Finite Group Databasebeta

← #30535

Subgroups — #30535

Order 256.

571 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic
3#2491C211yesmaximal, characteristic, complemented
4#2435C211yesmaximal, characteristic, complemented
5#2054C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#2012C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1972C211yesmaximal, characteristic, complemented
13#1926C211yesmaximal, characteristic, complemented
14#1924C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic, complemented
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic, complemented
21C2×#124—21no—
22C2×#124C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26#489—24no—
27#447—24no—
28C2×#106C2211yescharacteristic, complemented
29C2×C4⋊C8C2211yescharacteristic, complemented
30C2×#103C2211yescharacteristic, complemented
31C2×#103C2211yescharacteristic, complemented
32C2×#103C2211yescharacteristic, complemented
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic, complemented
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×#97C2211yescharacteristic, complemented
43#394—21no—
44#394—21no—
45#394—21no—
46#393—21no—
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic, complemented
50#390C2211yescharacteristic, complemented
51#388—21no—
52#380C2211yescharacteristic
53#379—21no—
54#379—21no—
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×Q8C2311yescharacteristic
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C2×SD16—41no—
77C2×SD16—41no—
78C2×SD16—41no—
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—41no—
83C2×D8—41no—
84C2×D8—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C4⋊Q8—22no—
90C4⋊Q8—41no—
91#124—22no—
92#124—22no—
93#124—41no—
94#124—41no—
95#124—44no—
96C22⋊Q8—41no—
97C22⋊Q8—41no—
98#121—22no—
99#121—41no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
122C2×C22⋊C4D411yescharacteristic
123C2×C42—21no—
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126#106—41no—
127C4⋊C8—41no—
128#103—22no—
129#103—22no—
130#103—41no—
131#102—22no—
132#102—22no—
133#102—41no—
134#102—41no—
135#102—41no—
136#102—41no—
137#102D414yes—
138C22⋊C8—22no—
139C22⋊C8—41no—
140#97—22no—
141#95—21no—
142#95—21no—
143#95D411yescharacteristic
144#95D411yescharacteristic
145C24—21no—
146C24—21no—
147C24—21no—
148C24—42no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24—81no—
154C24—81no—
155C24C2×D411yescharacteristic
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—81no—
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—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—44no—
189C2×D4—42no—
190C2×D4—81no—
191C2×D4C2×D412yes—
192C2×D4#4112yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2411yescharacteristic
210SD16—44no—
211D8—44no—
212D8—44no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8C2×D412yes—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4C2×D412yes—
235C22⋊C4—22no—
236C22⋊C4—41no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—81no—
248C22⋊C4—81no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C22⋊C4—81no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—42no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—22no—
265C23—48no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23—84no—
289C23—84no—
290C23—84no—
291C23AGL(1,ℤ/8)12yes—
292C23C22×D411yescharacteristic
293Q8—41no—
294Q8—41no—
295Q8—41no—
296Q8—41no—
297Q8—84no—
298D4—22no—
299D4—22no—
300D4—22no—
301D4—22no—
302D4—48no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—41no—
309D4—41no—
310D4—41no—
311D4—84no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
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×C4C22×D411yescharacteristic
347C2×C4C2×#4111yescharacteristic
348C2×C4H(5,𝔽2)11yescharacteristic
349C2×C4H(5,𝔽2)11yescharacteristic
350C2×C4H(5,𝔽2)11yescharacteristic
351C2×C4H(5,𝔽2)11yescharacteristic
352C8—22no—
353C8—41no—
354C8—41no—
355C8—41no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—816no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#53511yescharacteristic
386C22D4211yescharacteristic
387C22#54611yescharacteristic
388C22C2×AGL(1,ℤ/8)11yescharacteristic
389C22#57511yescharacteristic
390C22#57611yescharacteristic
391C22#57711yescharacteristic
392C4—41no—
393C4—41no—
394C4—42no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4#53511yescharacteristic
403C4#53811yescharacteristic
404C4#54611yescharacteristic
405C4H(3,𝔽4)11yescharacteristic
406C2—42no—
407C2—82no—
408C2—82no—
409C2—161no—
410C2#267011yescharacteristic
411C2#321311yescharacteristic
412C2#322411yescharacteristic
413C2#326411yescharacteristic
414C2#327511yescharacteristic
415C2#330611yescharacteristic
416C2#331911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
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×#489C2semidirect
3. #2491C2semidirect
4. #2435C2semidirect
5. #2054C2semidirect
6. #2035C2semidirect
7. #2023C2semidirect
8. #2012C2semidirect
9. #2012C2semidirect
10. #2000C2semidirect
11. #1983C2semidirect
12. #1972C2semidirect
13. #1926C2semidirect
14. #1924C2semidirect
15. #1786C2semidirect
17. C22×SD16C22semidirect
20. C2×C4⋊Q8C22semidirect
28. C2×#106C22semidirect
29. C2×C4⋊C8C22semidirect
30. C2×#103C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
39. C2×#102C22semidirect
42. C2×#97C22semidirect
49. #390C22semidirect
50. #390C22semidirect
55. C4×C4⋊C4C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4472
#24912
#24352
#20542
#20352
#20232
#20122
#20002
#19832
#19722
#19262
#19242
#17862