Finite Group Databasebeta

← #16261

Subgroups — #16261

Order 256.

1201 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#121C211yesmaximal, characteristic
2C2×#396C212yesmaximal, complemented
3C2×#394C211yesmaximal, characteristic, complemented
4C2×#394C212yesmaximal, complemented
5C2×#385C211yesmaximal, characteristic, complemented
6C2×#380C211yesmaximal, characteristic, complemented
7#1444C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1439C212yesmaximal, complemented
10#1438C211yesmaximal, characteristic, complemented
11#1438C212yesmaximal, complemented
12C23×D4C2212yes—
13C23×D4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C2×#121—24no—
16C2×#121—28no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2212yes—
21C22×C4⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2212yes—
30#396—28no—
31#394—28no—
32#394—24no—
33#385—24no—
34#380—24no—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2211yescharacteristic
39C2×#95C2211yescharacteristic
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—22no—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—42no—
54C22×D4—42no—
55C22×D4—42no—
56C23×C4—24no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—42no—
60C23×C4D414yes—
61C23×C4C2311yescharacteristic
62C23×C4C2312yes—
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2312yes—
66C23×C4C2311yescharacteristic
67C23×C4C2312yes—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70#121—232no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—24no—
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—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4D418yes—
106C2×C22⋊C4D414yescomplemented
107#95—22no—
108#95—22no—
109#95—22no—
110#95—24no—
111#95—24no—
112#95—42no—
113#95D414yescomplemented
114C24—21no—
115C24—22no—
116C24—21no—
117C24—21no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—21no—
122C24—22no—
123C24—22no—
124C24—22no—
125C24—22no—
126C24—22no—
127C24—22no—
128C24—22no—
129C24—22no—
130C24—22no—
131C24—22no—
132C24—22no—
133C24—22no—
134C24—22no—
135C24—22no—
136C24C2×D414yes—
137C24C2×D412yes—
138C24C2×D412yes—
139C24#4112yes—
140C24C2411yescharacteristic
141C2×D4—28no—
142C2×D4—216no—
143C2×D4—48no—
144C2×D4—48no—
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—21no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—28no—
181C22×C4—28no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—44no—
200C22×C4—48no—
201C22×C4—48no—
202C22×C4C2×D412yes—
203C22×C4C2×D414yes—
204C22×C4C2×D414yes—
205C22×C4C2×D412yes—
206C22×C4C2×D412yes—
207C22×C4C2×D414yes—
208C22×C4C2×D412yes—
209C22×C4C2×D412yes—
210C22×C4#4112yes—
211C4⋊C4—28no—
212C4⋊C4—48no—
213C4⋊C4—48no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C22⋊C4—28no—
217C22⋊C4—216no—
218C22⋊C4—48no—
219C22⋊C4—48no—
220C22⋊C4—48no—
221C22⋊C4—48no—
222C22⋊C4—44no—
223C22⋊C4—48no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—24no—
232C23—22no—
233C23—24no—
234C23—24no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—24no—
239C23—24no—
240C23—24no—
241C23—24no—
242C23—24no—
243C23—24no—
244C23—24no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
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#12114yes—
282C23C22×D411yescharacteristic
283C23C22×D412yes—
284C23C22×D411yescharacteristic
285C23C22×D411yescharacteristic
286C23C2×#4111yescharacteristic
287C23C2×#4111yescharacteristic
288C23H(5,𝔽2)12yes—
289C23H(5,𝔽2)12yes—
290C23H(5,𝔽2)11yescharacteristic
291C232−1+412yes—
292C232−1+411yescharacteristic
293D4—416no—
294D4—432no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—416no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
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—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4#12014yes—
348C2×C4#12114yes—
349C22—24no—
350C22—24no—
351C22—24no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—44no—
359C22—42no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—44no—
364C22—44no—
365C22—44no—
366C22—44no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22C2×#12011yescharacteristic
376C22C2×#12111yescharacteristic
377C22C2×#12111yescharacteristic
378C22#53411yescharacteristic
379C22#53511yescharacteristic
380C22#53611yescharacteristic
381C22#53612yes—
382C22#53711yescharacteristic
383C22#53712yes—
384C22#53712yes—
385C22#53712yes—
386C22#53912yes—
387C22#54212yes—
388C22D4212yes—
389C22D4212yes—
390C22D4211yescharacteristic
391C22#54612yes—
392C22#54612yes—
393C22#54711yescharacteristic
394C22#54811yescharacteristic
395C22#55311yescharacteristic
396C22#56012yes—
397C22#56212yes—
398C4—24no—
399C4—24no—
400C4—42no—
401C4—44no—
402C4—42no—
403C4—44no—
404C4—81no—
405C4—82no—
406C4—81no—
407C2—44no—
408C2—84no—
409C2#240811yescharacteristic
410C2#241011yescharacteristic
411C2#241611yescharacteristic
412C2#261611yescharacteristic
413C2#261811yescharacteristic
414C2#262011yescharacteristic
415C2#267212yes—
416C2#267512yes—
417C2#268112yes—
418C2#269112yes—
419C2#281311yescharacteristic
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×#396C2semidirect
3. C2×#394C2semidirect
4. C2×#394C2semidirect
5. C2×#385C2semidirect
6. C2×#380C2semidirect
7. #1444C2semidirect
8. #1441C2semidirect
9. #1439C2semidirect
10. #1438C2semidirect
11. #1438C2semidirect
106. C2×C22⋊C4D4semidirect
113. #95D4semidirect

Maximal subgroups

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

subgroupindex
C22×#1212
C2×#3962
C2×#3942
C2×#3852
C2×#3802
#14442
#14412
#14392
#14382