Finite Group Databasebeta

← #24893

Subgroups — #24893

Order 256.

616 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#447C211yesmaximal, characteristic
2C2×C4×D8C211yesmaximal, characteristic, complemented
3#2422C211yesmaximal, characteristic, complemented
4D4×C22⋊C4C211yesmaximal, characteristic, complemented
5#2093C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2054C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1934C211yesmaximal, characteristic, complemented
10#1923C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1879C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#123—21no—
20C2×#121C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic, complemented
24C22×C22⋊C4—21no—
25#447—24no—
26C4×D8—24no—
27C2×#107C2211yescharacteristic, complemented
28C2×#107C2211yescharacteristic, complemented
29C2×C4⋊C8C2211yescharacteristic, complemented
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic, complemented
37C2×#102C2211yescharacteristic, complemented
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic, complemented
41#397—21no—
42#394—21no—
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic, complemented
46#389C2211yescharacteristic, complemented
47#386—21no—
48#386—21no—
49#385—21no—
50#384C2211yescharacteristic, complemented
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#379—21no—
54C4×C22⋊C4C2211yescharacteristic
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—22no—
65C22×D4—22no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77C2×D8—22no—
78C2×D8—22no—
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—41no—
82C2×D8D414yes—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87#123—44no—
88#121—22no—
89#121—41no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—41no—
94C4×D4—41no—
95C4×D4—41no—
96C2×C4⋊C4—21no—
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—44no—
117C2×C22⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—22no—
122#107—41no—
123C4⋊C8—41no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129#102—22no—
130#102—41no—
131#102—41no—
132C22⋊C8—22no—
133C22⋊C8—41no—
134C4×C8—22no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95D411yescharacteristic
139#95D411yescharacteristic
140C24—21no—
141C24—24no—
142C24—21no—
143C24—21no—
144C24—42no—
145C24—42no—
146C24—42no—
147C24—42no—
148C24—81no—
149C24—81no—
150C24C2×D411yescharacteristic
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—42no—
166C2×D4—42no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4C2×D412yes—
175C2×D4C2×D412yes—
176C2×D4#4112yes—
177C2×D4#4112yes—
178C2×D4#4112yes—
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—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2411yescharacteristic
205D8—28no—
206D8—44no—
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—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8C2×D412yes—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—81no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
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⋊C4D814yescomplemented
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—48no—
262C23—48no—
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—41no—
278C23—41no—
279C23—84no—
280C23—84no—
281C23—84no—
282C23—84no—
283C23—84no—
284C23C2×D812yes—
285C23C22×D411yescharacteristic
286D4—22no—
287D4—22no—
288D4—22no—
289D4—22no—
290D4—22no—
291D4—22no—
292D4—22no—
293D4—48no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—84no—
299D4#12314yescomplemented
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
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—41no—
331C2×C4—41no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4C2×D812yes—
336C2×C4C2×D812yes—
337C2×C4C22×D411yescharacteristic
338C2×C4C2×#4111yescharacteristic
339C2×C4C2×#4111yescharacteristic
340C2×C4C2×#4111yescharacteristic
341C2×C4H(5,𝔽2)11yescharacteristic
342C2×C4H(5,𝔽2)11yescharacteristic
343C8—22no—
344C8—41no—
345C8—41no—
346C8—41no—
347C22—22no—
348C22—24no—
349C22—22no—
350C22—22no—
351C22—42no—
352C22—41no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—816no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22#52511yescharacteristic
377C22D4211yescharacteristic
378C22#54611yescharacteristic
379C22C22×D811yescharacteristic
380C22#57511yescharacteristic
381C22#57611yescharacteristic
382C22#57711yescharacteristic
383C4—22no—
384C4—22no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4C2×#12311yescharacteristic
394C4#53811yescharacteristic
395C4#54611yescharacteristic
396C4#55111yescharacteristic
397C2—42no—
398C2—44no—
399C2—82no—
400C2—82no—
401C2#241811yescharacteristic
402C2#310011yescharacteristic
403C2#311811yescharacteristic
404C2#326711yescharacteristic
405C2D4×D811yescharacteristic
406C2#329411yescharacteristic
407C2#332211yescharacteristic
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×C4×D8C2semidirect
3. #2422C2semidirect
4. D4×C22⋊C4C2semidirect
5. #2093C2semidirect
6. #2075C2semidirect
7. #2054C2semidirect
8. #2033C2semidirect
9. #1934C2semidirect
10. #1923C2semidirect
11. #1879C2semidirect
12. #1879C2semidirect
13. #1862C2semidirect
14. #1802C2semidirect
15. #1786C2semidirect
23. C2×C4×D4C22semidirect
27. C2×#107C22semidirect
28. C2×#107C22semidirect
29. C2×C4⋊C8C22semidirect
36. C2×#102C22semidirect
37. C2×#102C22semidirect
40. C2×C4×C8C22semidirect
45. #390C22semidirect
46. #389C22semidirect
50. #384C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #336C22semidirect
60. #336C22semidirect
245. C22⋊C4D8semidirect
299. D4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#4472
C2×C4×D82
#24222
D4×C22⋊C42
#20932
#20752
#20542
#20332
#19342
#19232
#18792
#18622
#18022
#17862