Finite Group Databasebeta

← #31016

Subgroups — #31016

Order 256.

621 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#480C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3#2416C211yesmaximal, characteristic, complemented
4#2408C211yesmaximal, characteristic, complemented
5#2075C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic, complemented
7#2054C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1999C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1954C211yesmaximal, characteristic, complemented
13#1934C211yesmaximal, characteristic, complemented
14#1874C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121—21no—
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic, complemented
24C2×#120—21no—
25C2×C4×D4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic, complemented
27#480—24no—
28#447—24no—
29C2×#107C2211yescharacteristic, complemented
30C2×#107C2211yescharacteristic, complemented
31C2×C4⋊C8C2211yescharacteristic, complemented
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic, complemented
39C2×#102C2211yescharacteristic, complemented
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic, complemented
43C2×C22⋊C8C2211yescharacteristic, complemented
44C2×C4×C8C2211yescharacteristic, complemented
45#396—21no—
46#394—21no—
47#394—21no—
48#392—21no—
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#390C2211yescharacteristic, complemented
52#389C2211yescharacteristic, complemented
53#386—21no—
54#380C2211yescharacteristic
55#380C2211yescharacteristic
56#379—21no—
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61C25—21no—
62C22×D4—22no—
63C22×D4—22no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—41no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4D411yescharacteristic
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C2×D8—22no—
81C2×D8—22no—
82C2×D8—22no—
83C2×D8—41no—
84C2×D8—41no—
85C2×D8—41no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90#121—22no—
91#121—22no—
92#121—22no—
93#121—41no—
94#121—41no—
95#121—41no—
96#121—41no—
97#121—41no—
98#121—44no—
99#120—44no—
100C4×D4—41no—
101C4×D4—41no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
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—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4D411yescharacteristic
125C2×C22⋊C4D411yescharacteristic
126C2×C42C2311yescharacteristic
127#107—22no—
128#107—22no—
129C4⋊C8—41no—
130#102—22no—
131#102—22no—
132#102—41no—
133#102—41no—
134#102—41no—
135#102—41no—
136#102—41no—
137#102D414yescomplemented
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C4×C8—41no—
143#95—21no—
144#95—21no—
145#95—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—42no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24—42no—
154C24—81no—
155C24—81no—
156C24C2×D411yescharacteristic
157C24C2×D411yescharacteristic
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—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4—81no—
195C2×D4D814yescomplemented
196C2×D4C2×D412yes—
197C2×D4#4112yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—81no—
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2411yescharacteristic
223D8—44no—
224D8—44no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8C2×D412yes—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4C2×D412yes—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—81no—
262C22⋊C4—81no—
263C22⋊C4—81no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—22no—
280C23—48no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—84no—
301C23—84no—
302C23—84no—
303C23—84no—
304C23—84no—
305C23—84no—
306C23C2×D812yes—
307C23C2×D812yes—
308C23C22×D411yescharacteristic
309D4—22no—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—44no—
314D4—48no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—84no—
324D4—84no—
325D4—84no—
326D4—84no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—84no—
362C2×C4C2×D812yes—
363C2×C4C22×D411yescharacteristic
364C2×C4C2×#4111yescharacteristic
365C2×C4H(5,𝔽2)11yescharacteristic
366C2×C4H(5,𝔽2)11yescharacteristic
367C2×C4H(5,𝔽2)11yescharacteristic
368C2×C4H(5,𝔽2)11yescharacteristic
369C8—22no—
370C8—41no—
371C8—41no—
372C8—41no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—816no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22#53411yescharacteristic
410C22D4211yescharacteristic
411C22#54611yescharacteristic
412C22C22×D811yescharacteristic
413C22#57511yescharacteristic
414C22#57611yescharacteristic
415C22#57711yescharacteristic
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—42no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4#53511yescharacteristic
426C4#53811yescharacteristic
427C4#54611yescharacteristic
428C4#56011yescharacteristic
429C2—42no—
430C2—42no—
431C2—82no—
432C2—82no—
433C2—161no—
434C2#267811yescharacteristic
435C2#318611yescharacteristic
436C2#319811yescharacteristic
437C2#327511yescharacteristic
438C2D4×D811yescharacteristic
439C2#329411yescharacteristic
440C2#332211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#480C2semidirect
2. C2×#447C2semidirect
3. #2416C2semidirect
4. #2408C2semidirect
5. #2075C2semidirect
6. #2067C2semidirect
7. #2054C2semidirect
8. #2033C2semidirect
9. #1999C2semidirect
10. #1999C2semidirect
11. #1970C2semidirect
12. #1954C2semidirect
13. #1934C2semidirect
14. #1874C2semidirect
15. #1786C2semidirect
23. C2×#121C22semidirect
26. C22×C4⋊C4C22semidirect
29. C2×#107C22semidirect
30. C2×#107C22semidirect
31. C2×C4⋊C8C22semidirect
38. C2×#102C22semidirect
39. C2×#102C22semidirect
42. C2×C22⋊C8C22semidirect
43. C2×C22⋊C8C22semidirect
44. C2×C4×C8C22semidirect
51. #390C22semidirect
52. #389C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #340C22semidirect
137. #102D4semidirect
195. C2×D4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
C2×#4472
#24162
#24082
#20752
#20672
#20542
#20332
#19992
#19702
#19542
#19342
#18742
#17862