Finite Group Databasebeta

← #14019

Subgroups — #14019

Order 256.

1269 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#388C212yesmaximal, complemented
3C2×#386C211yesmaximal, characteristic
4C2×#386C211yesmaximal, characteristic
5C2×#386C211yesmaximal, characteristic
6C2×C4×C22⋊C4C211yesmaximal, characteristic, complemented
7#1442C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1436C212yesmaximal, complemented
10C22⋊#95C211yesmaximal, characteristic, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C24×C4C2211yescharacteristic
16C22×C4⋊C4—24no—
17C22×C4⋊C4—28no—
18C22×C4⋊C4C2212yes—
19C22×C4⋊C4C2212yes—
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C42C2212yes—
28#388—28no—
29#386—24no—
30#386—24no—
31#386C418yescomplemented
32C4×C22⋊C4—24no—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C2×#95C2212yes—
41C2×#95C2212yes—
42C25C2311yescharacteristic
43C25C2311yescharacteristic
44C22×D4—22no—
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4C2×C414yes—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—24no—
54C23×C4—24no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—24no—
59C23×C4—24no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4C2×C414yes—
65C23×C4D414yes—
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2312yes—
69C23×C4C2311yescharacteristic
70C23×C4C2312yes—
71C23×C4C2311yescharacteristic
72C23×C4C2312yes—
73C23×C4C2312yes—
74C23×C4C2311yescharacteristic
75C2×C4⋊C4—232no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4D418yescomplemented
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4C2×C414yes—
99C2×C22⋊C4C2×C418yescomplemented
100C2×C22⋊C4D414yes—
101C2×C42—24no—
102C2×C42—24no—
103C2×C42—42no—
104#95—24no—
105#95—24no—
106#95—22no—
107#95—22no—
108#95—24no—
109#95—24no—
110#95—24no—
111#95—42no—
112#95C2×C418yescomplemented
113C24—22no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—21no—
119C24—21no—
120C24—22no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—22no—
126C24—22no—
127C24C22×C412yes—
128C24C22×C412yes—
129C24C2×D412yes—
130C24C2×D412yes—
131C24#4112yes—
132C24#4112yes—
133C24#4112yes—
134C24#4112yes—
135C24C2411yescharacteristic
136C2×D4—28no—
137C2×D4—28no—
138C2×D4—28no—
139C2×D4—44no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4—44no—
143C22×C4—28no—
144C22×C4—216no—
145C22×C4—28no—
146C22×C4—28no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—216no—
155C22×C4—28no—
156C22×C4—28no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—28no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—44no—
197C22×C4—48no—
198C22×C4—48no—
199C22×C4—48no—
200C22×C4—48no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4C22×C412yes—
204C22×C4C22×C414yes—
205C22×C4C22×C414yes—
206C22×C4C2×D414yes—
207C22×C4C2×D412yes—
208C22×C4C2×D414yes—
209C22×C4C2×D414yes—
210C22×C4C2×D412yes—
211C22×C4C2×D414yes—
212C22×C4#4114yes—
213C4⋊C4—216no—
214C4⋊C4—216no—
215C22⋊C4—28no—
216C22⋊C4—28no—
217C22⋊C4—216no—
218C22⋊C4—48no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C42—48no—
223C23—24no—
224C23—24no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—22no—
229C23—22no—
230C23—24no—
231C23—24no—
232C23—24no—
233C23—24no—
234C23—24no—
235C23—24no—
236C23—24no—
237C23—24no—
238C23—24no—
239C23—22no—
240C23—24no—
241C23—22no—
242C23—22no—
243C23—24no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23#11714yes—
263C23C4×D414yes—
264C23C4×D414yes—
265C23#12114yes—
266C23#12314yes—
267C23#12414yes—
268C23#12614yes—
269C23C23×C411yescharacteristic
270C23C22×D411yescharacteristic
271C23C22×D412yes—
272C23C22×D411yescharacteristic
273C23C2×#4111yescharacteristic
274C23C2×#4111yescharacteristic
275C23C2×#4111yescharacteristic
276C23C2×#4112yes—
277C23C2×#4111yescharacteristic
278C23H(5,𝔽2)12yes—
279C232−1+412yes—
280D4—416no—
281C2×C4—232no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—416no—
308C2×C4—416no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4C4×D418yes—
338C2×C4#12014yes—
339C22—24no—
340C22—24no—
341C22—28no—
342C22—28no—
343C22—24no—
344C22—24no—
345C22—24no—
346C22—28no—
347C22—28no—
348C22—24no—
349C22—42no—
350C22—44no—
351C22—44no—
352C22—42no—
353C22—42no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—82no—
359C22—82no—
360C22#38818yescomplemented
361C22C2×#11711yescharacteristic
362C22C2×C4×D411yescharacteristic
363C22C2×C4×D412yes—
364C22C2×C4×D411yescharacteristic
365C22#52012yes—
366C22C2×#12011yescharacteristic
367C22C2×#12111yescharacteristic
368C22C2×#12311yescharacteristic
369C22#52511yescharacteristic
370C22#52512yes—
371C22C2×#12411yescharacteristic
372C22C2×#12611yescharacteristic
373C22#53411yescharacteristic
374C22#53611yescharacteristic
375C22#53912yes—
376C22D4212yes—
377C22#54612yes—
378C22#54712yes—
379C22#54812yes—
380C22#54812yes—
381C22#55212yes—
382C22#55312yes—
383C22#55312yes—
384C4—24no—
385C4—28no—
386C4—44no—
387C4—42no—
388C4—44no—
389C4—44no—
390C4—42no—
391C4—81no—
392C4—81no—
393C2—28no—
394C2—82no—
395C2C2×#38811yescharacteristic
396C2C4×#12011yescharacteristic
397C2#232211yescharacteristic
398C2#235212yes—
399C2#236112yes—
400C2#241011yescharacteristic
401C2#241311yescharacteristic
402C2#243611yescharacteristic
403C2#244312yes—
404C2#248112yes—
405C2#259111yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC23×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4⋊C4C2semidirect
2. C2×#388C2semidirect
6. C2×C4×C22⋊C4C2semidirect
7. #1442C2semidirect
8. #1439C2semidirect
9. #1436C2semidirect
10. C22⋊#95C2semidirect
11. C22⋊#95C2semidirect
31. #386C4semidirect
81. C2×C4⋊C4D4semidirect
99. C2×C22⋊C4C2×C4semidirect
112. #95C2×C4semidirect
360. C22#388semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C42
C2×#3882
C2×#3862
C2×C4×C22⋊C42
#14422
#14392
#14362
C22⋊#952