Finite Group Databasebeta

← #30992

Subgroups — #30992

Order 256.

565 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#463C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic
3#2435C211yesmaximal, characteristic, complemented
4#2420C211yesmaximal, characteristic, complemented
5#2073C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1958C211yesmaximal, characteristic, complemented
13#1923C211yesmaximal, characteristic, complemented
14#1879C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic, complemented
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic, complemented
25#463—24no—
26#447—24no—
27C2×#107C2211yescharacteristic, complemented
28C2×C4⋊C8C2211yescharacteristic, complemented
29C2×#103C2211yescharacteristic, complemented
30C2×#103C2211yescharacteristic, complemented
31C2×#103C2211yescharacteristic, complemented
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic, complemented
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic, complemented
42#397—21no—
43#394—21no—
44#394—21no—
45#394—21no—
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic, complemented
49#389C2211yescharacteristic, complemented
50#386—21no—
51#382C2211yescharacteristic, complemented
52#381—21no—
53#380C2211yescharacteristic
54#379—21no—
55#379—21no—
56C4×C22⋊C4C2211yescharacteristic
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61C25—21no—
62C22×Q8C2311yescharacteristic
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C2×SD16—22no—
77C2×SD16—41no—
78C2×SD16—41no—
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—41no—
82C2×D8—41no—
83C2×D8—41no—
84C2×D8D414yes—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89#124—22no—
90#124—22no—
91#124—41no—
92#124—41no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95#121—22no—
96#121—41no—
97C4×D4—22no—
98C4×D4—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
120C2×C22⋊C4D411yescharacteristic
121C2×C42C2311yescharacteristic
122C2×C42C2311yescharacteristic
123#107—41no—
124C4⋊C8—22no—
125#103—22no—
126#103—41no—
127#103—41no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—41no—
132#102—41no—
133#102—41no—
134#102—41no—
135C22⋊C8—22no—
136C22⋊C8—41no—
137C4×C8—41no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24—21no—
146C24—21no—
147C24—42no—
148C24—42no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—81no—
153C24—81no—
154C24C2×D411yescharacteristic
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—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—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—81no—
186C2×D4C2×D412yes—
187C2×D4C2×D412yes—
188C2×D4#4112yes—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—81no—
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2411yescharacteristic
209SD16—44no—
210D8—44no—
211D8—44no—
212C2×C8—21no—
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×C8C2×D412yes—
224C4⋊C4—21no—
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—81no—
234C22⋊C4—22no—
235C22⋊C4—41no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—81no—
244C22⋊C4—81no—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C22⋊C4—81no—
248C22⋊C4—81no—
249C22⋊C4—81no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—48no—
264C23—41no—
265C23—41no—
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—84no—
283C23—84no—
284C23—84no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23#13512yes—
289C23C22×D411yescharacteristic
290Q8—41no—
291Q8—41no—
292Q8—41no—
293Q8—41no—
294D4—22no—
295D4—22no—
296D4—22no—
297D4—22no—
298D4—22no—
299D4—22no—
300D4—48no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—84no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
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—84no—
343C2×C4C22×D411yescharacteristic
344C2×C4C2×#4111yescharacteristic
345C2×C4H(5,𝔽2)11yescharacteristic
346C2×C4H(5,𝔽2)11yescharacteristic
347C2×C4H(5,𝔽2)11yescharacteristic
348C2×C4H(5,𝔽2)11yescharacteristic
349C8—22no—
350C8—41no—
351C8—41no—
352C8—41no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—42no—
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#53511yescharacteristic
383C22D4211yescharacteristic
384C22#54611yescharacteristic
385C22C2×#13511yescharacteristic
386C22#57511yescharacteristic
387C22#57611yescharacteristic
388C22#57611yescharacteristic
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4#53411yescharacteristic
400C4#53811yescharacteristic
401C4#54611yescharacteristic
402C4#56011yescharacteristic
403C2—42no—
404C2—82no—
405C2—82no—
406C2—82no—
407C2#267811yescharacteristic
408C2#321111yescharacteristic
409C2#322011yescharacteristic
410C2#327311yescharacteristic
411C2#328011yescharacteristic
412C2#329711yescharacteristic
413C2#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×#463C2semidirect
3. #2435C2semidirect
4. #2420C2semidirect
5. #2073C2semidirect
6. #2067C2semidirect
7. #2042C2semidirect
8. #2036C2semidirect
9. #2012C2semidirect
10. #2000C2semidirect
11. #1970C2semidirect
12. #1958C2semidirect
13. #1923C2semidirect
14. #1879C2semidirect
15. #1786C2semidirect
17. C22×SD16C22semidirect
24. C2×C4×D4C22semidirect
27. C2×#107C22semidirect
28. C2×C4⋊C8C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
31. C2×#103C22semidirect
38. C2×#102C22semidirect
41. C2×C4×C8C22semidirect
48. #390C22semidirect
49. #389C22semidirect
51. #382C22semidirect
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×#4632
C2×#4472
#24352
#24202
#20732
#20672
#20422
#20362
#20122
#20002
#19702
#19582
#19232
#18792
#17862