Finite Group Databasebeta

← #14912

Subgroups — #14912

Order 256.

1157 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123C211yesmaximal, characteristic
2C2×#386C211yesmaximal, characteristic
3C2×#386C211yesmaximal, characteristic
4C2×#385C211yesmaximal, characteristic, complemented
5C2×#381C211yesmaximal, characteristic, complemented
6C22×#95C211yesmaximal, characteristic, complemented
7#1442C212yesmaximal, complemented
8#1441C212yesmaximal, complemented
9#1440C212yesmaximal, complemented
10#1439C212yesmaximal, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C24×C4C2211yescharacteristic
16C2×#123—24no—
17C2×#123—24no—
18C2×#123—24no—
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29#386—24no—
30#386—24no—
31#385—24no—
32#381—24no—
33C2×#95—24no—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C2×#95C2212yes—
42C2×#95C2212yes—
43C25C2311yescharacteristic
44C25C2311yescharacteristic
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4D414yes—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—24no—
61C23×C4—24no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—42no—
67C23×C4—42no—
68C23×C4D414yes—
69C23×C4C2312yes—
70C23×C4C2311yescharacteristic
71C23×C4C2312yes—
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2312yes—
77C23×C4C2312yes—
78#123—232no—
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—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—42no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4D414yes—
108C2×C22⋊C4D414yes—
109#95—24no—
110#95—24no—
111#95—24no—
112#95—24no—
113#95—24no—
114#95—24no—
115#95—24no—
116#95—24no—
117#95—42no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—22no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—22no—
126C24—22no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—22no—
131C24—22no—
132C24C2×D412yes—
133C24C2×D412yes—
134C24C2×D412yes—
135C24C2×D412yes—
136C24C2×D412yes—
137C24#4112yes—
138C24#4112yes—
139C24#4112yes—
140C24C2411yescharacteristic
141C2×D4—28no—
142C2×D4—28no—
143C2×D4—28no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4—44no—
147C2×D4—44no—
148C22×C4—28no—
149C22×C4—28no—
150C22×C4—28no—
151C22×C4—28no—
152C22×C4—28no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
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—22no—
204C22×C4—48no—
205C22×C4—48no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4C2×D414yes—
217C22×C4C2×D412yes—
218C22×C4C2×D412yes—
219C22×C4C2×D412yes—
220C22×C4C2×D412yes—
221C22×C4C2×D412yes—
222C22×C4#4114yes—
223C22×C4#4114yes—
224C4⋊C4—28no—
225C4⋊C4—28no—
226C4⋊C4—48no—
227C22⋊C4—28no—
228C22⋊C4—28no—
229C22⋊C4—28no—
230C22⋊C4—28no—
231C22⋊C4—48no—
232C22⋊C4—48no—
233C22⋊C4—44no—
234C23—22no—
235C23—22no—
236C23—24no—
237C23—24no—
238C23—22no—
239C23—22no—
240C23—24no—
241C23—24no—
242C23—24no—
243C23—24no—
244C23—24no—
245C23—24no—
246C23—24no—
247C23—24no—
248C23—24no—
249C23—24no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—24no—
254C23—24no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—42no—
262C23—42no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23#12014yes—
274C23#12114yes—
275C23#12114yes—
276C23#12114yes—
277C23#12314yes—
278C23#12314yes—
279C23#12414yes—
280C23C22×D411yescharacteristic
281C23C22×D411yescharacteristic
282C23C22×D411yescharacteristic
283C23C22×D411yescharacteristic
284C23C2×#4111yescharacteristic
285C23C2×#4111yescharacteristic
286C23C2×#4111yescharacteristic
287C23C2×#4112yes—
288C23C2×#4112yes—
289C23H(5,𝔽2)12yes—
290C232−1+412yes—
291D4—416no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—416no—
315C2×C4—416no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
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—416no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22—24no—
359C22—24no—
360C22—28no—
361C22—28no—
362C22—28no—
363C22—28no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—44no—
368C22—44no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—82no—
374C22—82no—
375C22#39418yes—
376C22C2×#12011yescharacteristic
377C22C2×#12111yescharacteristic
378C22C2×#12111yescharacteristic
379C22C2×#12111yescharacteristic
380C22C2×#12311yescharacteristic
381C22C2×#12311yescharacteristic
382C22#52511yescharacteristic
383C22#52511yescharacteristic
384C22C2×#12411yescharacteristic
385C22#52912yes—
386C22#52912yes—
387C22#53511yescharacteristic
388C22#53611yescharacteristic
389C22#53912yes—
390C22#53912yes—
391C22#54612yes—
392C22#54612yes—
393C22#54612yes—
394C22#54712yes—
395C22#54712yes—
396C22#54712yes—
397C22#55112yes—
398C22#55212yes—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—44no—
404C4—42no—
405C4—44no—
406C4—44no—
407C4—82no—
408C2—28no—
409C2—82no—
410C2C2×#39411yescharacteristic
411C2#240411yescharacteristic
412C2#240611yescharacteristic
413C2#241111yescharacteristic
414C2#241311yescharacteristic
415C2#241912yes—
416C2#242512yes—
417C2#242512yes—
418C2#248812yes—
419C2#262311yescharacteristic
420C2#285311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC23×D4C22
LCSG ▷ C24 ▷ 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. C2×#381C2semidirect
6. C22×#95C2semidirect
7. #1442C2semidirect
8. #1441C2semidirect
9. #1440C2semidirect
10. #1439C2semidirect
11. C22⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1232
C2×#3862
C2×#3852
C2×#3812
C22×#952
#14422
#14412
#14402
#14392
C22⋊#952