Finite Group Databasebeta

← #14747

Subgroups — #14747

Order 256.

1365 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#121C211yesmaximal, characteristic
2C2×#394C211yesmaximal, characteristic, complemented
3C2×#386C212yesmaximal, complemented
4C2×#386C212yesmaximal, complemented
5C2×#385C211yesmaximal, characteristic, complemented
6C22×#95C211yesmaximal, characteristic, complemented
7#1442C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1440C212yesmaximal, complemented
10#1439C212yesmaximal, complemented
11#1438C211yesmaximal, characteristic, complemented
12C23×D4C2212yes—
13C23×D4C2211yescharacteristic
14C24×C4C2212yes—
15C24×C4C2211yescharacteristic
16C2×#121—28no—
17C2×#121—24no—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2211yescharacteristic
28#394—24no—
29#386—28no—
30#386—28no—
31#385—24no—
32C2×#95—24no—
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2211yescharacteristic
39C2×#95C2212yes—
40C2×#95C2212yes—
41C2×#95C2211yescharacteristic
42C25C2312yes—
43C25C2311yescharacteristic
44C22×D4—24no—
45C22×D4—24no—
46C22×D4—24no—
47C22×D4—24no—
48C22×D4—24no—
49C22×D4—24no—
50C22×D4—24no—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—22no—
54C22×D4—42no—
55C22×D4D418yes—
56C23×C4—24no—
57C23×C4—24no—
58C23×C4—24no—
59C23×C4—22no—
60C23×C4—24no—
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4D414yes—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2312yes—
74C23×C4C2312yes—
75C23×C4C2312yes—
76C23×C4C2311yescharacteristic
77#121—232no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
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—24no—
92C2×C22⋊C4—24no—
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—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4D414yescomplemented
105#95—24no—
106#95—22no—
107#95—22no—
108#95—22no—
109#95—24no—
110#95—24no—
111#95—24no—
112#95—24no—
113#95D414yescomplemented
114C24—21no—
115C24—21no—
116C24—22no—
117C24—22no—
118C24—21no—
119C24—21no—
120C24—22no—
121C24—22no—
122C24—22no—
123C24—22no—
124C24—22no—
125C24—22no—
126C24—22no—
127C24—22no—
128C24—22no—
129C24—22no—
130C24—22no—
131C24—22no—
132C24—22no—
133C24—22no—
134C24C2×D414yes—
135C24C2×D414yes—
136C24C2×D412yes—
137C24C2×D412yes—
138C24#4112yes—
139C24#4114yes—
140C24C2411yescharacteristic
141C2×D4—28no—
142C2×D4—216no—
143C2×D4—216no—
144C2×D4—216no—
145C2×D4—48no—
146C2×D4—48no—
147C2×D4—44no—
148C2×D4—48no—
149C2×D4—48no—
150C2×D4—48no—
151C2×D4—48no—
152C2×D4—44no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—216no—
156C22×C4—28no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—28no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—44no—
201C22×C4—48no—
202C22×C4—48no—
203C22×C4—48no—
204C22×C4—48no—
205C22×C4—48no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4C2×D412yes—
210C22×C4C2×D412yes—
211C22×C4C2×D414yes—
212C22×C4C2×D414yes—
213C22×C4C2×D412yes—
214C22×C4C2×D412yes—
215C22×C4C2×D412yes—
216C22×C4#4114yes—
217C22×C4#4112yes—
218C4⋊C4—28no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C22⋊C4—216no—
222C22⋊C4—28no—
223C22⋊C4—48no—
224C22⋊C4—48no—
225C22⋊C4—48no—
226C22⋊C4—48no—
227C22⋊C4—44no—
228C23—22no—
229C23—24no—
230C23—24no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—24no—
235C23—24no—
236C23—24no—
237C23—22no—
238C23—24no—
239C23—24no—
240C23—24no—
241C23—22no—
242C23—24no—
243C23—22no—
244C23—22no—
245C23—24no—
246C23—24no—
247C23—24no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—24no—
252C23—24no—
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#12014yes—
282C23#12114yes—
283C23#12118yes—
284C23#12318yes—
285C23#12414yes—
286C23C22×D411yescharacteristic
287C23C22×D412yes—
288C23C22×D411yescharacteristic
289C23C22×D411yescharacteristic
290C23C2×#4111yescharacteristic
291C23C2×#4112yes—
292C23C2×#4111yescharacteristic
293C23C2×#4112yes—
294C23H(5,𝔽2)12yes—
295C23H(5,𝔽2)11yescharacteristic
296C232−1+411yescharacteristic
297D4—416no—
298D4—432no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—416no—
321C2×C4—432no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
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—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4#12114yes—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22—28no—
359C22—28no—
360C22—24no—
361C22—28no—
362C22—28no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—44no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22#39418yescomplemented
380C22C2×#12011yescharacteristic
381C22C2×#12111yescharacteristic
382C22C2×#12111yescharacteristic
383C22C2×#12112yes—
384C22C2×#12312yes—
385C22#52512yes—
386C22C2×#12411yescharacteristic
387C22#52912yes—
388C22#53211yescharacteristic
389C22#53411yescharacteristic
390C22#53711yescharacteristic
391C22#53812yes—
392C22#53912yes—
393C22D4212yes—
394C22D4211yescharacteristic
395C22#54612yes—
396C22#54612yes—
397C22#54612yes—
398C22#54711yescharacteristic
399C22#54712yes—
400C22#54811yescharacteristic
401C22#55311yescharacteristic
402C22#55312yes—
403C4—24no—
404C4—44no—
405C4—42no—
406C4—42no—
407C4—44no—
408C4—44no—
409C4—42no—
410C4—81no—
411C4—81no—
412C2—28no—
413C2—84no—
414C2C2×#39411yescharacteristic
415C2#241011yescharacteristic
416C2#241611yescharacteristic
417C2#242412yes—
418C2#242512yes—
419C2#243011yescharacteristic
420C2#244312yes—
421C2#244412yes—
422C2#254311yescharacteristic
423C2#263911yescharacteristic
424C2#285011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC22×#121C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#394C2semidirect
3. C2×#386C2semidirect
4. C2×#386C2semidirect
5. C2×#385C2semidirect
6. C22×#95C2semidirect
7. #1442C2semidirect
8. #1441C2semidirect
9. #1440C2semidirect
10. #1439C2semidirect
11. #1438C2semidirect
104. C2×C22⋊C4D4semidirect
113. #95D4semidirect
379. C22#394semidirect

Maximal subgroups

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

subgroupindex
C22×#1212
C2×#3942
C2×#3862
C2×#3852
C22×#952
#14422
#14412
#14402
#14392
#14382