Finite Group Databasebeta

← #30400

Subgroups — #30400

Order 256.

616 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#466C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3#2535C211yesmaximal, characteristic, complemented
4#2408C211yesmaximal, characteristic, complemented
5#2092C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic, complemented
7#2073C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1869C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C23×C8C2211yescharacteristic, complemented
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic, complemented
24C2×#120—21no—
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic, complemented
27#466—24no—
28#447—24no—
29C2×#107C2211yescharacteristic, complemented
30C2×#106C2211yescharacteristic, complemented
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic, complemented
38C2×#102C2211yescharacteristic, complemented
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic, complemented
42#397—21no—
43#394—21no—
44#392—21no—
45#392—21no—
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#386—21no—
49#386—21no—
50#382C2211yescharacteristic, complemented
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#380C2211yescharacteristic, complemented
54#379—21no—
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4D411yescharacteristic
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—41no—
83C2×D8—41no—
84C2×D8—41no—
85C2×D8D414yescomplemented
86C22×C8—22no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93#121—22no—
94#121—22no—
95#121—22no—
96#121—22no—
97#121—41no—
98#121—41no—
99#121—41no—
100#121—41no—
101#120—44no—
102C4×D4—41no—
103C4×D4—41no—
104C4×D4—41no—
105C4×D4—41no—
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4D411yescharacteristic
125C2×C22⋊C4D411yescharacteristic
126C2×C42C2311yescharacteristic
127#107—22no—
128#106—41no—
129#102—22no—
130#102—22no—
131#102—22no—
132#102—22no—
133#102—41no—
134#102—41no—
135#102—41no—
136#102—41no—
137C22⋊C8—22no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—42no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24—81no—
154C24—81no—
155C24C2×D411yescharacteristic
156C24C2×D411yescharacteristic
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—44no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4C2×D412yes—
195C2×D4C2×D412yes—
196C2×D4#4112yes—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
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—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2411yescharacteristic
222D8—44no—
223D8—44no—
224C2×C8—21no—
225C2×C8—21no—
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—44no—
236C2×C8C2×D412yes—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—81no—
259C22⋊C4—81no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C42—21no—
263C42—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—48no—
280C23—41no—
281C23—41no—
282C23—41no—
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—41no—
298C23—84no—
299C23—84no—
300C23—84no—
301C23—84no—
302C23—84no—
303C23—84no—
304C23#13512yes—
305C23AGL(1,ℤ/8)12yes—
306C23C22×D411yescharacteristic
307D4—22no—
308D4—22no—
309D4—22no—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—44no—
314D4—48no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—84no—
322D4—84no—
323D4—84no—
324D4—84no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
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—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—84no—
361C2×C4C22×D411yescharacteristic
362C2×C4C2×#4111yescharacteristic
363C2×C4H(5,𝔽2)11yescharacteristic
364C2×C4H(5,𝔽2)11yescharacteristic
365C2×C4H(5,𝔽2)11yescharacteristic
366C2×C4H(5,𝔽2)11yescharacteristic
367C8—22no—
368C8—41no—
369C8—41no—
370C8—41no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—816no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22#53411yescharacteristic
408C22D4211yescharacteristic
409C22#54611yescharacteristic
410C22C2×#13511yescharacteristic
411C22C2×AGL(1,ℤ/8)11yescharacteristic
412C22#57611yescharacteristic
413C22#57611yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4#53411yescharacteristic
424C4#53811yescharacteristic
425C4#54611yescharacteristic
426C4#56011yescharacteristic
427C2—42no—
428C2—42no—
429C2—82no—
430C2—82no—
431C2—82no—
432C2#266811yescharacteristic
433C2#318811yescharacteristic
434C2#319911yescharacteristic
435C2#326411yescharacteristic
436C2#328011yescharacteristic
437C2#329711yescharacteristic
438C2#331811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×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
1. C2×#466C2semidirect
2. C2×#447C2semidirect
3. #2535C2semidirect
4. #2408C2semidirect
5. #2092C2semidirect
6. #2076C2semidirect
7. #2073C2semidirect
8. #2034C2semidirect
9. #2034C2semidirect
10. #1999C2semidirect
11. #1893C2semidirect
12. #1879C2semidirect
13. #1879C2semidirect
14. #1869C2semidirect
15. #1786C2semidirect
19. C23×C8C22semidirect
23. C2×#121C22semidirect
26. C2×C4×D4C22semidirect
29. C2×#107C22semidirect
30. C2×#106C22semidirect
37. C2×#102C22semidirect
38. C2×#102C22semidirect
41. C2×C22⋊C8C22semidirect
50. #382C22semidirect
53. #380C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
85. C2×D8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4662
C2×#4472
#25352
#24082
#20922
#20762
#20732
#20342
#19992
#18932
#18792
#18692
#17862