Finite Group Databasebeta

← #14015

Subgroups — #14015

Order 256.

1475 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4C211yesmaximal, characteristic
2C23×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#386C211yesmaximal, characteristic
4C2×#386C211yesmaximal, characteristic
5C2×#385C212yesmaximal, complemented
6C2×#385C211yesmaximal, characteristic, complemented
7#1444C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1436C212yesmaximal, complemented
10C22⋊#95C211yesmaximal, characteristic, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C24×C4C2211yescharacteristic
16C24×C4C2211yescharacteristic
17C2×C4×D4—24no—
18C2×C4×D4—24no—
19C2×C4×D4—24no—
20C22×C4⋊C4—24no—
21C22×C4⋊C4—28no—
22C22×C4⋊C4C2212yes—
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2212yes—
25C22×C4⋊C4C2212yes—
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2212yes—
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2212yes—
31C22×C22⋊C4C2211yescharacteristic
32C22×C42C2211yescharacteristic
33#386—24no—
34#386C418yescomplemented
35#385—28no—
36#385—24no—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C2×#95C2212yes—
42C2×#95C2212yes—
43C2×#95C2212yes—
44C25C2311yescharacteristic
45C25C2311yescharacteristic
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4C2×C414yes—
51C22×D4D414yes—
52C22×D4Q814yes—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—24no—
56C23×C4—24no—
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—22no—
66C23×C4—22no—
67C23×C4—24no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4C2×C414yescomplemented
72C23×C4D414yes—
73C23×C4C2312yes—
74C23×C4C2312yes—
75C23×C4C2311yescharacteristic
76C23×C4C2312yes—
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C23×C4C2312yes—
81C23×C4C2311yescharacteristic
82C4×D4—232no—
83C2×C4⋊C4—232no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4D418yescomplemented
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4C2×C414yescomplemented
112C2×C22⋊C4C2×C418yescomplemented
113C2×C22⋊C4D414yes—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—22no—
117#95—24no—
118#95—24no—
119#95—24no—
120#95—24no—
121#95—24no—
122#95—42no—
123#95C2×C418yescomplemented
124C24—22no—
125C24—22no—
126C24—22no—
127C24—22no—
128C24—22no—
129C24—21no—
130C24—21no—
131C24—22no—
132C24—21no—
133C24—21no—
134C24—22no—
135C24—22no—
136C24C22×C412yes—
137C24C22×C412yes—
138C24C2×D412yes—
139C24C2×D412yes—
140C24C2×D412yes—
141C24C2×D412yes—
142C24C2×Q812yes—
143C24C2×Q812yes—
144C24#4112yes—
145C24#4112yes—
146C24C2411yescharacteristic
147C2×D4—28no—
148C2×D4—28no—
149C2×D4—28no—
150C2×D4—28no—
151C2×D4—28no—
152C2×D4—28no—
153C2×D4C4⋊C4116yescomplemented
154C22×C4—28no—
155C22×C4—216no—
156C22×C4—28no—
157C22×C4—28no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—216no—
170C22×C4—28no—
171C22×C4—28no—
172C22×C4—28no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—28no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—44no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4—48no—
210C22×C4—48no—
211C22×C4—48no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4C22×C414yes—
217C22×C4C22×C412yes—
218C22×C4C22×C414yes—
219C22×C4C2×D412yes—
220C22×C4C2×D412yes—
221C22×C4C2×D414yes—
222C22×C4C2×D412yes—
223C22×C4C2×D412yes—
224C22×C4C2×D414yes—
225C22×C4C2×D414yes—
226C22×C4C2×D412yes—
227C22×C4C2×Q812yes—
228C22×C4#4114yes—
229C4⋊C4—216no—
