Finite Group Databasebeta

← #25116

Subgroups — #25116

Order 256.

660 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#466C211yesmaximal, characteristic, complemented
2C2×#447C211yesmaximal, characteristic, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4C2×#418C211yesmaximal, characteristic, complemented
5#2414C211yesmaximal, characteristic, complemented
6#2408C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#2034C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1972C211yesmaximal, characteristic, complemented
12#1936C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic, complemented
14#1893C211yesmaximal, characteristic, complemented
15#1854C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C23×C8C2211yescharacteristic, complemented
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic, complemented
25C2×#120—21no—
26C2×C4×D4C2211yescharacteristic
27C2×#117C2211yescharacteristic
28#466—24no—
29#447—24no—
30#435—24no—
31C2×#107C2211yescharacteristic, complemented
32C2×#106C2211yescharacteristic, complemented
33C2×C4⋊C8C2211yescharacteristic, complemented
34#418—24no—
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic, complemented
42C2×#102C2211yescharacteristic, complemented
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic, complemented
45C2×#97C2211yescharacteristic, complemented
46#394—21no—
47#394—21no—
48#392—21no—
49#392—21no—
50#390—21no—
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#390C2211yescharacteristic, complemented
54#389C2211yescharacteristic, complemented
55#388—21no—
56#380C2211yescharacteristic
57#380C2211yescharacteristic, complemented
58#379—21no—
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63C25—21no—
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—22no—
71C22×D4—21no—
72C22×D4—22no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4D411yescharacteristic
76C22×D4C2311yescharacteristic
77C22×D4C2311yescharacteristic
78C22×D4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C2×D8—22no—
82C2×D8—22no—
83C2×D8—22no—
84C2×D8—22no—
85C2×D8—41no—
86C2×D8—41no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—41no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—41no—
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97#121—22no—
98#121—22no—
99#121—22no—
100#121—22no—
101#121—22no—
102#121—41no—
103#121—41no—
104#121—41no—
105#120—44no—
106C4×D4—22no—
107C4×D4—22no—
108#117—22no—
109#117—41no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4D411yescharacteristic
131C2×C22⋊C4D411yescharacteristic
132C2×C42—21no—
133C2×C42C2311yescharacteristic
134#107—22no—
135#106—41no—
136C4⋊C8—22no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—41no—
144#102D414yescomplemented
145C22⋊C8—22no—
146C22⋊C8D414yescomplemented
147#97—22no—
148#95—21no—
149#95—21no—
150#95—21no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—42no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—42no—
159C24—81no—
160C24—81no—
161C24C2×D411yescharacteristic
162C24C2×D411yescharacteristic
163C2×D4—28no—
164C2×D4—22no—
165C2×D4—22no—
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—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—41no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—81no—
197C2×D4—81no—
198C2×D4—81no—
199C2×D4—81no—
200C2×D4—81no—
201C2×D4C2×D412yes—
202C2×D4#4112yes—
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—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D412yes—
223C22×C4#4112yes—
224C22×C4C2411yescharacteristic
225D8—44no—
226D8—44no—
227M16—44no—
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×C8—21no—
237C2×C8—44no—
238C2×C8C2×D412yes—
239C2×C8C2×D412yes—
240C2×C8C2×D412yes—
241C4⋊C4—22no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—81no—
252C4⋊C4C2×D412yes—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—81no—
265C22⋊C4—81no—
266C22⋊C4—81no—
267C22⋊C4—81no—
268C42—21no—
269C42—21no—
270C42—42no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—22no—
284C23—22no—
285C23—48no—
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—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—84no—
306C23—84no—
307C23—84no—
308C23—84no—
309C23—84no—
310C23—84no—
311C23#12112yes—
312C23AGL(1,ℤ/8)12yes—
313C23C22×D411yescharacteristic
314D4—22no—
315D4—22no—
316D4—22no—
317D4—22no—
318D4—48no—
319D4—44no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—41no—
324D4—41no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—84no—
329D4—84no—
330D4—84no—
331D4—84no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
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—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4#12112yes—
367C2×C4C22×D411yescharacteristic
368C2×C4C22×D411yescharacteristic
369C2×C4C2×#4111yescharacteristic
370C2×C4C2×#4111yescharacteristic
371C2×C4H(5,𝔽2)11yescharacteristic
372C2×C4H(5,𝔽2)11yescharacteristic
373C8—22no—
374C8—22no—
375C8—22no—
376C8—41no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—816no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22C2×#12111yescharacteristic
414C22D4211yescharacteristic
415C22#54611yescharacteristic
416C22C2×AGL(1,ℤ/8)11yescharacteristic
417C22#57511yescharacteristic
418C22#57611yescharacteristic
419C22#57611yescharacteristic
420C4—22no—
421C4—41no—
422C4—42no—
423C4—41no—
424C4—41no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4#52511yescharacteristic
430C4#53811yescharacteristic
431C4D4211yescharacteristic
432C4#54611yescharacteristic
433C2—42no—
434C2—42no—
435C2—82no—
436C2—82no—
437C2—161no—
438C2#242411yescharacteristic
439C2#305511yescharacteristic
440C2#305811yescharacteristic
441C2#326411yescharacteristic
442C2#326811yescharacteristic
443C2#331811yescharacteristic
444C2#331911yescharacteristic
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×#466C2semidirect
2. C2×#447C2semidirect
3. C2×#435C2semidirect
4. C2×#418C2semidirect
5. #2414C2semidirect
6. #2408C2semidirect
7. #2067C2semidirect
8. #2054C2semidirect
9. #2034C2semidirect
10. #1999C2semidirect
11. #1972C2semidirect
12. #1936C2semidirect
13. #1895C2semidirect
14. #1893C2semidirect
15. #1854C2semidirect
20. C23×C8C22semidirect
24. C2×#121C22semidirect
31. C2×#107C22semidirect
32. C2×#106C22semidirect
33. C2×C4⋊C8C22semidirect
41. C2×#102C22semidirect
42. C2×#102C22semidirect
44. C2×C22⋊C8C22semidirect
45. C2×#97C22semidirect
53. #390C22semidirect
54. #389C22semidirect
57. #380C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #340C22semidirect
144. #102D4semidirect
146. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4662
C2×#4472
C2×#4352
C2×#4182
#24142
#24082
#20672
#20542
#20342
#19992
#19722
#19362
#18952
#18932
#18542