Finite Group Databasebeta

← #18618

Subgroups — #18618

Order 256.

542 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3519C211yesmaximal, characteristic, complemented
2#3247C211yesmaximal, characteristic, complemented
3#3147C211yesmaximal, characteristic, complemented
4#3109C211yesmaximal, characteristic, complemented
5#1634C211yesmaximal, characteristic, complemented
6#1632C211yesmaximal, characteristic, complemented
7#1631C211yesmaximal, characteristic, complemented
8#1629C211yesmaximal, characteristic, complemented
9#1624C211yesmaximal, characteristic, complemented
10#1622C211yesmaximal, characteristic, complemented
11#1621C211yesmaximal, characteristic, complemented
12#1619C211yesmaximal, characteristic, complemented
13#1551C211yesmaximal, characteristic, complemented
14#1494C211yesmaximal, characteristic, complemented
15C2×D4⋊C8C211yesmaximal, characteristic, complemented
16C2×#135—22no—
17#555C2211yescharacteristic
18#554C2211yescharacteristic
19#553—21no—
20#553—21no—
21#550C2211yescharacteristic
22#546—22no—
23#546—21no—
24D42—21no—
25D42C2211yescharacteristic
26#540—21no—
27#538—22no—
28#532C2211yescharacteristic
29#532C2211yescharacteristic, complemented
30#529C2211yescharacteristic
31#525—21no—
32C2×#121—21no—
33#518—21no—
34C2×C4×D4C2211yescharacteristic
35#495—22no—
36#494C2211yescharacteristic, complemented
37#493C2211yescharacteristic, complemented
38#492C2211yescharacteristic, complemented
39#492C2211yescharacteristic, complemented
40#481—21no—
41#478—21no—
42#477C2211yescharacteristic, complemented
43#476C2211yescharacteristic
44#476C2211yescharacteristic, complemented
45#476C2211yescharacteristic, complemented
46#466—21no—
47#465—21no—
48#463—21no—
49#460C2211yescharacteristic, complemented
50#459C2211yescharacteristic
51#459C2211yescharacteristic, complemented
52#459C2211yescharacteristic, complemented
53#447—21no—
54#442—21no—
55#442—21no—
56#432C2211yescharacteristic, complemented
57C2×C4⋊C8C2211yescharacteristic, complemented
58C2×C4×C8C2211yescharacteristic, complemented
59#333C2211yescharacteristic, complemented
60#333C2211yescharacteristic, complemented
61#331C2211yescharacteristic, complemented
62#331C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic
66D4⋊C8C2211yescharacteristic
67D4⋊C8C2211yescharacteristic, complemented
68D4⋊C8C2211yescharacteristic, complemented
69D4⋊C8C2211yescharacteristic, complemented
70D4⋊C8C2211yescharacteristic, complemented
71C2×#41—21no—
72C2×#41—21no—
73C2×#41D412yes—
74C22×D4—22no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4D411yescharacteristic
78C23×C4—21no—
79#135—48no—
80C2×Q16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—21no—
84C2×D8—22no—
85C2×D8—21no—
86C2×D8—21no—
87C2×D8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C4⋊Q8C2311yescharacteristic
91C4⋊Q8C2311yescharacteristic
92#127—21no—
93#127C2311yescharacteristic
94#127C2311yescharacteristic
95#126—22no—
96#126—21no—
97#125—21no—
98#125D411yescharacteristic
99#124—22no—
100#124—21no—
101#124D411yescharacteristic
102#124D412yes—
103#123—22no—
104#123—22no—
105#123—22no—
106#123—21no—
107C22⋊Q8—21no—
108C22⋊Q8—22no—
109C22⋊Q8D411yescharacteristic
110#121—22no—
111#121—22no—
112#121—22no—
113#121—22no—
114#121—21no—
115#121—22no—
116#121—21no—
117#121—21no—
118#121—21no—
119#121—21no—
120#121—21no—
121#121—21no—
122#121—21no—
123#121—21no—
124#121D411yescharacteristic
125#120—22no—
126#120—22no—
127#120—22no—
128C4×Q8C2311yescharacteristic
129C4×D4—22no—
130C4×D4—22no—
131C4×D4—22no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—21no—
139C4×D4—21no—
140C4×D4—21no—
141C4×D4D411yescharacteristic
142C4×D4D411yescharacteristic
143C4×D4C2311yescharacteristic
144C4×D4C2311yescharacteristic
145C4×D4C2311yescharacteristic
146#117—21no—
147#117—21no—
148#117—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4D411yescharacteristic
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—21no—
154C2×C42C2311yescharacteristic
155#107—21no—
156#106—21no—
