Finite Group Databasebeta

← #25101

Subgroups — #25101

Order 256.

624 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic
4C2×#418C211yesmaximal, characteristic, complemented
5#2435C211yesmaximal, characteristic, complemented
6#2363C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2054C211yesmaximal, characteristic, complemented
9#2042C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1873C211yesmaximal, characteristic, complemented
15#1851C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic, complemented
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic, complemented
26C2×#117C2211yescharacteristic
27#468—24no—
28#463—24no—
29#447—24no—
30C2×#107C2211yescharacteristic, complemented
31C2×C4⋊C8C2211yescharacteristic, complemented
32C2×C4⋊C8C2211yescharacteristic, complemented
33#418—24no—
34C2×#103C2211yescharacteristic, complemented
35C2×#103C2211yescharacteristic, complemented
36C2×#103C2211yescharacteristic, complemented
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic, complemented
44C2×C22⋊C8C2211yescharacteristic
45#394—21no—
46#394—21no—
47#390—21no—
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#390C2211yescharacteristic, complemented
51#389C2211yescharacteristic, complemented
52#388—21no—
53#386—21no—
54#379—21no—
55#379—21no—
56C4×C4⋊C4C2211yescharacteristic, complemented
57C4×C22⋊C4—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×Q8C2311yescharacteristic
65C22×D4—22no—
66C22×D4—22no—
67C22×D4—21no—
68C22×D4—22no—
69C22×D4—21no—
70C22×D4—22no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C23×C4—21no—
77C23×C4C2311yescharacteristic
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—41no—
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×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94#124—22no—
95#124—22no—
96#124—41no—
97#124—41no—
98C22⋊Q8—22no—
99C22⋊Q8—22no—
100#121—22no—
101#121D414yes—
102C4×D4—22no—
103C4×D4—41no—
104#117—22no—
105#117—41no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
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⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4D411yescharacteristic
127C2×C22⋊C4D411yescharacteristic
128C2×C42—21no—
129C2×C42—21no—
130C2×C42C2311yescharacteristic
131C2×C42C2311yescharacteristic
132#107—22no—
133C4⋊C8—22no—
134C4⋊C8—41no—
135#103—22no—
136#103—22no—
137#103—41no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—22no—
143#102—22no—
144#102D414yescomplemented
145C22⋊C8—22no—
146#95—21no—
147#95—21no—
148#95D411yescharacteristic
149#95D411yescharacteristic
150C24—21no—
151C24—21no—
152C24—21no—
153C24—42no—
154C24—42no—
155C24—42no—
156C24—42no—
157C24—42no—
158C24—81no—
159C24—81no—
160C24C2×D411yescharacteristic
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×D4—28no—
168C2×D4—24no—
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—21no—
183C2×D4—21no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—44no—
189C2×D4—42no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4C2×D412yes—
193C2×D4C2×D412yes—
194C2×D4#4112yes—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—81no—
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D412yes—
213C22×C4#4112yes—
214C22×C4C2411yescharacteristic
215SD16—44no—
216D8—44no—
217D8—44no—
218M16—44no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8C2×D412yes—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—42no—
238C4⋊C4—81no—
239C4⋊C4C2×D412yes—
240C4⋊C4C2×D412yes—
241C22⋊C4—24no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—81no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C22⋊C4—81no—
256C22⋊C4—81no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—42no—
262C42—81no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—48no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—84no—
292C23—84no—
293C23—84no—
294C23—84no—
295C23—84no—
296C23—84no—
297C23#12112yes—
298C23C22×D411yescharacteristic
299Q8—41no—
300Q8—41no—
301Q8—41no—
302Q8—41no—
303D4—22no—
304D4—22no—
305D4—22no—
306D4—22no—
307D4—22no—
308D4—22no—
309D4—48no—
310D4—41no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—84no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
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—84no—
351C2×C4#12112yes—
352C2×C4C22×D411yescharacteristic
353C2×C4C22×D411yescharacteristic
354C2×C4C2×#4111yescharacteristic
355C2×C4C2×#4111yescharacteristic
356C2×C4H(5,𝔽2)11yescharacteristic
357C2×C4H(5,𝔽2)11yescharacteristic
358C8—22no—
359C8—41no—
360C8—41no—
361C8—41no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—816no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22C2×#12111yescharacteristic
392C22D4211yescharacteristic
393C22#54611yescharacteristic
394C22#57511yescharacteristic
395C22#57511yescharacteristic
396C22#57611yescharacteristic
397C22#57611yescharacteristic
398C4—22no—
399C4—41no—
400C4—41no—
401C4—42no—
402C4—42no—
403C4—41no—
404C4—41no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4#52511yescharacteristic
409C4#53811yescharacteristic
410C4D4211yescharacteristic
411C4#54611yescharacteristic
412C2—42no—
413C2—82no—
414C2—82no—
415C2—82no—
416C2#242411yescharacteristic
417C2#304011yescharacteristic
418C2#304111yescharacteristic
419C2#326811yescharacteristic
420C2#327311yescharacteristic
421C2#331911yescharacteristic
422C2#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
1. C2×#468C2semidirect
2. C2×#463C2semidirect
4. C2×#418C2semidirect
5. #2435C2semidirect
6. #2363C2semidirect
7. #2067C2semidirect
8. #2054C2semidirect
9. #2042C2semidirect
10. #2012C2semidirect
11. #1983C2semidirect
12. #1925C2semidirect
13. #1879C2semidirect
14. #1873C2semidirect
15. #1851C2semidirect
17. C22×SD16C22semidirect
25. C2×C4×D4C22semidirect
30. C2×#107C22semidirect
31. C2×C4⋊C8C22semidirect
32. C2×C4⋊C8C22semidirect
34. C2×#103C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
43. C2×#102C22semidirect
50. #390C22semidirect
51. #389C22semidirect
56. C4×C4⋊C4C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect
144. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×#4632
C2×#4472
C2×#4182
#24352
#23632
#20672
#20542
#20422
#20122
#19832
#19252
#18792
#18732
#18512