Finite Group Databasebeta

← #22052

Subgroups — #22052

Order 256.

540 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#450C211yesmaximal, characteristic
2#2323C211yesmaximal, characteristic, complemented
3#2309C211yesmaximal, characteristic
4#1973C211yesmaximal, characteristic, complemented
5#1968C211yesmaximal, characteristic, complemented
6#1936C211yesmaximal, characteristic, complemented
7#1934C211yesmaximal, characteristic, complemented
8#1926C211yesmaximal, characteristic, complemented
9#1925C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1873C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1786C211yesmaximal, characteristic
15#1750C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic, complemented
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23#450C418yescomplemented
24C2×#107C2211yescharacteristic, complemented
25C2×#106C2211yescharacteristic, complemented
26C2×#103C2211yescharacteristic, complemented
27C2×#103C2211yescharacteristic, complemented
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×#97C2211yescharacteristic, complemented
37C2×C4×C8C2211yescharacteristic, complemented
38#391C2211yescharacteristic, complemented
39#390—21no—
40#390—21no—
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389—21no—
46#389C2211yescharacteristic, complemented
47#388—21no—
48#388—21no—
49#387—21no—
50#380C2211yescharacteristic
51#379—21no—
52#379—21no—
53C4×C4⋊C4C2211yescharacteristic, complemented
54#340C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#336C2211yescharacteristic, complemented
59#336C2211yescharacteristic, complemented
60C25—21no—
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4—22no—
64C22×D4—41no—
65C22×D4—41no—
66C22×D4—41no—
67C22×D4—41no—
68C22×D4C2×C414yes—
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C2×SD16—41no—
73C2×SD16—41no—
74C2×SD16—41no—
75C2×SD16—41no—
76C2×SD16C2×C414yes—
77C2×SD16C2×C414yes—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22⋊Q8—22no—
83C22⋊Q8C2×C414yes—
84C4×Q8—41no—
85C4×Q8—41no—
86#117—41no—
87#117—41no—
88#117—41no—
89#117—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
112C2×C22⋊C4D411yescharacteristic
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118#107—41no—
119#106—41no—
120#103—41no—
121#103—41no—
122#102—41no—
123#102—41no—
124#102—41no—
125#102—41no—
126#102C2×C414yes—
127#102C2×C414yes—
128C22⋊C8—22no—
129C22⋊C8C2×C414yes—
130#97—41no—
131C4×C8—41no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137C24—21no—
138C24—21no—
139C24—24no—
140C24—42no—
141C24—42no—
142C24—81no—
143C24—81no—
144C24—81no—
145C24—81no—
146C24C2×D411yescharacteristic
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8C22×C412yes—
153C2×D4—24no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—44no—
165C2×D4—44no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4C22×C412yes—
171C2×D4C22×C412yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4C22×C412yes—
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4#4112yes—
189C22×C4C2411yescharacteristic
190SD16—44no—
191SD16—44no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8C22×C412yes—
203C2×C8C22×C412yes—
204C4⋊C4—24no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—81no—
218C4⋊C4C22×C412yes—
219C22⋊C4—24no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—42no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—42no—
237C42—42no—
238C42—81no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—48no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—84no—
261C23—84no—
262C23—84no—
263C23—84no—
264C23—84no—
265C23—84no—
266C23C4×D412yes—
267C23C22×D411yescharacteristic
268Q8—22no—
269Q8—22no—
270Q8—41no—
271Q8—41no—
272D4—22no—
273D4—22no—
274D4—22no—
275D4—22no—
276D4—48no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4C4×D412yes—
318C2×C4C23×C411yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C2×C4H(5,𝔽2)11yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C8—22no—
325C8—22no—
326C8—41no—
327C8—41no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—816no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C2×C4×D411yescharacteristic
351C22#53411yescharacteristic
352C22#53511yescharacteristic
353C22#57611yescharacteristic
354C22#57711yescharacteristic
355C22#57711yescharacteristic
356C22#57711yescharacteristic
357C4—22no—
358C4—42no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—42no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4#51811yescharacteristic
369C4#51811yescharacteristic
370C4#53811yescharacteristic
371C4#53811yescharacteristic
372C2—42no—
373C2—82no—
374C2—82no—
375C2#232111yescharacteristic
376C2#294611yescharacteristic
377C2#294911yescharacteristic
378C2#319811yescharacteristic
379C2#320011yescharacteristic
380C2#322211yescharacteristic
381C2#322411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2323C2semidirect
4. #1973C2semidirect
5. #1968C2semidirect
6. #1936C2semidirect
7. #1934C2semidirect
8. #1926C2semidirect
9. #1925C2semidirect
10. #1880C2semidirect
11. #1880C2semidirect
12. #1873C2semidirect
13. #1854C2semidirect
15. #1750C2semidirect
20. C2×C4×Q8C22semidirect
23. #450C4semidirect
24. C2×#107C22semidirect
25. C2×#106C22semidirect
26. C2×#103C22semidirect
27. C2×#103C22semidirect
36. C2×#97C22semidirect
37. C2×C4×C8C22semidirect
38. #391C22semidirect
46. #389C22semidirect
53. C4×C4⋊C4C22semidirect
54. #340C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #336C22semidirect
59. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4502
#23232
#23092
#19732
#19682
#19362
#19342
#19262
#19252
#18802
#18732
#18542
#17862
#17502