Finite Group Databasebeta

← #22870

Subgroups — #22870

Order 256.

699 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×#447C211yesmaximal, characteristic
2C2×C4×D8C211yesmaximal, characteristic, complemented
3C22×#102C211yesmaximal, characteristic, complemented
4#2361C211yesmaximal, characteristic, complemented
5D4×C22⋊C4C211yesmaximal, characteristic, complemented
6#1970C211yesmaximal, characteristic, complemented
7#1934C211yesmaximal, characteristic, complemented
8#1925C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1852C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15C8×C22⋊C4C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic, complemented
25C22×C4⋊C4C2211yescharacteristic
26C22×C22⋊C4—21no—
27#447C418yescomplemented
28C4×D8—24no—
29C2×#107C2211yescharacteristic, complemented
30C2×#107C2211yescharacteristic, complemented
31C2×#102—24no—
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic, complemented
40C2×C22⋊C8C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic, complemented
42C2×C4×C8C2211yescharacteristic, complemented
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#390C2211yescharacteristic, complemented
46#389C2211yescharacteristic, complemented
47#388—21no—
48#386—21no—
49#386—21no—
50#386—21no—
51#385—21no—
52#380C2211yescharacteristic
53#379—21no—
54C4×C4⋊C4C2211yescharacteristic, complemented
55C4×C22⋊C4C2211yescharacteristic
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#336C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62C25—21no—
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—22no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4C2×C414yescomplemented
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—41no—
82C2×D8—41no—
83C2×D8C2×C414yescomplemented
84C2×D8C2×C414yescomplemented
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—41no—
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#121—22no—
93#121C2×C414yescomplemented
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—44no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
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⋊C4—44no—
122C2×C22⋊C4D411yescharacteristic
123C2×C22⋊C4D411yescharacteristic
124C2×C42—21no—
125C2×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127#107—22no—
128#107—41no—
129#102—22no—
130#102—22no—
131#102—22no—
132#102—22no—
133#102—41no—
134#102C2×C414yescomplemented
135#102C2×C414yescomplemented
136#102D414yescomplemented
137C22⋊C8C2×C414yescomplemented
138C4×C8—22no—
139C4×C8—41no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95D411yescharacteristic
144#95D411yescharacteristic
145C24—21no—
146C24—21no—
147C24—21no—
148C24—24no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24—81no—
154C24—81no—
155C24C2×D411yescharacteristic
156C2×D4—28no—
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—21no—
160C2×D4—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—44no—
173C2×D4—44no—
174C2×D4—44no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4C22×C412yes—
180C2×D4C22×C412yes—
181C2×D4C22×C412yes—
182C2×D4C2×D412yes—
183C2×D4#4112yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4C22×C412yes—
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4#4112yes—
212C22×C4C2411yescharacteristic
213D8—28no—
214D8—44no—
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×C8—44no—
225C2×C8C22×C412yes—
226C2×C8C22×C412yes—
227C2×C8C2×D412yes—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4C22×C412yes—
239C4⋊C4C2×D412yes—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C22⋊C4—81no—
252C22⋊C4D814yescomplemented
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—42no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—48no—
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—48no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23—84no—
289C23—84no—
290C23C4×D412yes—
291C23C2×D812yes—
292C23#13512yes—
293C23C22×D411yescharacteristic
294D4—22no—
295D4—22no—
296D4—22no—
297D4—22no—
298D4—22no—
299D4—22no—
300D4—22no—
301D4—48no—
302D4—48no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4C4×D414yescomplemented
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
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—84no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4C4×D412yes—
344C2×C4C2×D812yes—
345C2×C4C2×D812yes—
346C2×C4C23×C411yescharacteristic
347C2×C4C22×D411yescharacteristic
348C2×C4C2×#4111yescharacteristic
349C2×C4C2×#4111yescharacteristic
350C2×C4H(5,𝔽2)11yescharacteristic
351C2×C4H(5,𝔽2)11yescharacteristic
352C8—22no—
353C8—22no—
354C8—22no—
355C8—41no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—24no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—816no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22C4×D814yescomplemented
385C22C2×C4×D411yescharacteristic
386C22D4211yescharacteristic
387C22#54611yescharacteristic
388C22C22×D811yescharacteristic
389C22C2×#13511yescharacteristic
390C22#57611yescharacteristic
391C22#57711yescharacteristic
392C4—22no—
393C4—22no—
394C4—22no—
395C4—42no—
396C4—42no—
397C4—41no—
398C4—41no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4C2×C4×D411yescharacteristic
403C4#51811yescharacteristic
404C4#54611yescharacteristic
405C4#55111yescharacteristic
406C2—24no—
407C2—44no—
408C2—82no—
409C2—82no—
410C2#235811yescharacteristic
411C2C2×C4×D811yescharacteristic
412C2#294611yescharacteristic
413C2D4×D811yescharacteristic
414C2#328211yescharacteristic
415C2#329411yescharacteristic
416C2#329711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C2×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. C22×#102C2semidirect
4. #2361C2semidirect
5. D4×C22⋊C4C2semidirect
6. #1970C2semidirect
7. #1934C2semidirect
8. #1925C2semidirect
9. #1893C2semidirect
10. #1879C2semidirect
11. #1879C2semidirect
12. #1862C2semidirect
13. #1852C2semidirect
14. C4×#102C2semidirect
15. C8×C22⋊C4C2semidirect
24. C2×C4×D4C22semidirect
27. #447C4semidirect
29. C2×#107C22semidirect
30. C2×#107C22semidirect
38. C2×#102C22semidirect
39. C2×#102C22semidirect
41. C2×C4×C8C22semidirect
42. C2×C4×C8C22semidirect
45. #390C22semidirect
46. #389C22semidirect
54. C4×C4⋊C4C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #336C22semidirect
61. #336C22semidirect
70. C22×D4C2×C4semidirect
83. C2×D8C2×C4semidirect
84. C2×D8C2×C4semidirect
93. #121C2×C4semidirect
134. #102C2×C4semidirect
135. #102C2×C4semidirect
136. #102D4semidirect
137. C22⋊C8C2×C4semidirect
252. C22⋊C4D8semidirect
307. D4C4×D4semidirect
384. C22C4×D8semidirect

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
C22×#1022
#23612
D4×C22⋊C42
#19702
#19342
#19252
#18932
#18792
#18622
#18522
C4×#1022
C8×C22⋊C42