Finite Group Databasebeta

← #22414

Subgroups — #22414

Order 256.

819 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic
2C2×C8×D4C211yesmaximal, characteristic, complemented
3C2×#416C211yesmaximal, characteristic, complemented
4C22×#102C211yesmaximal, characteristic, complemented
5#2352C211yesmaximal, characteristic, complemented
6#1983C211yesmaximal, characteristic, complemented
7#1958C211yesmaximal, characteristic, complemented
8#1893C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1804C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C23×C8C2211yescharacteristic, complemented
20#546—24no—
21#546C418yescomplemented
22C2×#124C2211yescharacteristic
23C2×#123—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121—21no—
26C2×#121C2211yescharacteristic
27C2×#120—21no—
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×#117C2211yescharacteristic, complemented
31C22×C4⋊C4C2211yescharacteristic
32C22×C22⋊C4—21no—
33C8×D4—24no—
34C2×C4⋊C8C2211yescharacteristic, complemented
35#416—24no—
36C2×#103C2211yescharacteristic, complemented
37C2×#103C2211yescharacteristic, complemented
38C2×#103C2211yescharacteristic, complemented
39C2×#103C2211yescharacteristic, complemented
40C2×#102—24no—
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic, complemented
51C2×C4×C8C2211yescharacteristic, complemented
52#388—21no—
53#387—21no—
54#386—21no—
55#382C2211yescharacteristic, complemented
56#380C2211yescharacteristic
57#380C2211yescharacteristic, complemented
58C4×C4⋊C4C2211yescharacteristic, complemented
59C4×C22⋊C4—21no—
60#340C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63#340C2211yescharacteristic, complemented
64C25—21no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—41no—
69C2×#41—41no—
70C2×#41C2×C414yescomplemented
71C22×Q8C2311yescharacteristic
72C22×D4—22no—
73C22×D4—22no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—22no—
78C22×D4—41no—
79C22×D4—41no—
80C22×D4C2×C414yes—
81C22×D4D411yescharacteristic
82C22×D4C2311yescharacteristic
83C22×D4C2311yescharacteristic
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99#124—22no—
100#124C2×C414yescomplemented
101#123—28no—
102C22⋊Q8—22no—
103C22⋊Q8C2×C414yes—
104#121—22no—
105#121—28no—
106#121C2×C414yescomplemented
107#120—28no—
108C4×D4—22no—
109C4×D4C2×C414yes—
110C4×D4C2×C414yescomplemented
111C4×D4D414yes—
112#117—22no—
113#117—41no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4D411yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4D411yescharacteristic
134C2×C22⋊C4D411yescharacteristic
135C2×C42—21no—
136C2×C42—21no—
137C2×C42C2311yescharacteristic
138C4⋊C8—22no—
139#103—22no—
140#103—22no—
141#103—41no—
142#103—41no—
143#102—22no—
144#102—22no—
145#102—22no—
146#102—22no—
147#102—22no—
148#102—41no—
149#102—41no—
150#102D414yescomplemented
151C22⋊C8—22no—
152C22⋊C8—22no—
153C4×C8—22no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159C24—21no—
160C24—21no—
161C24—24no—
162C24—42no—
163C24—42no—
164C24—42no—
165C24—42no—
166C24—81no—
167C24—81no—
168C24C2×D411yescharacteristic
169C24C2×D411yescharacteristic
170#41—28no—
171#41—44no—
172#41—44no—
173#41—44no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8C22×C412yes—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—28no—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—24no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4C22⋊C414yes—
204C2×D4C22×C412yes—
205C2×D4C22×C412yes—
206C2×D4C2×D412yes—
207C2×D4#4112yes—
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—24no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4C22×C412yes—
232C22×C4C22×C412yes—
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D411yescharacteristic
