Finite Group Databasebeta

← #15264

Subgroups — #15264

Order 256.

1309 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124C211yesmaximal, characteristic, complemented
2C22×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#394C211yesmaximal, characteristic, complemented
4C2×#392C211yesmaximal, characteristic
5C2×#386C212yesmaximal, complemented
6#1446C211yesmaximal, characteristic, complemented
7#1439C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1438C212yesmaximal, complemented
10#1437C211yesmaximal, characteristic, complemented
11#1436C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C23×D4C2212yes—
15C24×C4C2211yescharacteristic
16C2×#124—24no—
17C2×#124—28no—
18C2×C22⋊Q8—24no—
19C2×C22⋊Q8—24no—
20C2×C22⋊Q8—24no—
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2212yes—
28C22×C22⋊C4C2212yes—
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2212yes—
31C22×C22⋊C4C2212yes—
32C22×C42C2211yescharacteristic
33#394—24no—
34#392—24no—
35#386—28no—
36C2×#95C2211yescharacteristic
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C2×#95C2212yes—
42C2×#95C2212yes—
43C25C2312yes—
44C25C2311yescharacteristic
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—24no—
48C22×Q8—42no—
49C22×Q8D414yescomplemented
50C22×D4—24no—
51C22×D4—24no—
52C22×D4—24no—
53C22×D4—24no—
54C22×D4—24no—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4—22no—
58C22×D4—24no—
59C22×D4—42no—
60C22×D4—42no—
61C22×D4—42no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—24no—
66C23×C4—42no—
67C23×C4C2312yes—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2312yes—
72C23×C4C2311yescharacteristic
73C23×C4C2312yes—
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76#124—232no—
77C22⋊Q8—232no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4D414yescomplemented
87C2×C22⋊C4—22no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4D414yes—
110C2×C22⋊C4D418yescomplemented
111C2×C42—24no—
112C2×C42—22no—
113#95—24no—
114#95—24no—
115#95—22no—
116#95—22no—
117#95—24no—
118#95—42no—
119#95—42no—
120C24—21no—
121C24—21no—
122C24—22no—
123C24—22no—
124C24—22no—
125C24—22no—
126C24—22no—
127C24—22no—
128C24—22no—
129C24—22no—
130C24—22no—
131C24—22no—
132C24—22no—
133C24—22no—
134C24—22no—
135C24—22no—
136C24—22no—
137C24—22no—
138C24—22no—
139C24—22no—
140C24—22no—
141C24C2×D414yes—
142C24C2×D412yes—
143C24#4114yes—
144C24C2411yescharacteristic
145C2×Q8—28no—
146C2×Q8—28no—
147C2×Q8—44no—
148C2×Q8—48no—
149C2×Q8—48no—
150C2×D4—28no—
151C2×D4—216no—
152C2×D4—48no—
153C2×D4—48no—
154C2×D4—48no—
155C2×D4—48no—
156C2×D4—44no—
157C2×D4—48no—
158C2×D4—48no—
159C2×D4—48no—
160C2×D4—48no—
161C2×D4—44no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—28no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—48no—
204C22×C4—48no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4C2×D414yes—
208C22×C4C2×D412yes—
209C22×C4C2×D412yes—
210C22×C4C2×D412yes—
211C22×C4C2×D412yes—
212C22×C4C2×D414yes—
213C22×C4C2×D412yes—
214C22×C4C2×D412yes—
215C22×C4C2×D412yes—
216C22×C4C2×D412yes—
217C22×C4#4114yes—
218C4⋊C4—28no—
219C4⋊C4—28no—
220C4⋊C4—28no—
221C22⋊C4—28no—
222C22⋊C4—216no—
223C22⋊C4—28no—
224C22⋊C4—216no—
225C22⋊C4—48no—
226C22⋊C4—48no—
227C22⋊C4—48no—
228C22⋊C4—48no—
229C42—28no—
230C23—22no—
231C23—24no—
232C23—22no—
233C23—24no—
234C23—24no—
235C23—24no—
236C23—24no—
237C23—24no—
238C23—24no—
239C23—24no—
240C23—24no—
241C23—24no—
242C23—24no—
243C23—24no—
244C23—24no—
245C23—24no—
246C23—24no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23#12414yescomplemented
286C23C22×D411yescharacteristic
287C23C22×D412yes—
288C23C22×D411yescharacteristic
289C23C22×D411yescharacteristic
290C23C2×#4112yes—
291C23C2×#4112yes—
292C23H(5,𝔽2)12yes—
293C23H(5,𝔽2)12yes—
294C232−1+412yes—
295Q8—416no—
296D4—416no—
297D4—432no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—416no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4#12014yes—
349C2×C4#12014yes—
350C22—24no—
351C22—28no—
352C22—42no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—44no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—44no—
364C22—44no—
365C22—44no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22C2×#12011yescharacteristic
375C22C2×#12011yescharacteristic
376C22#52512yes—
377C22C2×#12411yescharacteristic
378C22#53211yescharacteristic
379C22#53411yescharacteristic
380C22#53511yescharacteristic
381C22#53611yescharacteristic
382C22#53712yes—
383C22#53812yes—
384C22#54212yes—
385C22D4212yes—
386C22D4212yes—
387C22#54612yes—
388C22#54612yes—
389C22#54712yes—
390C22#54812yes—
391C22#54812yes—
392C22#55012yes—
393C22#55112yes—
394C22#56212yes—
395C4—24no—
396C4—24no—
397C4—42no—
398C4—42no—
399C4—44no—
400C4—42no—
401C4—42no—
402C4—44no—
403C4—82no—
404C2—44no—
405C2—84no—
406C2#240311yescharacteristic
407C2#240712yes—
408C2#243111yescharacteristic
409C2#243611yescharacteristic
410C2#246112yes—
411C2#247212yes—
412C2#267212yes—
413C2#267612yes—
414C2#273012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC2×#392C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#124C2semidirect
2. C22×C22⋊Q8C2semidirect
3. C2×#394C2semidirect
5. C2×#386C2semidirect
6. #1446C2semidirect
7. #1439C2semidirect
8. #1439C2semidirect
9. #1438C2semidirect
10. #1437C2semidirect
11. #1436C2semidirect
49. C22×Q8D4semidirect
86. C2×C4⋊C4D4semidirect
110. C2×C22⋊C4D4semidirect
285. C23#124semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C22×C22⋊Q82
C2×#3942
C2×#3922
C2×#3862
#14462
#14392
#14382
#14372
#14362