Finite Group Databasebeta

← #22405

Subgroups — #22405

Order 256.

834 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic
2C2×#410C211yesmaximal, characteristic, complemented
3C22×#99C211yesmaximal, characteristic, complemented
4#2342C212yesmaximal, complemented
5#1964C212yesmaximal, complemented
6#1899C212yesmaximal, complemented
7#1898C212yesmaximal, complemented
8#1799C211yesmaximal, characteristic, complemented
9#1782C211yesmaximal, characteristic, complemented
10#1781C211yesmaximal, characteristic, complemented
11C4×#99C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×D4C2211yescharacteristic
14#546—22no—
15#546—22no—
16#546—22no—
17#546—22no—
18C2×#124C411yescharacteristic, complemented
19C2×#123C411yescharacteristic, complemented
20C2×#123C2211yescharacteristic
21C2×C22⋊Q8C411yescharacteristic, complemented
22C2×#121C411yescharacteristic, complemented
23C2×#121C411yescharacteristic, complemented
24C2×#121C2211yescharacteristic
25C2×#120C411yescharacteristic, complemented
26C2×#120C2211yescharacteristic
27C2×C4×D4C411yescharacteristic, complemented
28C2×C4×D4C2211yescharacteristic
29C2×#117C2212yescomplemented
30C22×C22⋊C4C411yescharacteristic, complemented
31C22×C22⋊C4C2212yes—
32C22×C22⋊C4C2211yescharacteristic
33#410—24no—
34C2×#99—24no—
35C2×#99C2211yescharacteristic
36C2×#99C2211yescharacteristic
37C2×#99C2212yes—
38C2×#99C2211yescharacteristic
39C2×#99C2212yes—
40C2×#99C2211yescharacteristic
41C2×#99C2212yescomplemented
42C2×#99C2212yescomplemented
43#388—22no—
44#388—22no—
45#385—22no—
46#381C2212yescomplemented
47#380C2212yescomplemented
48#379C2212yes—
49C4×C22⋊C4—22no—
50C4×C22⋊C4C2212yescomplemented
51#342C2212yescomplemented
52#342C2212yescomplemented
53C25C2×C411yescharacteristic
54C25C2311yescharacteristic
55C2×#41—22no—
56C2×#41—22no—
57C2×#41—22no—
58C2×#41—22no—
59C2×#41—42no—
60C22×Q8C2×C411yescharacteristic, complemented
61C22×D4—22no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—42no—
66C22×D4C2×C411yescharacteristic
67C22×D4C2×C411yescharacteristic
68C22×D4C2×C411yescharacteristic
69C22×D4C2×C411yescharacteristic
70C22×D4C2×C411yescharacteristic, complemented
71C22×D4C2×C411yescharacteristic, complemented
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—22no—
76C23×C4C2×C411yescharacteristic
77C23×C4C2×C411yescharacteristic, complemented
78C23×C4C2×C411yescharacteristic, complemented
79C23×C4C2×C411yescharacteristic, complemented
80C23×C4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82#124—22no—
83#124—22no—
84#123—22no—
85#123—22no—
86#123—22no—
87#123—22no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90#121—22no—
91#121—22no—
92#121—22no—
93#121—22no—
94#121—22no—
95#121—22no—
96#120—22no—
97#120—22no—
98#120—22no—
99#120—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—21no—
106#117—24no—
107#117—42no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4C2×C411yescharacteristic, complemented
111C2×C4⋊C4C2×C411yescharacteristic, complemented
112C2×C4⋊C4C2×C411yescharacteristic, complemented
113C2×C4⋊C4C2×C411yescharacteristic, complemented
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4C2×C411yescharacteristic
124C2×C22⋊C4C2×C411yescharacteristic
125C2×C22⋊C4C2×C411yescharacteristic
126C2×C22⋊C4C2×C411yescharacteristic
127C2×C22⋊C4C2×C411yescharacteristic
128C2×C22⋊C4C2×C411yescharacteristic
129C2×C22⋊C4C2×C411yescharacteristic, complemented
130C2×C22⋊C4C2×C411yescharacteristic, complemented
131C2×C22⋊C4C2×C411yescharacteristic, complemented
132C2×C22⋊C4C2×C411yescharacteristic, complemented
133C2×C22⋊C4C2×C411yescharacteristic, complemented
134C2×C22⋊C4D412yes—
135C2×C22⋊C4D412yes—
136C2×C22⋊C4D412yes—
137C2×C22⋊C4D412yes—
138C2×C22⋊C4C2312yes—
139C2×C22⋊C4C2312yes—
140C2×C22⋊C4C2312yes—
141C2×C22⋊C4C2311yescharacteristic
142C2×C22⋊C4C2312yes—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42C2×C411yescharacteristic, complemented
146#99—22no—
147#99—22no—
148#99—22no—
149#99—24no—
150#99—24no—
151#99—42no—
152#99—42no—
153#99D414yescomplemented
154#95—22no—
155#95—22no—
156#95—22no—
157#95—22no—
158C24—21no—
159C24—21no—
160C24—22no—
161C24—21no—
162C24—21no—
163C24—22no—
164C24—41no—
165C24—42no—
166C24—42no—
167C24—81no—
168C24—81no—
169C24C22×C411yescharacteristic
170C24C22×C411yescharacteristic
171C24C22×C411yescharacteristic
172C24C22×C411yescharacteristic
173C24C22×C411yescharacteristic
174C24C2×D412yes—
175C24C2×D412yes—
176C24C2411yescharacteristic
177#41—44no—
178#41—44no—
179#41—44no—
180#41—44no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—41no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—42no—
