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4D411yescharacteristic
125C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
137C2×C22⋊C4D411yescharacteristic
138C2×C42—21no—
139C2×C42—21no—
140C2×C42C2311yescharacteristic
141C4⋊C8—22no—
142#103—22no—
143#103—22no—
144#103—22no—
145#103—22no—
146#103—22no—
147#103—41no—
148#103—41no—
149#103D414yescomplemented
150#102—22no—
151#102—22no—
152#102—41no—
153#102—41no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156#97—22no—
157#95—21no—
158#95—21no—
159#95—21no—
160C24—21no—
161C24C2×D411yescharacteristic
162C24C2×D411yescharacteristic
163#41—28no—
164#41—44no—
165#41—44no—
166#41—44no—
167C2×Q8—24no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—28no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—44no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8C22×C412yes—
183C2×Q8C22×C412yes—
184C2×Q8C2×D412yes—
185C2×Q8#4112yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4C22⋊C414yes—
200C2×D4C22×C412yes—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—24no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4C22×C412yes—
229C22×C4C22×C412yes—
230C22×C4C2×D411yescharacteristic
231C22×C4C2×D411yescharacteristic
232C22×C4C2×D411yescharacteristic
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D412yes—
235C22×C4C2×D412yes—
236C22×C4C2411yescharacteristic
237M16—44no—
238C2×C8—21no—
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—44no—
250C2×C8C2×D412yes—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—81no—
265C4⋊C4C22⋊C414yes—
266C4⋊C4C22×C412yes—
267C4⋊C4C2×D412yes—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4C22⋊C414yes—
275C22⋊C4C22⋊C414yes—
276C42—24no—
277C42—42no—
278C42C22×C412yes—
279C42C2×D412yes—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊C412yes—
298C23C2×C22⋊C412yes—
299C23AGL(1,ℤ/8)12yes—
300C23#13712yes—
301C23C22×D411yescharacteristic
302Q8—22no—
303Q8—22no—
304Q8—22no—
305Q8—22no—
306Q8—22no—
307Q8—48no—
308Q8—41no—
309Q8—41no—
310Q8C4×D414yescomplemented
311D4—28no—
312D4—22no—
313D4—22no—
314D4—48no—
315D4—48no—
316D4—41no—
317D4—41no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
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—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—48no—
341C2×C4—48no—
342C2×C4—41no—
343C2×C4—41no—
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—84no—
358C2×C4—84no—
359C2×C4C2×C22⋊C412yes—
360C2×C4C2×C22⋊C412yes—
361C2×C4C2×C22⋊C412yes—
362C2×C4C2×C22⋊C412yes—
363C2×C4C23×C411yescharacteristic
364C2×C4C22×D411yescharacteristic
365C2×C4C22×D411yescharacteristic
366C2×C4C2×#4111yescharacteristic
367C2×C4H(5,𝔽2)11yescharacteristic
368C2×C4H(5,𝔽2)11yescharacteristic
369C8—22no—
370C8—41no—
371C8—41no—
372C8—41no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22#41714yescomplemented
394C22C22×C22⋊C411yescharacteristic
395C22D4211yescharacteristic
396C22#54611yescharacteristic
397C22C2×AGL(1,ℤ/8)11yescharacteristic
398C22C2×#13711yescharacteristic
399C22#57711yescharacteristic
400C22#57811yescharacteristic
401C4—22no—
402C4—22no—
403C4—24no—
404C4—22no—
405C4—22no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—81no—
410C4—81no—
411C4—81no—
412C4C2×C4×D411yescharacteristic
413C4#51811yescharacteristic
414C4D4211yescharacteristic
415C4#54611yescharacteristic
416C2—24no—
417C2—42no—
418C2—82no—
419C2D4×C22⋊C411yescharacteristic
420C2C2×#41711yescharacteristic
421C2#289811yescharacteristic
422C2#326511yescharacteristic
423C2#327011yescharacteristic
424C2#330811yescharacteristic
425C2#331611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#548C2
LCSG ▷ C2×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×#435C2semidirect
3. C2×#417C2semidirect
4. C22×#103C2semidirect
5. D4×C4⋊C4C2semidirect
6. #1983C2semidirect
7. #1959C2semidirect
8. #1894C2semidirect
9. #1893C2semidirect
10. #1874C2semidirect
11. #1872C2semidirect
12. #1803C2semidirect
13. #1788C2semidirect
14. #1787C2semidirect
15. #1763C2semidirect
21. #548C4semidirect
69. C2×#41C2×C4semidirect
76. C22×Q8C2×C4semidirect
101. #124C2×C4semidirect
104. C22⋊Q8C2×C4semidirect
107. #121C2×C4semidirect
109. C4×Q8C2×C4semidirect
149. #103D4semidirect
310. Q8C4×D4semidirect
393. C22#417semidirect

Maximal subgroups

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

subgroupindex
C2×#5482
C2×#4352
C2×#4172
C22×#1032
D4×C4⋊C42
#19832
#19592
#18942
#18932
#18742
#18722
#18032
#17882
#17872
#17632