Finite Group Databasebeta

← #13996

Subgroups — #13996

Order 256.

1237 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123C211yesmaximal, characteristic
2C2×#386C211yesmaximal, characteristic
3C2×#386C211yesmaximal, characteristic
4C2×#385C211yesmaximal, characteristic, complemented
5C2×#381C211yesmaximal, characteristic, complemented
6C2×C4×C22⋊C4C211yesmaximal, characteristic, complemented
7C22×#95C211yesmaximal, characteristic, complemented
8#1445C212yesmaximal, complemented
9#1439C212yesmaximal, complemented
10#1437C212yesmaximal, complemented
11C4⋊#95C212yesmaximal, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C24×C4C2211yescharacteristic
16C2×#123—24no—
17C2×#123—24no—
18C2×#123C418yescomplemented
19C22×C4⋊C4C2212yes—
20C22×C4⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28C22×C42C2212yes—
29#386—24no—
30#386—24no—
31#385—24no—
32#381—24no—
33C4×C22⋊C4—24no—
34C2×#95—24no—
35C2×#95C2211yescharacteristic
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C2×#95C2212yes—
42C2×#95C2212yes—
43C2×#95C2211yescharacteristic
44C25C2311yescharacteristic
45C25C2311yescharacteristic
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4—22no—
52C22×D4C2×C414yes—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—42no—
67C23×C4—42no—
68C23×C4C2×C414yes—
69C23×C4D414yes—
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2312yes—
73C23×C4C2311yescharacteristic
74C23×C4C2312yes—
75C23×C4C2311yescharacteristic
76C23×C4C2312yes—
77C23×C4C2312yes—
78C23×C4C2311yescharacteristic
79#123—232no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4C2×C418yescomplemented
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4C2×C414yes—
101C2×C22⋊C4C2×C418yescomplemented
102C2×C22⋊C4D414yes—
103C2×C22⋊C4D414yes—
104C2×C22⋊C4D414yes—
105C2×C42—24no—
106C2×C42—24no—
107C2×C42—42no—
108#95—24no—
109#95—24no—
110#95—24no—
111#95—24no—
112#95—24no—
113#95—24no—
114#95—24no—
115#95—22no—
116#95—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—22no—
126C24—22no—
127C24—21no—
128C24—21no—
129C24—22no—
130C24—22no—
131C24C22×C412yes—
132C24C22×C412yes—
133C24C2×D412yes—
134C24C2×D412yes—
135C24C2×D412yes—
136C24C2×D412yes—
137C24#4112yes—
138C24#4112yes—
139C24C2411yescharacteristic
140C2×D4—28no—
141C2×D4—28no—
142C2×D4—28no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4—44no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—28no—
155C22×C4—28no—
156C22×C4—28no—
157C22×C4—28no—
158C22×C4—28no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—28no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—48no—
201C22×C4—48no—
202C22×C4—48no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4C22×C414yes—
210C22×C4C22×C412yes—
211C22×C4C22×C414yes—
212C22×C4C2×D414yes—
213C22×C4C2×D412yes—
214C22×C4C2×D414yes—
215C22×C4C2×D414yes—
216C22×C4C2×D412yes—
217C22×C4#4114yes—
218C22×C4#4114yes—
219C4⋊C4—216no—
220C4⋊C4—48no—
221C22⋊C4—28no—
222C22⋊C4—216no—
223C22⋊C4—48no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4C22⋊C4116yescomplemented
228C42—48no—
229C23—24no—
230C23—24no—
231C23—22no—
232C23—22no—
233C23—24no—
234C23—22no—
235C23—22no—
236C23—24no—
237C23—24no—
238C23—24no—
239C23—24no—
240C23—24no—
241C23—24no—
242C23—24no—
243C23—24no—
244C23—24no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—24no—
249C23—24no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23C2×C22⋊C414yes—
269C23C4×D414yes—
270C23C4×D414yes—
271C23#12114yes—
272C23#12114yes—
273C23#12414yes—
274C23#12714yes—
275C23C23×C411yescharacteristic
276C23C22×D411yescharacteristic
277C23C22×D411yescharacteristic
278C23C22×D411yescharacteristic
279C23C22×D411yescharacteristic
280C23C2×#4111yescharacteristic
281C23C2×#4111yescharacteristic
282C23C2×#4112yes—
283C23C2×#4112yes—
284C23H(5,𝔽2)12yes—
285C232−1+412yes—
286D4—416no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
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—416no—
313C2×C4—416no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—416no—
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×C4C2×C22⋊C418yes—
344C2×C4#12314yescomplemented
345C22—24no—
346C22—28no—
347C22—28no—
348C22—24no—
349C22—24no—
350C22—28no—
351C22—28no—
352C22—24no—
353C22—24no—
354C22—24no—
355C22—42no—
356C22—44no—
357C22—44no—
358C22—42no—
359C22—42no—
360C22—81no—
361C22—81no—
362C22—82no—
363C22—82no—
364C22—81no—
365C22—81no—
366C22#39018yes—
367C22C22×C22⋊C411yescharacteristic
368C22C2×C4×D411yescharacteristic
369C22C2×C4×D411yescharacteristic
370C22C4×#4112yes—
371C22#52012yes—
372C22C2×#12111yescharacteristic
373C22C2×#12111yescharacteristic
374C22C2×#12311yescharacteristic
375C22#52511yescharacteristic
376C22C2×#12411yescharacteristic
377C22#52912yes—
378C22C2×#12711yescharacteristic
379C22#53411yescharacteristic
380C22#53611yescharacteristic
381C22#53912yes—
382C22#54612yes—
383C22#54612yes—
384C22#54612yes—
385C22#54712yes—
386C22#54712yes—
387C22#54712yes—
388C22#55112yes—
389C22#55212yes—
390C4—28no—
391C4—24no—
392C4—42no—
393C4—44no—
394C4—44no—
395C4—44no—
396C4—42no—
397C4—81no—
398C4—81no—
399C2—28no—
400C2—82no—
401C2C2×#39011yescharacteristic
402C2C4×#12311yescharacteristic
403C2#232211yescharacteristic
404C2#233912yes—
405C2#235912yes—
406C2#241211yescharacteristic
407C2#241311yescharacteristic
408C2#242512yes—
409C2#248812yes—
410C2#250111yescharacteristic
411C2#253911yescharacteristic
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
4. C2×#385C2semidirect
5. C2×#381C2semidirect
6. C2×C4×C22⋊C4C2semidirect
7. C22×#95C2semidirect
8. #1445C2semidirect
9. #1439C2semidirect
10. #1437C2semidirect
11. C4⋊#95C2semidirect
18. C2×#123C4semidirect
85. C2×C4⋊C4C2×C4semidirect
101. C2×C22⋊C4C2×C4semidirect
227. C22⋊C4C22⋊C4semidirect
344. C2×C4#123semidirect

Maximal subgroups

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

subgroupindex
C22×#1232
C2×#3862
C2×#3852
C2×#3812
C2×C4×C22⋊C42
C22×#952
#14452
#14392
#14372
C4⋊#952