Finite Group Databasebeta

← #30539

Subgroups — #30539

Order 256.

618 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#490C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3#2490C211yesmaximal, characteristic, complemented
4#2433C211yesmaximal, characteristic, complemented
5#2054C211yesmaximal, characteristic, complemented
6#2034C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic, complemented
8#2011C211yesmaximal, characteristic, complemented
9#2011C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1982C211yesmaximal, characteristic, complemented
12#1972C211yesmaximal, characteristic, complemented
13#1926C211yesmaximal, characteristic, complemented
14#1924C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic, complemented
20C2×#127C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#125C2211yescharacteristic, complemented
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#120—21no—
26C2×#120—21no—
27#490—24no—
28#447—24no—
29C2×#106C2211yescharacteristic, complemented
30C2×C4⋊C8C2211yescharacteristic, complemented
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic, complemented
38C2×#102C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic, complemented
40C2×#102C2211yescharacteristic, complemented
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×#97C2211yescharacteristic, complemented
44#397—21no—
45#394—21no—
46#392—21no—
47#392—21no—
48#390—21no—
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#390C2211yescharacteristic, complemented
52#390C2211yescharacteristic, complemented
53#388—21no—
54#380C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic, complemented
56C4×C22⋊C4C2211yescharacteristic
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—21no—
70C22×D4—21no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C22×D4C2311yescharacteristic
78C22×D4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—22no—
83C2×D8—41no—
84C2×D8—41no—
85C2×D8—41no—
86C2×D8—41no—
87C2×D8—41no—
88C2×D8—41no—
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93#127—22no—
94#127—22no—
95#127—41no—
96#127—41no—
97#125—22no—
98#125—41no—
99#121—22no—
100#121—41no—
101#121—41no—
102#121—41no—
103#120—44no—
104#120—44no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
123C2×C22⋊C4D411yescharacteristic
124C2×C42—21no—
125C2×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127#106—41no—
128C4⋊C8—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—
137#102—41no—
138#102D414yescomplemented
139C22⋊C8—22no—
140C22⋊C8—41no—
141#97—22no—
142#95—21no—
143#95—21no—
144#95D411yescharacteristic
145#95D411yescharacteristic
146C24—21no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24—42no—
154C24—42no—
155C24—81no—
156C24—81no—
157C24C2×D411yescharacteristic
158C2×D4—21no—
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—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—42no—
181C2×D4—42no—
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—42no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4C2×D412yes—
206C2×D4#4112yes—
207C22×C4—21no—
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×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2411yescharacteristic
223D8—44no—
224D8—44no—
225D8—44no—
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×C8—21no—
237C2×C8C2×D412yes—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4C2×D412yes—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—81no—
258C22⋊C4—81no—
259C22⋊C4—81no—
260C22⋊C4—81no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—81no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—48no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
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—41no—
299C23—41no—
300C23—41no—
301C23—84no—
302C23—84no—
303C23—84no—
304C23—84no—
305C23—84no—
306C23—84no—
307C23AGL(1,ℤ/8)12yes—
308C23C22×D411yescharacteristic
309D4—22no—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—48no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—84no—
327D4—84no—
328D4—84no—
329D4—84no—
330D4—84no—
331D4—84no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
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—41no—
361C2×C4—41no—
362C2×C4AGL(1,ℤ/8)12yes—
363C2×C4C22×D411yescharacteristic
364C2×C4C2×#4111yescharacteristic
365C2×C4H(5,𝔽2)11yescharacteristic
366C2×C4H(5,𝔽2)11yescharacteristic
367C2×C4H(5,𝔽2)11yescharacteristic
368C2×C4H(5,𝔽2)11yescharacteristic
369C8—22no—
370C8—41no—
371C8—41no—
372C8—41no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—816no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
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—81no—
408C22—81no—
409C22#53511yescharacteristic
410C22D4211yescharacteristic
411C22#54611yescharacteristic
412C22C2×AGL(1,ℤ/8)11yescharacteristic
413C22C2×AGL(1,ℤ/8)11yescharacteristic
414C22#57611yescharacteristic
415C22#57611yescharacteristic
416C4—22no—
417C4—41no—
418C4—41no—
419C4—42no—
420C4—41no—
421C4—41no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4#53511yescharacteristic
426C4#53811yescharacteristic
427C4#54611yescharacteristic
428C4H(3,𝔽4)11yescharacteristic
429C2—42no—
430C2—82no—
431C2—82no—
432C2—161no—
433C2—161no—
434C2#267011yescharacteristic
435C2#321711yescharacteristic
436C2#322011yescharacteristic
437C2#326411yescharacteristic
438C2#327411yescharacteristic
439C2#331811yescharacteristic
440C2#331811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#490C2semidirect
2. C2×#447C2semidirect
3. #2490C2semidirect
4. #2433C2semidirect
5. #2054C2semidirect
6. #2034C2semidirect
7. #2025C2semidirect
8. #2011C2semidirect
9. #2011C2semidirect
10. #1999C2semidirect
11. #1982C2semidirect
12. #1972C2semidirect
13. #1926C2semidirect
14. #1924C2semidirect
15. #1786C2semidirect
19. C22×D8C22semidirect
22. C2×#125C22semidirect
29. C2×#106C22semidirect
30. C2×C4⋊C8C22semidirect
37. C2×#102C22semidirect
38. C2×#102C22semidirect
39. C2×#102C22semidirect
40. C2×#102C22semidirect
43. C2×#97C22semidirect
51. #390C22semidirect
52. #390C22semidirect
55. C4×C4⋊C4C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
138. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4902
C2×#4472
#24902
#24332
#20542
#20342
#20252
#20112
#19992
#19822
#19722
#19262
#19242
#17862