157#106—21no—
158#106—21no—
159C4⋊C8—21no—
160C4⋊C8C2311yescharacteristic
161C4⋊C8C2311yescharacteristic
162C4⋊C8C2311yescharacteristic
163C4⋊C8C2311yescharacteristic
164#103—21no—
165#102—21no—
166#102—21no—
167#102—21no—
168#102—21no—
169#102—21no—
170#102—21no—
171#102—21no—
172C22⋊C8—21no—
173#97—21no—
174C4×C8—21no—
175C4×C8C2311yescharacteristic
176C4×C8C2311yescharacteristic
177C24—22no—
178C24—21no—
179#41—44no—
180#41—44no—
181#41—81no—
182#41—81no—
183C2×Q8—21no—
184C2×Q8C2×D411yescharacteristic
185C2×Q8C2×D412yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4—81no—
205C2×D4—81no—
206C2×D4—84no—
207C2×D4C2×D411yescharacteristic
208C2×D4C2×D411yescharacteristic
209C2×D4C2×D411yescharacteristic
210C2×D4C2×D412yes—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226Q16—44no—
227SD16—44no—
228SD16—44no—
229SD16—81no—
230D8—44no—
231D8—42no—
232D8—42no—
233D8—81no—
234C2×C8—24no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—81no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4C2×D411yescharacteristic
254C4⋊C4C2×D411yescharacteristic
255C4⋊C4C2×D411yescharacteristic
256C4⋊C4C2×D411yescharacteristic
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—22no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—22no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C42—22no—
276C42—21no—
277C42C2×D411yescharacteristic
278C42C2×D411yescharacteristic
279C42C2×D411yescharacteristic
280C42C2411yescharacteristic
281C23—21no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—42no—
286C23—44no—
287C23—42no—
288C23—41no—
289C23—81no—
290C23—82no—
291C23—82no—
292C23#12011yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—42no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—42no—
304D4—42no—
305D4—44no—
306D4—42no—
307D4—44no—
308D4—41no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—82no—
317D4—82no—
318D4—82no—
319D4—82no—
320D4—82no—
321D4#13512yescomplemented
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4#12011yescharacteristic
348C2×C4#12011yescharacteristic
349C2×C4#12011yescharacteristic
350C2×C4C22×D411yescharacteristic
351C2×C4C22×D411yescharacteristic
352C2×C4C22×D411yescharacteristic
353C8—22no—
354C8—22no—
355C8—41no—
356C8—41no—
357C8—41no—
358C8—41no—
359C22—21no—
360C22—21no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—42no—
367C22—41no—
368C22—42no—
369C22—41no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—84no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—164no—
378C22#45312yes—
379C22C2×#12011yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—42no—
390C4—81no—
391C4C2×#13511yescharacteristic
392C4#57511yescharacteristic
393C4#57611yescharacteristic
394C4#57711yescharacteristic
395C2—22no—
396C2—42no—
397C2—81no—
398C2—82no—
399C2—81no—
400C2—161no—
401C2#300211yescharacteristic
402C2#301211yescharacteristic
403C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3519C2semidirect
2. #3247C2semidirect
3. #3147C2semidirect
4. #3109C2semidirect
5. #1634C2semidirect
6. #1632C2semidirect
7. #1631C2semidirect
8. #1629C2semidirect
9. #1624C2semidirect
10. #1622C2semidirect
11. #1621C2semidirect
12. #1619C2semidirect
13. #1551C2semidirect
14. #1494C2semidirect
15. C2×D4⋊C8C2semidirect
29. #532C22semidirect
36. #494C22semidirect
37. #493C22semidirect
38. #492C22semidirect
39. #492C22semidirect
42. #477C22semidirect
44. #476C22semidirect
45. #476C22semidirect
49. #460C22semidirect
51. #459C22semidirect
52. #459C22semidirect
56. #432C22semidirect
57. C2×C4⋊C8C22semidirect
58. C2×C4×C8C22semidirect
59. #333C22semidirect
60. #333C22semidirect
61. #331C22semidirect
62. #331C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
67. D4⋊C8C22semidirect
68. D4⋊C8C22semidirect
69. D4⋊C8C22semidirect
70. D4⋊C8C22semidirect
321. D4#135semidirect

Maximal subgroups

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

subgroupindex
#35192
#32472
#31472
#31092
#16342
#16322
#16312
#16292
#16242
#16222
#16212
#16192
#15512
#14942
C2×D4⋊C82