230C4⋊C4—28no—
231C4⋊C4—216no—
232C4⋊C4—48no—
233C22⋊C4—28no—
234C22⋊C4—28no—
235C22⋊C4—28no—
236C22⋊C4—216no—
237C22⋊C4—48no—
238C22⋊C4—44no—
239C42—28no—
240C23—24no—
241C23—24no—
242C23—22no—
243C23—22no—
244C23—24no—
245C23—22no—
246C23—22no—
247C23—24no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—24no—
252C23—24no—
253C23—24no—
254C23—24no—
255C23—24no—
256C23—24no—
257C23—22no—
258C23—22no—
259C23—24no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—24no—
266C23—24no—
267C23—24no—
268C23—41no—
269C23—41no—
270C23—42no—
271C23—41no—
272C23—41no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23C2×C4⋊C414yes—
279C23C2×C4⋊C414yescomplemented
280C23C4×D414yes—
281C23C4×D414yes—
282C23#12014yes—
283C23C22⋊Q814yes—
284C23C22⋊Q814yes—
285C23#12314yes—
286C23C23×C411yescharacteristic
287C23C22×D411yescharacteristic
288C23C22×D411yescharacteristic
289C23C22×D411yescharacteristic
290C23C22×D412yes—
291C23C22×Q811yescharacteristic
292C23C2×#4111yescharacteristic
293C23C2×#4111yescharacteristic
294C23C2×#4112yes—
295C23H(5,𝔽2)12yes—
296C232−1+412yes—
297D4—232no—
298C2×C4—232no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—416no—
329C2×C4—416no—
330C2×C4—416no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
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—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4C2×C4⋊C414yescomplemented
355C2×C4C4×D418yescomplemented
356C2×C4#12014yes—
357C22—24no—
358C22—24no—
359C22—28no—
360C22—28no—
361C22—24no—
362C22—24no—
363C22—24no—
364C22—28no—
365C22—28no—
366C22—24no—
367C22—28no—
368C22—24no—
369C22—42no—
370C22—42no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—44no—
376C22—44no—
377C22—42no—
378C22#38518yescomplemented
379C22C22×C4⋊C411yescharacteristic
380C22C2×C4×D411yescharacteristic
381C22C2×C4×D412yes—
382C22C2×C4×D411yescharacteristic
383C22#52012yes—
384C22C2×#12011yescharacteristic
385C22C2×#12011yescharacteristic
386C22C2×C22⋊Q811yescharacteristic
387C22C2×C22⋊Q811yescharacteristic
388C22C2×#12311yescharacteristic
389C22#52511yescharacteristic
390C22#52512yes—
391C22#53411yescharacteristic
392C22#53611yescharacteristic
393C22#53912yes—
394C22D4212yes—
395C22D4212yes—
396C22#54712yes—
397C22#54712yes—
398C22#54812yes—
399C22D4×Q812yes—
400C22#55312yes—
401C22#55412yes—
402C4—24no—
403C4—24no—
404C4—28no—
405C4—44no—
406C4—42no—
407C4—42no—
408C4—44no—
409C4—42no—
410C4—82no—
411C2—28no—
412C2—44no—
413C2C2×#38511yescharacteristic
414C2C4×#12011yescharacteristic
415C2#232211yescharacteristic
416C2D4×C4⋊C412yes—
417C2#236112yes—
418C2#240311yescharacteristic
419C2#240611yescharacteristic
420C2#240911yescharacteristic
421C2#244312yes—
422C2#244912yes—
423C2#257911yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC23×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C23×C4⋊C4C2semidirect
5. C2×#385C2semidirect
6. C2×#385C2semidirect
7. #1444C2semidirect
8. #1439C2semidirect
9. #1436C2semidirect
10. C22⋊#95C2semidirect
11. C22⋊#95C2semidirect
34. #386C4semidirect
71. C23×C4C2×C4semidirect
95. C2×C4⋊C4D4semidirect
111. C2×C22⋊C4C2×C4semidirect
112. C2×C22⋊C4C2×C4semidirect
123. #95C2×C4semidirect
153. C2×D4C4⋊C4semidirect
279. C23C2×C4⋊C4semidirect
354. C2×C4C2×C4⋊C4semidirect
355. C2×C4C4×D4semidirect
378. C22#385semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C23×C4⋊C42
C2×#3862
C2×#3852
#14442
#14392
#14362
C22⋊#952