Finite Group Databasebeta

← #14759

Subgroups — #14759

Order 256.

1207 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#393C211yesmaximal, characteristic, complemented
3C2×#386C211yesmaximal, characteristic
4C2×#386C212yesmaximal
5C22×#95C211yesmaximal, characteristic, complemented
6#1446C211yesmaximal, characteristic, complemented
7#1442C212yesmaximal, complemented
8#1442C211yesmaximal, characteristic, complemented
9#1439C211yesmaximal, characteristic, complemented
10#1439C212yesmaximal, complemented
11C22⋊#95C212yesmaximal, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C24×C4C2212yes—
15C24×C4C2211yescharacteristic
16C2×C22⋊Q8—24no—
17C2×C22⋊Q8—28no—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2212yes—
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26#393—24no—
27#386—24no—
28#386—28no—
29C2×#95—24no—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2211yescharacteristic
41C25C2311yescharacteristic
42C25C2311yescharacteristic
43C22×Q8—24no—
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—42no—
48C22×D4—22no—
49C22×D4—24no—
50C22×D4—24no—
51C22×D4—22no—
52C22×D4Q814yescomplemented
53C23×C4—24no—
54C23×C4—24no—
55C23×C4—24no—
56C23×C4—22no—
57C23×C4—24no—
58C23×C4—22no—
59C23×C4—24no—
60C23×C4—24no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4—42no—
64C23×C4D414yes—
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2312yes—
68C23×C4C2312yes—
69C23×C4C2311yescharacteristic
70C23×C4C2312yes—
71C23×C4C2311yescharacteristic
72C23×C4C2312yes—
73C23×C4C2311yescharacteristic
74C22⋊Q8—232no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4D418yes—
96#95—24no—
97#95—22no—
98#95—24no—
99#95—24no—
100#95—22no—
101#95—22no—
102#95—24no—
103#95—24no—
104#95—42no—
105#95—42no—
106#95D414yescomplemented
107C24—21no—
108C24—21no—
109C24—22no—
110C24—22no—
111C24—21no—
112C24—21no—
113C24—21no—
114C24—21no—
115C24—22no—
116C24—21no—
117C24—22no—
118C24—22no—
119C24—21no—
120C24—22no—
121C24—21no—
122C24—21no—
123C24C2×D412yes—
124C24C2×D414yes—
125C24C2×Q812yes—
126C24C2×Q812yes—
127C24#4114yes—
128C24#4112yes—
129C24C2411yescharacteristic
130C2×Q8—28no—
131C2×Q8—48no—
132C2×Q8—48no—
133C2×Q8—44no—
134C2×Q8—44no—
135C2×D4—28no—
136C2×D4—216no—
137C2×D4—44no—
138C2×D4—44no—
139C2×D4—48no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—216no—
143C22×C4—28no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—28no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—44no—
188C22×C4—48no—
189C22×C4—48no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4C2×D414yes—
197C22×C4C2×D414yes—
198C22×C4C2×D414yes—
199C22×C4C2×D412yes—
200C22×C4C2×D412yes—
201C22×C4C2×D412yes—
202C22×C4C2×Q812yes—
203C22×C4#4114yes—
204C22×C4#4112yes—
205C4⋊C4—28no—
206C4⋊C4—216no—
207C22⋊C4—216no—
208C22⋊C4—48no—
209C22⋊C4—48no—
210C22⋊C4—44no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C23—22no—
214C23—24no—
215C23—24no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—24no—
220C23—24no—
221C23—24no—
222C23—22no—
223C23—24no—
224C23—24no—
225C23—24no—
226C23—22no—
227C23—24no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—22no—
232C23—24no—
233C23—22no—
234C23—24no—
235C23—22no—
236C23—41no—
237C23—41no—
238C23—42no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23#12014yes—
255C23C22⋊Q814yes—
256C23C22⋊Q818yes—
257C23#12414yes—
258C23#12418yes—
259C23C22×D411yescharacteristic
260C23C22×D412yes—
261C23C22×D411yescharacteristic
262C23C22×Q811yescharacteristic
263C23C2×#4111yescharacteristic
264C23C2×#4112yes—
265C23C2×#4111yescharacteristic
266C23C2×#4112yes—
267C23H(5,𝔽2)12yes—
268C23H(5,𝔽2)11yescharacteristic
269C232−1+411yescharacteristic
270Q8—416no—
271D4—416no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—416no—
294C2×C4—432no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4#12114yes—
327C22—24no—
328C22—24no—
329C22—24no—
330C22—24no—
331C22—24no—
332C22—24no—
333C22—28no—
334C22—28no—
335C22—28no—
336C22—28no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—44no—
341C22—44no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—82no—
347C22—82no—
348C22#39318yescomplemented
349C22C2×#12011yescharacteristic
350C22C2×#12111yescharacteristic
351C22C2×C22⋊Q811yescharacteristic
352C22C2×C22⋊Q812yes—
353C22#52512yes—
354C22C2×#12411yescharacteristic
355C22C2×#12412yes—
356C22#52912yes—
357C22#53211yescharacteristic
358C22#53411yescharacteristic
359C22#53711yescharacteristic
360C22#53812yes—
361C22#53912yes—
362C22D4212yes—
363C22#54612yes—
364C22#54611yescharacteristic
365C22#54612yes—
366C22#54712yes—
367C22D4×Q811yescharacteristic
368C22#55112yes—
369C22#55211yescharacteristic
370C22#55412yes—
371C22#55411yescharacteristic
372C4—24no—
373C4—42no—
374C4—42no—
375C4—44no—
376C4—44no—
377C4—44no—
378C4—44no—
379C4—81no—
380C4—81no—
381C2—28no—
382C2—82no—
383C2C2×#39311yescharacteristic
384C2#241011yescharacteristic
385C2#241711yescharacteristic
386C2#243011yescharacteristic
387C2#244912yes—
388C2#245012yes—
389C2#247212yes—
390C2#248812yes—
391C2#256411yescharacteristic
392C2#263911yescharacteristic
393C2#285011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC23×D4C22
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×C22⋊Q8C2semidirect
2. C2×#393C2semidirect
5. C22×#95C2semidirect
6. #1446C2semidirect
7. #1442C2semidirect
8. #1442C2semidirect
9. #1439C2semidirect
10. #1439C2semidirect
11. C22⋊#95C2semidirect
52. C22×D4Q8semidirect
106. #95D4semidirect
348. C22#393semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#3932
C2×#3862
C22×#952
#14462
#14422
#14392
C22⋊#952