Finite Group Databasebeta

← #30519

Subgroups — #30519

Order 256.

639 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#486C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3#2491C211yesmaximal, characteristic, complemented
4#2433C211yesmaximal, characteristic, complemented
5#2054C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#2011C211yesmaximal, characteristic, complemented
9#2011C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1982C211yesmaximal, characteristic, complemented
12#1970C211yesmaximal, characteristic, complemented
13#1925C211yesmaximal, characteristic, complemented
14#1923C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic, complemented
20C2×C4⋊Q8C2211yescharacteristic, complemented
21C2×#127C2211yescharacteristic
22C2×#127C2211yescharacteristic
23C2×#124—21no—
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#120—21no—
27C2×#120—21no—
28#486—24no—
29#447—24no—
30C2×#107C2211yescharacteristic, complemented
31C2×C4⋊C8C2211yescharacteristic, complemented
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic, complemented
40C2×#102C2211yescharacteristic, complemented
41C2×#102C2211yescharacteristic, complemented
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic, complemented
45#394—21no—
46#393—21no—
47#392—21no—
48#392—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×Q8—21no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4C2311yescharacteristic
77C22×D4C2311yescharacteristic
78C22×D4C2311yescharacteristic
79C22×D4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C2×D8—22no—
82C2×D8—22no—
83C2×D8—22no—
84C2×D8—41no—
85C2×D8—41no—
86C2×D8—41no—
87C2×D8—41no—
88C2×D8—41no—
89C2×D8—41no—
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C4⋊Q8—22no—
95C4⋊Q8—41no—
96#127—22no—
97#127—22no—
98#127—41no—
99#127—41no—
100#124—44no—
101#121—22no—
102#121—41no—
103#121—41no—
104#121—41no—
105#120—44no—
106#120—44no—
107C2×C4⋊C4—21no—
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⋊C4D411yescharacteristic
124C2×C22⋊C4D411yescharacteristic
125C2×C42—21no—
126C2×C42C2311yescharacteristic
127C2×C42C2311yescharacteristic
128#107—41no—
129C4⋊C8—41no—
130#102—22no—
131#102—22no—
132#102—22no—
133#102—22no—
134#102—41no—
135#102—41no—
136#102—41no—
137#102—41no—
138#102—41no—
139#102D414yescomplemented
140C22⋊C8—22no—
141C22⋊C8—41no—
142C4×C8—22no—
143#95—21no—
144#95—21no—
145#95D411yescharacteristic
146#95D411yescharacteristic
147C24—21no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—42no—
152C24—42no—
153C24—42no—
154C24—42no—
155C24—42no—
156C24—81no—
157C24—81no—
158C24C2×D411yescharacteristic
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—81no—
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—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—81no—
205C2×D4—81no—
206C2×D4—81no—
207C2×D4—81no—
208C2×D4—81no—
209C2×D4C2×D412yes—
210C2×D4#4112yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226C22×C4C2411yescharacteristic
227D8—44no—
228D8—44no—
229D8—44no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8C2×D412yes—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—42no—
250C4⋊C4C2×D412yes—
251C22⋊C4—22no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—81no—
263C22⋊C4D814yescomplemented
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—42no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—22no—
276C23—22no—
277C23—48no—
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—41no—
302C23—41no—
303C23—41no—
304C23—84no—
305C23—84no—
306C23—84no—
307C23—84no—
308C23—84no—
309C23—84no—
310C23C2×D812yes—
311C23C22×D411yescharacteristic
312Q8—84no—
313D4—22no—
314D4—22no—
315D4—22no—
316D4—22no—
317D4—48no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—41no—
330D4—84no—
331D4—84no—
332D4—84no—
333D4—84no—
334D4—84no—
335D4—84no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
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×C4—41no—
363C2×C4—41no—
364C2×C4C2×D812yes—
365C2×C4C2×D812yes—
366C2×C4AGL(1,ℤ/8)12yes—
367C2×C4C22×D411yescharacteristic
368C2×C4C2×#4111yescharacteristic
369C2×C4H(5,𝔽2)11yescharacteristic
370C2×C4H(5,𝔽2)11yescharacteristic
371C2×C4H(5,𝔽2)11yescharacteristic
372C2×C4H(5,𝔽2)11yescharacteristic
373C8—22no—
374C8—41no—
375C8—41no—
376C8—41no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—816no—
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—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22#53511yescharacteristic
414C22D4211yescharacteristic
415C22#54611yescharacteristic
416C22C22×D811yescharacteristic
417C22C2×AGL(1,ℤ/8)11yescharacteristic
418C22#57611yescharacteristic
419C22#57711yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—41no—
424C4—42no—
425C4—41no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4#53511yescharacteristic
430C4#53811yescharacteristic
431C4#54611yescharacteristic
432C4H(3,𝔽4)11yescharacteristic
433C2—42no—
434C2—82no—
435C2—82no—
436C2—161no—
437C2—161no—
438C2#267011yescharacteristic
439C2#320511yescharacteristic
440C2#322411yescharacteristic
441C2#326511yescharacteristic
442C2D4×D811yescharacteristic
443C2#329411yescharacteristic
444C2#332511yescharacteristic
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×#486C2semidirect
2. C2×#447C2semidirect
3. #2491C2semidirect
4. #2433C2semidirect
5. #2054C2semidirect
6. #2033C2semidirect
7. #2023C2semidirect
8. #2011C2semidirect
9. #2011C2semidirect
10. #1999C2semidirect
11. #1982C2semidirect
12. #1970C2semidirect
13. #1925C2semidirect
14. #1923C2semidirect
15. #1786C2semidirect
19. C22×D8C22semidirect
20. C2×C4⋊Q8C22semidirect
30. C2×#107C22semidirect
31. C2×C4⋊C8C22semidirect
38. C2×#102C22semidirect
39. C2×#102C22semidirect
40. C2×#102C22semidirect
41. C2×#102C22semidirect
44. C2×C4×C8C22semidirect
51. #390C22semidirect
52. #390C22semidirect
55. C4×C4⋊C4C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
139. #102D4semidirect
263. C22⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4862
C2×#4472
#24912
#24332
#20542
#20332
#20232
#20112
#19992
#19822
#19702
#19252
#19232
#17862