Finite Group Databasebeta

← #21458

Subgroups — #21458

Order 256.

808 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#518C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic
3C2×#442C211yesmaximal, characteristic, complemented
4C2×C4×D8C211yesmaximal, characteristic, complemented
5#2324C211yesmaximal, characteristic, complemented
6#1879C211yesmaximal, characteristic, complemented
7#1879C211yesmaximal, characteristic, complemented
8#1875C211yesmaximal, characteristic, complemented
9#1864C211yesmaximal, characteristic, complemented
10#1862C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic, complemented
12#1786C211yesmaximal, characteristic, complemented
13#1762C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20#518—24no—
21#518—24no—
22C2×C4×D4—21no—
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic, complemented
29C2×#117C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33#447C418yescomplemented
34#442—24no—
35C4×D8—24no—
36C2×#107C2211yescharacteristic, complemented
37C2×#106C2211yescharacteristic, complemented
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic, complemented
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×#97C2211yescharacteristic, complemented
49C2×C4×C8C2211yescharacteristic, complemented
50#390—21no—
51#390C2211yescharacteristic, complemented
52#389C2211yescharacteristic, complemented
53#388—21no—
54#386—21no—
55#385—21no—
56#380C2211yescharacteristic
57C4×C4⋊C4C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63#336C2211yescharacteristic, complemented
64C25—21no—
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—22no—
72C22×D4C2×C414yescomplemented
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4C2311yescharacteristic
82C2×D8—22no—
83C2×D8—22no—
84C2×D8—22no—
85C2×D8—22no—
86C2×D8C2×C414yescomplemented
87C2×D8C2×C414yescomplemented
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#121—22no—
93#121C2×C414yescomplemented
94C4×D4—28no—
95C4×D4—28no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4D414yescomplemented
105C4×D4D414yescomplemented
106#117—22no—
107#117—22no—
108#117—22no—
109#117—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C22⋊C4—28no—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4D411yescharacteristic
131C2×C22⋊C4D411yescharacteristic
132C2×C42—21no—
133C2×C42—21no—
134C2×C42C2311yescharacteristic
135C2×C42C2311yescharacteristic
136#107—22no—
137#106—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144#102C2×C414yescomplemented
145#102C2×C414yescomplemented
146C22⋊C8—22no—
147C22⋊C8C2×C414yescomplemented
148#97—22no—
149C4×C8—22no—
150#95—21no—
151#95—21no—
152#95—21no—
153C24—24no—
154C24—24no—
155C24—21no—
156C24—21no—
157C24—24no—
158C24—21no—
159C24—81no—
160C24—81no—
161C24—81no—
162C24—81no—
163C24C2×D411yescharacteristic
164C2×D4—24no—
165C2×D4—24no—
166C2×D4—28no—
167C2×D4—28no—
168C2×D4—28no—
169C2×D4—28no—
170C2×D4—24no—
171C2×D4—24no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—81no—
184C2×D4—81no—
185C2×D4C22×C412yes—
186C2×D4C22×C412yes—
187C2×D4C22×C412yes—
188C2×D4C2×D412yes—
189C2×D4C2×D412yes—
190C2×D4#4112yes—
191C2×D4#4112yes—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4—81no—
216C22×C4C22×C412yes—
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4#4112yes—
223C22×C4C2411yescharacteristic
224D8—28no—
225D8—28no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8C22×C412yes—
237C2×C8C22×C412yes—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—81no—
248C4⋊C4C22×C412yes—
249C4⋊C4C2×D412yes—
250C4⋊C4C2×D412yes—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—42no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C22⋊C4—81no—
263C42—24no—
264C42—21no—
265C42—21no—
266C42—81no—
267C42C2×D412yes—
268C42C2×D412yes—
269C23—216no—
270C23—22no—
271C23—22no—
272C23—21no—
273C23—22no—
274C23—22no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—22no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—84no—
298C23—84no—
299C23—84no—
300C23—84no—
301C23—84no—
302C23—84no—
303C23C4×D412yes—
304C23C22×D411yescharacteristic
305D4—216no—
306D4—22no—
307D4—22no—
308D4—22no—
309D4—22no—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—22no—
314D4—41no—
315D4—41no—
316D4—84no—
317D4C4×D414yes—
318D4C4×D414yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
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—84no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4C4×D412yes—
356C2×C4#12012yes—
357C2×C4#12012yes—
358C2×C4C23×C411yescharacteristic
359C2×C4C22×D411yescharacteristic
360C2×C4C22×D411yescharacteristic
361C2×C4C2×#4111yescharacteristic
362C2×C4C2×#4111yescharacteristic
363C2×C4C2×#4111yescharacteristic
364C8—22no—
365C8—22no—
366C8—41no—
367C8—41no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—24no—
372C22—24no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—41no—
386C22—42no—
387C22—42no—
388C22—41no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—816no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22C2×C4×D411yescharacteristic
398C22C2×#12011yescharacteristic
399C22#52511yescharacteristic
400C22#57611yescharacteristic
401C22#57611yescharacteristic
402C22#57611yescharacteristic
403C22#57711yescharacteristic
404C4—22no—
405C4—22no—
406C4—22no—
407C4—42no—
408C4—41no—
409C4—42no—
410C4—41no—
411C4—42no—
412C4—81no—
413C4—81no—
414C4C2×C4×D411yescharacteristic
415C4C2×C4×D411yescharacteristic
416C4#52511yescharacteristic
417C4#52511yescharacteristic
418C2—42no—
419C2—44no—
420C2—44no—
421C2—82no—
422C2C4×#12011yescharacteristic
423C2#294611yescharacteristic
424C2#294811yescharacteristic
425C2#300811yescharacteristic
426C2#301211yescharacteristic
427C2#311811yescharacteristic
428C2#312211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#518C2semidirect
3. C2×#442C2semidirect
4. C2×C4×D8C2semidirect
5. #2324C2semidirect
6. #1879C2semidirect
7. #1879C2semidirect
8. #1875C2semidirect
9. #1864C2semidirect
10. #1862C2semidirect
11. #1854C2semidirect
12. #1786C2semidirect
13. #1762C2semidirect
14. C4×#102C2semidirect
15. #1750C2semidirect
28. C2×C4×D4C22semidirect
33. #447C4semidirect
36. C2×#107C22semidirect
37. C2×#106C22semidirect
44. C2×#102C22semidirect
45. C2×#102C22semidirect
48. C2×#97C22semidirect
49. C2×C4×C8C22semidirect
51. #390C22semidirect
52. #389C22semidirect
57. C4×C4⋊C4C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #336C22semidirect
63. #336C22semidirect
72. C22×D4C2×C4semidirect
86. C2×D8C2×C4semidirect
87. C2×D8C2×C4semidirect
93. #121C2×C4semidirect
104. C4×D4D4semidirect
105. C4×D4D4semidirect
144. #102C2×C4semidirect
145. #102C2×C4semidirect
147. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5182
C2×#4472
C2×#4422
C2×C4×D82
#23242
#18792
#18752
#18642
#18622
#18542
#17862
#17622
C4×#1022
#17502