235C22×C4C2×D411yescharacteristic
236C22×C4C2×D411yescharacteristic
237C22×C4C2×D412yes—
238C22×C4C2×D412yes—
239C22×C4C2411yescharacteristic
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—
251C2×C8—44no—
252C2×C8—44no—
253C2×C8C2×D412yes—
254C4⋊C4—24no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—42no—
264C4⋊C4C22⋊C414yes—
265C4⋊C4C22×C412yes—
266C4⋊C4C2×D412yes—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—81no—
275C22⋊C4C22⋊C414yes—
276C22⋊C4C22⋊C414yes—
277C42—42no—
278C42—81no—
279C42C22×C412yes—
280C42C2×D412yes—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—22no—
286C23—21no—
287C23—22no—
288C23—21no—
289C23—22no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—22no—
295C23—48no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—48no—
307C23—84no—
308C23—84no—
309C23—84no—
310C23—84no—
311C23—84no—
312C23C2×C22⋊C412yes—
313C23C2×C22⋊C412yes—
314C23#13512yes—
315C23#13512yes—
316C23C22×D411yescharacteristic
317Q8—22no—
318Q8—22no—
319Q8—41no—
320Q8—41no—
321D4—28no—
322D4—22no—
323D4—22no—
324D4—22no—
325D4—22no—
326D4—22no—
327D4—48no—
328D4—48no—
329D4—41no—
330D4—41no—
331D4—84no—
332D4C4×D414yes—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—48no—
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—84no—
369C2×C4—84no—
370C2×C4C2×C22⋊C412yes—
371C2×C4C2×C22⋊C412yes—
372C2×C4C2×C22⋊C412yes—
373C2×C4C2×C22⋊C412yes—
374C2×C4C23×C411yescharacteristic
375C2×C4C22×D411yescharacteristic
376C2×C4C22×D411yescharacteristic
377C2×C4C2×#4111yescharacteristic
378C2×C4H(5,𝔽2)11yescharacteristic
379C2×C4H(5,𝔽2)11yescharacteristic
380C8—22no—
381C8—41no—
382C8—41no—
383C8—41no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—24no—
392C22—22no—
393C22—22no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—42no—
400C22—41no—
401C22—42no—
402C22—41no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—816no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22#41614yescomplemented
413C22C22×C22⋊C411yescharacteristic
414C22D4211yescharacteristic
415C22#54611yescharacteristic
416C22C2×#13511yescharacteristic
417C22C2×#13511yescharacteristic
418C22#57611yescharacteristic
419C22#57711yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—81no—
428C4—81no—
429C4—81no—
430C4C2×C4×D411yescharacteristic
431C4#51811yescharacteristic
432C4D4211yescharacteristic
433C4#54611yescharacteristic
434C2—24no—
435C2—42no—
436C2—44no—
437C2—82no—
438C2D4×C22⋊C411yescharacteristic
439C2C2×#41611yescharacteristic
440C2#289711yescharacteristic
441C2#328011yescharacteristic
442C2#328211yescharacteristic
443C2#329611yescharacteristic
444C2#329711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#546C2
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×C8×D4C2semidirect
3. C2×#416C2semidirect
4. C22×#102C2semidirect
5. #2352C2semidirect
6. #1983C2semidirect
7. #1958C2semidirect
8. #1893C2semidirect
9. #1892C2semidirect
10. #1875C2semidirect
11. #1872C2semidirect
12. #1804C2semidirect
13. #1788C2semidirect
14. #1786C2semidirect
15. C4×#102C2semidirect
19. C23×C8C22semidirect
21. #546C4semidirect
30. C2×#117C22semidirect
34. C2×C4⋊C8C22semidirect
36. C2×#103C22semidirect
37. C2×#103C22semidirect
38. C2×#103C22semidirect
39. C2×#103C22semidirect
50. C2×C22⋊C8C22semidirect
51. C2×C4×C8C22semidirect
55. #382C22semidirect
57. #380C22semidirect
58. C4×C4⋊C4C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #340C22semidirect
70. C2×#41C2×C4semidirect
100. #124C2×C4semidirect
106. #121C2×C4semidirect
110. C4×D4C2×C4semidirect
150. #102D4semidirect
412. C22#416semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×C8×D42
C2×#4162
C22×#1022
#23522
#19832
#19582
#18932
#18922
#18752
#18722
#18042
#17882
#17862
C4×#1022