212C2×D4—81no—
213C2×D4—81no—
214C2×D4C2×D412yes—
215C2×D4#4112yes—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—81no—
240C22×C4—82no—
241C22×C4—81no—
242C22×C4C22×C411yescharacteristic
243C22×C4C22×C411yescharacteristic
244C22×C4C22×C411yescharacteristic
245C22×C4C22×C411yescharacteristic
246C22×C4C22×C411yescharacteristic
247C22×C4C22×C411yescharacteristic
248C22×C4C22×C411yescharacteristic
249C22×C4C22×C411yescharacteristic
250C22×C4C22×C411yescharacteristic
251C22×C4C2×D412yes—
252C22×C4C2×D412yes—
253C22×C4C2×D412yes—
254C22×C4C2×D412yes—
255C4⋊C4—22no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—44no—
261C4⋊C4—82no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—22no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—82no—
284C22⋊C4C22⋊C411yescharacteristic
285C22⋊C4C22⋊C411yescharacteristic
286C22⋊C4C22⋊C411yescharacteristic
287C22⋊C4C22⋊C411yescharacteristic
288C22⋊C4C22⋊C412yes—
289C22⋊C4C22⋊C412yes—
290C22⋊C4C22⋊C412yes—
291C22⋊C4C22⋊C412yes—
292C22⋊C4C22⋊C412yes—
293C22⋊C4C22⋊C412yes—
294C22⋊C4C2×D414yes—
295C42—21no—
296C42—21no—
297C42—44no—
298C42—82no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—22no—
304C23—22no—
305C23—24no—
306C23—22no—
307C23—21no—
308C23—21no—
309C23—22no—
310C23—22no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—42no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—41no—
328C23—41no—
329C23—42no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23—82no—
338C23—82no—
339C23—84no—
340C23—84no—
341C23—84no—
342C23C2×C22⋊C411yescharacteristic
343C23C2×C22⋊C412yes—
344C23C2×C22⋊C411yescharacteristic
345C23C4×D412yes—
346C23C23×C411yescharacteristic
347C23C22×D412yes—
348C23C22×D411yescharacteristic
349C23C2×#4111yescharacteristic
350C23H(5,𝔽2)12yes—
351Q8—44no—
352D4—24no—
353D4—24no—
354D4—44no—
355D4—44no—
356D4—44no—
357D4—42no—
358D4—42no—
359D4—42no—
360D4—42no—
361D4—42no—
362D4—42no—
363D4—82no—
364D4—82no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—88no—
410C2×C4C2×C22⋊C411yescharacteristic
411C2×C4C2×C22⋊C411yescharacteristic
412C2×C4C2×C22⋊C412yes—
413C2×C4C2×C22⋊C411yescharacteristic
414C2×C4C2×C22⋊C411yescharacteristic
415C2×C4C2×C22⋊C412yes—
416C2×C4C4×D412yes—
417C22—21no—
418C22—22no—
419C22—21no—
420C22—22no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—21no—
425C22—22no—
426C22—21no—
427C22—22no—
428C22—22no—
429C22—21no—
430C22—41no—
431C22—41no—
432C22—42no—
433C22—44no—
434C22—44no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—82no—
448C22—82no—
449C22—81no—
450C22—81no—
451C22—84no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—84no—
457C22#41012yescomplemented
458C22C22×C22⋊C411yescharacteristic
459C22C2×C4×D411yescharacteristic
460C22#51811yescharacteristic
461C22D4212yes—
462C22#54612yes—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—22no—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—44no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—82no—
476C4—82no—
477C4—82no—
478C2—22no—
479C2—22no—
480C2—21no—
481C2—21no—
482C2—22no—
483C2—42no—
484C2—41no—
485C2—42no—
486C2—81no—
487C2—81no—
488C2D4×C22⋊C411yescharacteristic
489C2C2×#41011yescharacteristic
490C2#288311yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC24C24
prime radicalC2×#546C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#410C2semidirect
3. C22×#99C2semidirect
4. #2342C2semidirect
5. #1964C2semidirect
6. #1899C2semidirect
7. #1898C2semidirect
8. #1799C2semidirect
9. #1782C2semidirect
10. #1781C2semidirect
11. C4×#99C2semidirect
18. C2×#124C4semidirect
19. C2×#123C4semidirect
21. C2×C22⋊Q8C4semidirect
22. C2×#121C4semidirect
23. C2×#121C4semidirect
25. C2×#120C4semidirect
27. C2×C4×D4C4semidirect
29. C2×#117C22semidirect
30. C22×C22⋊C4C4semidirect
41. C2×#99C22semidirect
42. C2×#99C22semidirect
46. #381C22semidirect
47. #380C22semidirect
50. C4×C22⋊C4C22semidirect
51. #342C22semidirect
52. #342C22semidirect
60. C22×Q8C2×C4semidirect
70. C22×D4C2×C4semidirect
71. C22×D4C2×C4semidirect
77. C23×C4C2×C4semidirect
78. C23×C4C2×C4semidirect
79. C23×C4C2×C4semidirect
110. C2×C4⋊C4C2×C4semidirect
111. C2×C4⋊C4C2×C4semidirect
112. C2×C4⋊C4C2×C4semidirect
113. C2×C4⋊C4C2×C4semidirect
129. C2×C22⋊C4C2×C4semidirect
130. C2×C22⋊C4C2×C4semidirect
131. C2×C22⋊C4C2×C4semidirect
132. C2×C22⋊C4C2×C4semidirect
133. C2×C22⋊C4C2×C4semidirect
145. C2×C42C2×C4semidirect
153. #99D4semidirect
457. C22#410semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4102
C22×#992
#23422
#19642
#18992
#18982
#17992
#17822
#17812
C4×#992