Finite Group Databasebeta

← #24429

Subgroups — #24429

Order 256.

754 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#448C211yesmaximal, characteristic, complemented
4C2×#448C211yesmaximal, characteristic, complemented
5C22×#103C211yesmaximal, characteristic, complemented
6#2416C211yesmaximal, characteristic
7#2043C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2013C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1872C211yesmaximal, characteristic, complemented
13#1869C211yesmaximal, characteristic, complemented
14#1865C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C23×C8C2211yescharacteristic
22D4×Q8—24no—
23D4×Q8—24no—
24C2×C4⋊Q8—21no—
25C2×C4⋊Q8C2211yescharacteristic
26C2×C22⋊Q8—21no—
27C2×C22⋊Q8—21no—
28C2×C22⋊Q8C2211yescharacteristic
29C2×C22⋊Q8C2211yescharacteristic
30C2×#121—21no—
31C2×#121C2211yescharacteristic
32C2×#121C2211yescharacteristic
33C2×C4×Q8C2211yescharacteristic
34C2×C4×D4—21no—
35C2×C4×D4C2211yescharacteristic
36C22×C4⋊C4C2211yescharacteristic
37#465—24no—
38#448—24no—
39#448—24no—
40C2×#106C2211yescharacteristic
41C2×#103—24no—
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#103C2211yescharacteristic
47C2×#103C2211yescharacteristic
48C2×#103C2211yescharacteristic
49C2×#103C2211yescharacteristic
50C2×#103C2211yescharacteristic
51C2×#102C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic
54C2×C22⋊C8C2211yescharacteristic
55#396—21no—
56#394—21no—
57#390C2211yescharacteristic
58#389C2211yescharacteristic
59#386—21no—
60#380C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63#336C2211yescharacteristic
64#336C2211yescharacteristic
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—21no—
68C22×Q8—21no—
69C22×Q8—22no—
70C22×Q8—22no—
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8D414yescomplemented
79C22×Q8C2311yescharacteristic
80C22×Q8C2311yescharacteristic
81C22×Q8C2311yescharacteristic
82C22×D4—21no—
83C22×D4—21no—
84C22×D4D411yescharacteristic
85C22×D4C2311yescharacteristic
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C2×SD16—22no—
92C2×SD16—22no—
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—41no—
98C2×SD16—41no—
99C2×SD16—41no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—41no—
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107C4⋊Q8—28no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C22⋊Q8—28no—
111C22⋊Q8—28no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C22⋊Q8—22no—
115C22⋊Q8D414yescomplemented
116#121—22no—
117#121—22no—
118#121—22no—
119#121—41no—
120#121—44no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×D4—28no—
124C4×D4—22no—
125C4×D4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4D411yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4D411yescharacteristic
148C2×C22⋊C4D411yescharacteristic
149C2×C42—21no—
150C2×C42C2311yescharacteristic
151#106—22no—
152#103—22no—
153#103—22no—
154#103—22no—
155#103—22no—
156#103—22no—
157#103—22no—
158#103—22no—
159#103—41no—
160#103D414yescomplemented
161#102—22no—
162C22⋊C8—22no—
163C22⋊C8—22no—
164C22⋊C8—41no—
165#95—21no—
166#95—21no—
167C24—21no—
168C24C2×D411yescharacteristic
169C24C2×D411yescharacteristic
170C2×Q8—24no—
171C2×Q8—24no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—28no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—44no—
191C2×Q8—44no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×Q8—44no—
196C2×Q8—44no—
197C2×Q8C2×D412yes—
198C2×Q8C2×D412yes—
199C2×Q8C2×D412yes—
200C2×Q8C2×D412yes—
201C2×Q8#4112yes—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—81no—
216C2×D4SD1614yescomplemented
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—81no—
243C22×C4C2×D411yescharacteristic
244C22×C4C2×D411yescharacteristic
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D412yes—
247C22×C4C2×D412yes—
248C22×C4C2×D411yescharacteristic
249C22×C4C2411yescharacteristic
250SD16—44no—
251SD16—44no—
252SD16—44no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—44no—
265C2×C8C2×D412yes—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4C2×D412yes—
281C4⋊C4C2×D412yes—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—81no—
291C42—24no—
292C42—21no—
293C42—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23#12012yes—
312C23C2×SD1612yes—
313C23C2×SD1612yes—
314C23#13712yes—
315C23C22×D411yescharacteristic
316Q8—22no—
317Q8—22no—
318Q8—22no—
319Q8—22no—
320Q8—22no—
321Q8—22no—
322Q8—22no—
323Q8—48no—
324Q8—48no—
325Q8—41no—
326Q8—41no—
327Q8—41no—
328Q8—41no—
329Q8#12114yescomplemented
330D4—28no—
331D4—41no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—84no—
336D4—84no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—84no—
379C2×C4#12012yes—
380C2×C4C2×SD1612yes—
381C2×C4C22×D411yescharacteristic
382C2×C4C22×D411yescharacteristic
383C2×C4C22×D411yescharacteristic
384C2×C4C2×#4111yescharacteristic
385C2×C4H(5,𝔽2)11yescharacteristic
386C2×C4H(5,𝔽2)11yescharacteristic
387C8—22no—
388C8—41no—
389C8—41no—
390C8—41no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22#44814yescomplemented
412C22C2×#12011yescharacteristic
413C22D4211yescharacteristic
414C22#54611yescharacteristic
415C22C22×SD1611yescharacteristic
416C22C2×#13711yescharacteristic
417C22#57711yescharacteristic
418C22#57811yescharacteristic
419C4—22no—
420C4—24no—
421C4—22no—
422C4—41no—
423C4—42no—
424C4—42no—
425C4—41no—
426C4—42no—
427C4—42no—
428C4—81no—
429C4—81no—
430C4C2×#12111yescharacteristic
431C4#53411yescharacteristic
432C4D4211yescharacteristic
433C4#54611yescharacteristic
434C2—24no—
435C2—42no—
436C2—161no—
437C2#241011yescharacteristic
438C2C2×#44811yescharacteristic
439C2#301311yescharacteristic
440C2#327011yescharacteristic
441C2D4×SD1611yescharacteristic
442C2#329811yescharacteristic
443C2#332011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2416C2
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×D4×Q8C2semidirect
2. C2×#465C2semidirect
3. C2×#448C2semidirect
4. C2×#448C2semidirect
5. C22×#103C2semidirect
7. #2043C2semidirect
8. #2035C2semidirect
9. #2013C2semidirect
10. #2000C2semidirect
11. #1880C2semidirect
12. #1872C2semidirect
13. #1869C2semidirect
14. #1865C2semidirect
15. #1787C2semidirect
78. C22×Q8D4semidirect
115. C22⋊Q8D4semidirect
160. #103D4semidirect
216. C2×D4SD16semidirect
329. Q8#121semidirect
411. C22#448semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#4652
C2×#4482
C22×#1032
#24162
#20432
#20352
#20132
#20002
#18802
#18722
#18692
#18652
#17872