Finite Group Databasebeta

← #14259

Subgroups — #14259

Order 256.

1049 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#386C211yesmaximal, characteristic
2C2×#386C211yesmaximal, characteristic
3C2×#386C211yesmaximal, characteristic
4C2×#385C211yesmaximal, characteristic, complemented
5C22×#95C211yesmaximal, characteristic, complemented
6#1443C212yesmaximal, complemented
7#1442C212yesmaximal, complemented
8#1440C212yesmaximal, complemented
9#1439C212yesmaximal, complemented
10C22⋊#95C211yesmaximal, characteristic, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C24×C4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17C22×C22⋊C4C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24#386—24no—
25#386—24no—
26#386C418yescomplemented
27#385—24no—
28C2×#95—24no—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C25C2311yescharacteristic
41C25C2311yescharacteristic
42C22×D4—22no—
43C22×D4—22no—
44C22×D4—22no—
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4C2×C414yes—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—42no—
56C23×C4—42no—
57C23×C4—42no—
58C23×C4—42no—
59C23×C4—42no—
60C23×C4—42no—
61C23×C4C2×C414yes—
62C23×C4D414yes—
63C23×C4D414yes—
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2312yes—
67C23×C4C2312yes—
68C23×C4C2312yes—
69C23×C4C2311yescharacteristic
70C23×C4C2312yes—
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C2×C4⋊C4—24no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4C2×C414yes—
94C2×C22⋊C4C2×C418yescomplemented
95#95—24no—
96#95—24no—
97#95—24no—
98#95—22no—
99#95—22no—
100#95—24no—
101#95—42no—
102#95—42no—
103#95—42no—
104#95—42no—
105#95C2×C418yescomplemented
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—21no—
111C24—21no—
112C24—21no—
113C24—21no—
114C24—21no—
115C24—21no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24C22×C412yes—
121C24C22×C412yes—
122C24C2×D412yes—
123C24C2×D412yes—
124C24#4112yes—
125C24#4112yes—
126C24#4112yes—
127C24#4112yes—
128C24C2411yescharacteristic
129C2×D4—28no—
130C2×D4—28no—
131C2×D4—28no—
132C2×D4—44no—
133C2×D4—44no—
134C2×D4—44no—
135C2×D4—44no—
136C22×C4—28no—
137C22×C4—28no—
138C22×C4—28no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
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—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
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—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—48no—
189C22×C4—48no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—48no—
197C22×C4—48no—
198C22×C4—48no—
199C22×C4—48no—
200C22×C4C22×C412yes—
201C22×C4C22×C414yes—
202C22×C4C22×C414yes—
203C22×C4C2×D412yes—
204C22×C4C2×D412yes—
205C22×C4C2×D414yes—
206C4⋊C4—48no—
207C22⋊C4—28no—
208C22⋊C4—48no—
209C22⋊C4—48no—
210C22⋊C4—44no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C22⋊C4—44no—
214C23—24no—
215C23—24no—
216C23—22no—
217C23—22no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—24no—
225C23—24no—
226C23—24no—
227C23—24no—
228C23—24no—
229C23—24no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—24no—
234C23—24no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23#11714yes—
254C23C4×D414yes—
255C23C4×D414yes—
256C23#12114yes—
257C23#12314yes—
258C23#12414yes—
259C23#12614yes—
260C23C23×C411yescharacteristic
261C23C22×D411yescharacteristic
262C23C22×D411yescharacteristic
263C23C2×#4111yescharacteristic
264C23C2×#4111yescharacteristic
265C23C2×#4111yescharacteristic
266C23C2×#4111yescharacteristic
267C23H(5,𝔽2)12yes—
268C23H(5,𝔽2)12yes—
269C23H(5,𝔽2)12yes—
270C232−1+412yes—
271D4—416no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—416no—
291C2×C4—416no—
292C2×C4—416no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C22—28no—
331C22—24no—
332C22—24no—
333C22—28no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22—24no—
338C22—28no—
339C22—28no—
340C22—44no—
341C22—42no—
342C22—42no—
343C22—44no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—82no—
350C22—82no—
351C22#38818yes—
352C22C2×#11711yescharacteristic
353C22C2×C4×D411yescharacteristic
354C22C2×C4×D411yescharacteristic
355C22#51812yes—
356C22#52012yes—
357C22C2×#12111yescharacteristic
358C22C2×#12311yescharacteristic
359C22C2×#12411yescharacteristic
360C22C2×#12611yescharacteristic
361C22#53411yescharacteristic
362C22#53411yescharacteristic
363C22#53412yes—
364C22#53511yescharacteristic
365C22#53611yescharacteristic
366C22#53712yes—
367C22#53812yes—
368C22#53812yes—
369C22#53812yes—
370C22#53812yes—
371C22#53912yes—
372C22#53912yes—
373C22#53912yes—
374C22#53912yes—
375C4—44no—
376C4—44no—
377C4—44no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—81no—
382C4—81no—
383C4—82no—
384C2—28no—
385C2—82no—
386C2C2×#38811yescharacteristic
387C2#231812yes—
388C2#232111yescharacteristic
389C2#232211yescharacteristic
390C2#232512yes—
391C2#261711yescharacteristic
392C2#261811yescharacteristic
393C2#262211yescharacteristic
394C2#262311yescharacteristic
395C2#263912yes—
396C2#264312yes—
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
4. C2×#385C2semidirect
5. C22×#95C2semidirect
6. #1443C2semidirect
7. #1442C2semidirect
8. #1440C2semidirect
9. #1439C2semidirect
10. C22⋊#95C2semidirect
11. C22⋊#95C2semidirect
26. #386C4semidirect
94. C2×C22⋊C4C2×C4semidirect
105. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3862
C2×#3852
C22×#952
#14432
#14422
#14402
#14392
C22⋊#952