Finite Group Databasebeta

← #24539

Subgroups — #24539

Order 256.

599 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#447C211yesmaximal, characteristic
2C2×#418C211yesmaximal, characteristic, complemented
3#2500C211yesmaximal, characteristic, complemented
4D4×C22⋊C4C211yesmaximal, characteristic, complemented
5#2073C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#2054C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1851C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C22⋊C4—21no—
25#447—24no—
26C2×C4⋊C8C2211yescharacteristic, complemented
27C2×C4⋊C8C2211yescharacteristic, complemented
28#418—24no—
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic, complemented
36C2×#102C2211yescharacteristic, complemented
37C2×C22⋊C8C2211yescharacteristic
38#394—21no—
39#394—21no—
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic, complemented
43#390C2211yescharacteristic, complemented
44#388—21no—
45#388—21no—
46#386—21no—
47#386—21no—
48#384C2211yescharacteristic, complemented
49#382C2211yescharacteristic, complemented
50#380C2211yescharacteristic
51#379—21no—
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic, complemented
54#340C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
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—22no—
67C22×D4—41no—
68C22×D4—41no—
69C22×D4D414yes—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—22no—
79C2×D8—22no—
80C2×D8—41no—
81C2×D8—41no—
82C2×M16—22no—
83C2×M16—41no—
84C2×M16—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89#121—22no—
90#121—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—41no—
94C4×D4—41no—
95#117—22no—
96#117—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
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⋊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—44no—
118C2×C22⋊C4D411yescharacteristic
119C2×C22⋊C4D411yescharacteristic
120C2×C42—21no—
121C2×C42C2311yescharacteristic
122C2×C42C2311yescharacteristic
123C4⋊C8—41no—
124C4⋊C8—41no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—41no—
132#102—41no—
133C22⋊C8—22no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95D411yescharacteristic
138#95D411yescharacteristic
139C24—21no—
140C24—21no—
141C24—24no—
142C24—21no—
143C24—42no—
144C24—42no—
145C24—42no—
146C24—42no—
147C24—81no—
148C24—81no—
149C24C2×D411yescharacteristic
150C2×D4—28no—
151C2×D4—24no—
152C2×D4—21no—
153C2×D4—21no—
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—21no—
165C2×D4—21no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4C2×D412yes—
175C2×D4C2×D412yes—
176C2×D4#4112yes—
177C2×D4#4112yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D412yes—
198C22×C4C2×D411yescharacteristic
199C22×C4#4112yes—
200C22×C4C2411yescharacteristic
201D8—44no—
202D8—44no—
203M16—44no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C22⋊C4—22no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—42no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—48no—
259C23—48no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
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—84no—
278C23—84no—
279C23—84no—
280C23—84no—
281C23—84no—
282C23#12112yes—
283C23C22×D411yescharacteristic
284D4—22no—
285D4—22no—
286D4—22no—
287D4—22no—
288D4—22no—
289D4—48no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—84no—
297D4#12314yes—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4#12112yes—
333C2×C4AGL(1,ℤ/8)12yes—
334C2×C4C22×D411yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4C2×#4111yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C4H(5,𝔽2)11yescharacteristic
340C8—41no—
341C8—41no—
342C8—41no—
343C8—41no—
344C22—22no—
345C22—24no—
346C22—22no—
347C22—22no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—41no—
352C22—41no—
353C22—42no—
354C22—41no—
355C22—42no—
356C22—42no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—816no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22C2×#12111yescharacteristic
374C22#54611yescharacteristic
375C22#54611yescharacteristic
376C22C2×AGL(1,ℤ/8)11yescharacteristic
377C22#57511yescharacteristic
378C22#57611yescharacteristic
379C22#57611yescharacteristic
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4C2×#12311yescharacteristic
391C4#53411yescharacteristic
392C4#55111yescharacteristic
393C4#55111yescharacteristic
394C2—42no—
395C2—44no—
396C2—82no—
397C2—161no—
398C2#241211yescharacteristic
399C2#303611yescharacteristic
400C2#304111yescharacteristic
401C2#331811yescharacteristic
402C2#331911yescharacteristic
403C2#332211yescharacteristic
404C2#332511yescharacteristic
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
2. C2×#418C2semidirect
3. #2500C2semidirect
4. D4×C22⋊C4C2semidirect
5. #2073C2semidirect
6. #2071C2semidirect
7. #2054C2semidirect
8. #2054C2semidirect
9. #1895C2semidirect
10. #1879C2semidirect
11. #1879C2semidirect
12. #1851C2semidirect
13. #1825C2semidirect
14. #1804C2semidirect
15. #1802C2semidirect
26. C2×C4⋊C8C22semidirect
27. C2×C4⋊C8C22semidirect
35. C2×#102C22semidirect
36. C2×#102C22semidirect
42. #390C22semidirect
43. #390C22semidirect
48. #384C22semidirect
49. #382C22semidirect
53. #340C22semidirect
54. #340C22semidirect
55. #340C22semidirect
56. #340C22semidirect
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×#4472
C2×#4182
#25002
D4×C22⋊C42
#20732
#20712
#20542
#18952
#18792
#18512
#18252
#18042
#18022