Finite Group Databasebeta

← #14928

Subgroups — #14928

Order 256.

1215 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124C211yesmaximal, characteristic, complemented
2C2×#395C211yesmaximal, characteristic, complemented
3C2×#394C211yesmaximal, characteristic, complemented
4C2×#393C211yesmaximal, characteristic, complemented
5C2×#392C211yesmaximal, characteristic
6C2×#390C212yesmaximal, complemented
7C2×#388C211yesmaximal, characteristic, complemented
8#1439C212yesmaximal, complemented
9#1438C212yesmaximal, complemented
10#1437C212yesmaximal, complemented
11C4×#95C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C23×D4C2212yes—
15C2×#124—28no—
16C2×#124—24no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2212yes—
28C22×C42C2212yes—
29C22×C42C2211yescharacteristic
30#395—24no—
31#394—24no—
32#393—24no—
33#392—24no—
34#390—28no—
35#388—24no—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C2×#95C2211yescharacteristic
41C2×#95C2211yescharacteristic
42C2×#95C2211yescharacteristic
43C25C2312yes—
44C25C2311yescharacteristic
45C22×Q8—24no—
46C22×Q8—22no—
47C22×Q8—22no—
48C22×Q8—22no—
49C22×Q8—42no—
50C22×D4—24no—
51C22×D4—24no—
52C22×D4—24no—
53C22×D4—22no—
54C22×D4—24no—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4—24no—
58C22×D4—24no—
59C22×D4—42no—
60C22×D4—42no—
61C22×D4—42no—
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2312yes—
65C23×C4C2311yescharacteristic
66C23×C4C2312yes—
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2312yes—
71#124—232no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—42no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—22no—
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—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—22no—
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⋊C4D418yescomplemented
104C2×C42—24no—
105C2×C42—24no—
106C2×C42—24no—
107C2×C42—42no—
108C2×C42D414yescomplemented
109#95—22no—
110#95—22no—
111#95—24no—
112#95—24no—
113#95—24no—
114#95D414yes—
115#95D414yescomplemented
116C24—21no—
117C24—21no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—22no—
122C24—22no—
123C24—21no—
124C24—21no—
125C24—22no—
126C24—22no—
127C24—22no—
128C24—22no—
129C24—22no—
130C24—22no—
131C24—22no—
132C24—22no—
133C24—22no—
134C24—22no—
135C24—22no—
136C24—22no—
137C24—21no—
138C24—22no—
139C24—21no—
140C24C2×D414yes—
141C24#4112yes—
142C24C2411yescharacteristic
143C2×Q8—28no—
144C2×Q8—48no—
145C2×Q8—48no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×D4—28no—
149C2×D4—216no—
150C2×D4—48no—
151C2×D4—48no—
152C2×D4—48no—
153C2×D4—48no—
154C2×D4—48no—
155C2×D4—48no—
156C2×D4—44no—
157C2×D4—48no—
158C2×D4—48no—
159C2×D4—44no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
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—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4C2×D414yes—
202C22×C4C2×D412yes—
203C22×C4C2×D414yes—
204C22×C4C2×D412yes—
205C22×C4C2×D414yes—
206C22×C4C2×D414yes—
207C22×C4C2×D412yes—
208C22×C4C2×D412yes—
209C22×C4C2×D412yes—
210C22×C4#4112yes—
211C22×C4#4114yes—
212C22×C4#4114yes—
213C4⋊C4—48no—
214C4⋊C4—44no—
215C22⋊C4—28no—
216C22⋊C4—216no—
217C22⋊C4—216no—
218C22⋊C4—48no—
219C22⋊C4—48no—
220C22⋊C4—48no—
221C22⋊C4—44no—
222C22⋊C4—48no—
223C22⋊C4—44no—
224C42—28no—
225C42—48no—
226C23—24no—
227C23—22no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—24no—
233C23—22no—
234C23—24no—
235C23—24no—
236C23—22no—
237C23—24no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—41no—
257C23—42no—
258C23—41no—
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—41no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—41no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23C22×D411yescharacteristic
285C23C22×D412yes—
286C23C22×D411yescharacteristic
287C23C22×D411yescharacteristic
288C23C2×#4111yescharacteristic
289C23C2×#4112yes—
290C23C2×#4111yescharacteristic
291C23C2×#4112yes—
292C23H(5,𝔽2)12yes—
293C23H(5,𝔽2)11yescharacteristic
294C232−1+411yescharacteristic
295Q8—416no—
296D4—416no—
297D4—432no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
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—22no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4#12014yes—
349C2×C4#12418yescomplemented
350C2×C4#12714yes—
351C22—42no—
352C22—44no—
353C22—42no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—42no—
362C22—44no—
363C22—82no—
364C22—81no—
365C22—82no—
366C22—82no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—82no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22C2×#12011yescharacteristic
378C22#52511yescharacteristic
379C22#52512yes—
380C22C2×#12412yes—
381C22C2×#12711yescharacteristic
382C22#53211yescharacteristic
383C22#53211yescharacteristic
384C22#53212yes—
385C22#53212yes—
386C22#53511yescharacteristic
387C22#53711yescharacteristic
388C22#53812yes—
389C22#54212yes—
390C22D4212yes—
391C22D4212yes—
392C22#54612yes—
393C22#54611yescharacteristic
394C22#54611yescharacteristic
395C22#54712yes—
396C22#54811yescharacteristic
397C22#54811yescharacteristic
398C22#55112yes—
399C22#55512yes—
400C4—24no—
401C4—24no—
402C4—28no—
403C4—42no—
404C4—44no—
405C4—42no—
406C4—44no—
407C4—81no—
408C4—81no—
409C2—82no—
410C2—84no—
411C2#238811yescharacteristic
412C2#243011yescharacteristic
413C2#243311yescharacteristic
414C2#246112yes—
415C2#247212yes—
416C2#249112yes—
417C2#254011yescharacteristic
418C2#255512yes—
419C2#261011yescharacteristic
420C2#266211yescharacteristic
421C2#285811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC2×#392C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#124C2semidirect
2. C2×#395C2semidirect
3. C2×#394C2semidirect
4. C2×#393C2semidirect
6. C2×#390C2semidirect
7. C2×#388C2semidirect
8. #1439C2semidirect
9. #1438C2semidirect
10. #1437C2semidirect
11. C4×#95C2semidirect
103. C2×C22⋊C4D4semidirect
108. C2×C42D4semidirect
115. #95D4semidirect
349. C2×C4#124semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C2×#3952
C2×#3942
C2×#3932
C2×#3922
C2×#3902
C2×#3882
#14392
#14382
#14372
C4×#952