Finite Group Databasebeta

← #14927

Subgroups — #14927

Order 256.

1157 conjugacy classes of subgroups.

The subgroup lattice has 409 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×#381C211yesmaximal, characteristic, complemented
5C2×#380C211yesmaximal, 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⋊C4C2212yes—
20C22×C4⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28#386—24no—
29#386—24no—
30#381—24no—
31#380—24no—
32C2×#95—24no—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C2×#95C2212yes—
41C2×#95C2211yescharacteristic
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—22no—
61C23×C4—24no—
62C23×C4—24no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4D414yes—
68C23×C4D414yes—
69C23×C4C2312yes—
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2312yes—
73C23×C4C2312yes—
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2312yes—
77C23×C4C2311yescharacteristic
78#123—232no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—42no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—24no—
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—24no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4D414yes—
104#95—24no—
105#95—24no—
106#95—24no—
107#95—24no—
108#95—24no—
109#95—22no—
110#95—22no—
111#95—24no—
112#95—24no—
113#95—42no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—21no—
119C24—21no—
120C24—21no—
121C24—22no—
122C24—22no—
123C24—21no—
124C24—21no—
125C24—21no—
126C24—22no—
127C24—22no—
128C24C2×D412yes—
129C24C2×D412yes—
130C24C2×D412yes—
131C24C2×D412yes—
132C24C2×D412yes—
133C24#4112yes—
134C24#4112yes—
135C24#4112yes—
136C24C2411yescharacteristic
137C2×D4—28no—
138C2×D4—28no—
139C2×D4—28no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4—44no—
143C2×D4—44no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—28no—
148C22×C4—28no—
149C22×C4—28no—
150C22×C4—28no—
151C22×C4—28no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—28no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—48no—
201C22×C4—48no—
202C22×C4—48no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4C2×D412yes—
210C22×C4C2×D414yes—
211C22×C4C2×D412yes—
212C22×C4C2×D414yes—
213C22×C4C2×D412yes—
214C22×C4#4114yes—
215C22×C4#4114yes—
216C4⋊C4—216no—
217C4⋊C4—48no—
218C22⋊C4—216no—
219C22⋊C4—28no—
220C22⋊C4—48no—
221C22⋊C4—44no—
222C22⋊C4—44no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—22no—
230C23—22no—
231C23—24no—
232C23—24no—
233C23—24no—
234C23—24no—
235C23—24no—
236C23—24no—
237C23—24no—
238C23—24no—
239C23—24no—
240C23—24no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—24no—
245C23—24no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—42no—
253C23—42no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23#12014yes—
265C23#12114yes—
266C23#12114yes—
267C23#12114yes—
268C23#12314yes—
269C23#12314yes—
270C23#12414yes—
271C23C22×D411yescharacteristic
272C23C22×D411yescharacteristic
273C23C22×D411yescharacteristic
274C23C22×D411yescharacteristic
275C23C2×#4111yescharacteristic
276C23C2×#4111yescharacteristic
277C23C2×#4111yescharacteristic
278C23C2×#4112yes—
279C23C2×#4112yes—
280C23H(5,𝔽2)12yes—
281C232−1+412yes—
282D4—416no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—416no—
303C2×C4—416no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—416no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4#12314yescomplemented
341C22—24no—
342C22—24no—
343C22—24no—
344C22—24no—
345C22—28no—
346C22—28no—
347C22—24no—
348C22—28no—
349C22—28no—
350C22—24no—
351C22—42no—
352C22—44no—
353C22—44no—
354C22—42no—
355C22—42no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—82no—
361C22—82no—
362C22#39418yes—
363C22C2×#12011yescharacteristic
364C22C2×#12111yescharacteristic
365C22C2×#12111yescharacteristic
366C22C2×#12111yescharacteristic
367C22C2×#12311yescharacteristic
368C22C2×#12311yescharacteristic
369C22C2×#12311yescharacteristic
370C22#52511yescharacteristic
371C22C2×#12411yescharacteristic
372C22#52912yes—
373C22#53212yes—
374C22#53411yescharacteristic
375C22#53611yescharacteristic
376C22#53712yes—
377C22#53912yes—
378C22#54612yes—
379C22#54612yes—
380C22#54712yes—
381C22#54712yes—
382C22#55112yes—
383C22#55112yes—
384C22#55212yes—
385C22#55212yes—
386C4—24no—
387C4—44no—
388C4—42no—
389C4—42no—
390C4—44no—
391C4—44no—
392C4—81no—
393C4—81no—
394C4—82no—
395C2—28no—
396C2—82no—
397C2C2×#39411yescharacteristic
398C2#241211yescharacteristic
399C2#241311yescharacteristic
400C2#241912yes—
401C2#243012yes—
402C2#248611yescharacteristic
403C2#248812yes—
404C2#249812yes—
405C2#253411yescharacteristic
406C2#261811yescharacteristic
407C2#285411yescharacteristic
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×#381C2semidirect
5. C2×#380C2semidirect
6. C22×#95C2semidirect
7. #1442C2semidirect
8. #1441C2semidirect
9. #1440C2semidirect
10. #1439C2semidirect
11. C22⋊#95C2semidirect
340. C2×C4#123semidirect

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×#3812
C2×#3802
C22×#952
#14422
#14412
#14402
#14392
C22⋊#952