Finite Group Databasebeta

← #14764

Subgroups — #14764

Order 256.

1207 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#121C211yesmaximal, characteristic
2C2×#394C211yesmaximal, characteristic, complemented
3C2×#388C212yesmaximal, complemented
4C2×#385C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊C4C211yesmaximal, characteristic, complemented
6#1442C211yesmaximal, characteristic, complemented
7#1441C211yesmaximal, characteristic, complemented
8#1438C212yesmaximal, complemented
9#1438C212yesmaximal, complemented
10#1438C211yesmaximal, characteristic, complemented
11#1437C212yesmaximal, complemented
12C23×D4C2211yescharacteristic
13C23×D4C2212yes—
14C24×C4C2211yescharacteristic
15C2×#121—24no—
16C2×#121—28no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2211yescharacteristic
28C22×C42C2212yes—
29#394—24no—
30#388—28no—
31#385—24no—
32C4×C22⋊C4—24no—
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2211yescharacteristic
41C25C2312yes—
42C25C2311yescharacteristic
43C22×D4—24no—
44C22×D4—24no—
45C22×D4—24no—
46C22×D4—24no—
47C22×D4—24no—
48C22×D4—24no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4—22no—
52C22×D4—42no—
53C22×D4—42no—
54C22×D4—42no—
55C23×C4—24no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—42no—
59C23×C4D414yes—
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C23×C4C2312yes—
63C23×C4C2312yes—
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2312yes—
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69#121—232no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
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—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4D418yes—
100C2×C42—24no—
101C2×C42—24no—
102C2×C42—42no—
103#95—22no—
104#95—22no—
105#95—22no—
106#95—24no—
107#95—24no—
108#95—42no—
109#95—42no—
110#95D414yescomplemented
111C24—21no—
112C24—22no—
113C24—21no—
114C24—21no—
115C24—22no—
116C24—21no—
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×D412yes—
136C24#4114yes—
137C24C2411yescharacteristic
138C2×D4—28no—
139C2×D4—216no—
140C2×D4—48no—
141C2×D4—48no—
142C2×D4—48no—
143C2×D4—48no—
144C2×D4—44no—
145C2×D4—48no—
146C2×D4—48no—
147C2×D4—48no—
148C2×D4—48no—
149C2×D4—44no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—28no—
184C22×C4—28no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—44no—
196C22×C4—48no—
197C22×C4—48no—
198C22×C4C2×D414yes—
199C22×C4C2×D414yes—
200C22×C4C2×D412yes—
201C22×C4C2×D414yes—
202C22×C4C2×D412yes—
203C22×C4C2×D412yes—
204C22×C4#4112yes—
205C22×C4#4114yes—
206C22×C4#4112yes—
207C22×C4#4112yes—
208C4⋊C4—28no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C22⋊C4—216no—
212C22⋊C4—28no—
213C22⋊C4—216no—
214C22⋊C4—216no—
215C22⋊C4—48no—
216C22⋊C4—48no—
217C22⋊C4—48no—
218C22⋊C4—44no—
219C42—48no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—24no—
224C23—24no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—24no—
233C23—24no—
234C23—24no—
235C23—24no—
236C23—24no—
237C23—24no—
238C23—24no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
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#12414yes—
278C23C22×D411yescharacteristic
279C23C22×D412yes—
280C23C22×D411yescharacteristic
281C23C2×#4111yescharacteristic
282C23C2×#4111yescharacteristic
283C23C2×#4112yes—
284C23C2×#4112yes—
285C23C2×#4111yescharacteristic
286C23H(5,𝔽2)12yes—
287C23H(5,𝔽2)11yescharacteristic
288C23H(5,𝔽2)11yescharacteristic
289D4—416no—
290D4—432no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—416no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
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—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4#12014yes—
344C2×C4#12114yes—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22C2×#12011yescharacteristic
372C22C2×#12111yescharacteristic
373C22#52511yescharacteristic
374C22#52512yes—
375C22#52512yes—
376C22C2×#12411yescharacteristic
377C22#52912yes—
378C22#53211yescharacteristic
379C22#53212yes—
380C22#53411yescharacteristic
381C22#53511yescharacteristic
382C22#53812yes—
383C22#54012yes—
384C22D4212yes—
385C22#54612yes—
386C22#54611yescharacteristic
387C22#54612yes—
388C22#54612yes—
389C22#55011yescharacteristic
390C22#55312yes—
391C22#55312yes—
392C22#55311yescharacteristic
393C22#55511yescharacteristic
394C4—24no—
395C4—24no—
396C4—42no—
397C4—42no—
398C4—44no—
399C4—44no—
400C4—44no—
401C4—81no—
402C4—81no—
403C2—44no—
404C2—84no—
405C2#238811yescharacteristic
406C2#241011yescharacteristic
407C2#241411yescharacteristic
408C2#243511yescharacteristic
409C2#245612yes—
410C2#246412yes—
411C2#247612yes—
412C2#255912yes—
413C2#259111yescharacteristic
414C2#266111yescharacteristic
415C2#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×#388C2semidirect
4. C2×#385C2semidirect
5. C2×C4×C22⋊C4C2semidirect
6. #1442C2semidirect
7. #1441C2semidirect
8. #1438C2semidirect
9. #1438C2semidirect
10. #1438C2semidirect
11. #1437C2semidirect
110. #95D4semidirect

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×#3882
C2×#3852
C2×C4×C22⋊C42
#14422
#14412
#14382
#14372