Finite Group Databasebeta

← #14921

Subgroups — #14921

Order 256.

1081 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#386C211yesmaximal, characteristic
2C2×#386C211yesmaximal, characteristic
3C2×#386C211yesmaximal, characteristic
4C2×#385C211yesmaximal, characteristic, complemented
5C2×#380C211yesmaximal, characteristic, complemented
6C22×#95C211yesmaximal, characteristic, complemented
7#1443C212yesmaximal, complemented
8#1441C212yesmaximal, complemented
9#1439C212yesmaximal, complemented
10#1439C212yesmaximal, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C24×C4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2212yes—
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25#386—24no—
26#386—24no—
27#386—24no—
28#385—24no—
29#380—24no—
30C2×#95—24no—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C25C2311yescharacteristic
43C22×D4—22no—
44C22×D4—22no—
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4Q814yes—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—24no—
56C23×C4—24no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—42no—
60C23×C4—42no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4D414yes—
64C23×C4D414yes—
65C23×C4C2311yescharacteristic
66C23×C4C2312yes—
67C23×C4C2311yescharacteristic
68C23×C4C2312yes—
69C23×C4C2312yes—
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2312yes—
73C23×C4C2311yescharacteristic
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
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—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4D414yes—
99#95—24no—
100#95—24no—
101#95—24no—
102#95—24no—
103#95—24no—
104#95—22no—
105#95—24no—
106#95—22no—
107#95—24no—
108#95—42no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—21no—
114C24—21no—
115C24—21no—
116C24—22no—
117C24—22no—
118C24—21no—
119C24—21no—
120C24—21no—
121C24—22no—
122C24—22no—
123C24C2×D412yes—
124C24C2×D412yes—
125C24C2×D412yes—
126C24C2×Q812yes—
127C24C2×Q812yes—
128C24#4112yes—
129C24#4112yes—
130C24#4112yes—
131C24C2411yescharacteristic
132C2×D4—28no—
133C2×D4—28no—
134C2×D4—28no—
135C2×D4—44no—
136C2×D4—44no—
137C2×D4—44no—
138C2×D4—44no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—28no—
143C22×C4—28no—
144C22×C4—28no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—28no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
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—44no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—48no—
201C22×C4—44no—
202C22×C4C2×D412yes—
203C22×C4C2×D414yes—
204C22×C4C2×D414yes—
205C22×C4C2×D412yes—
206C22×C4C2×Q812yes—
207C22×C4#4114yes—
208C22×C4#4114yes—
209C4⋊C4—48no—
210C4⋊C4—48no—
211C22⋊C4—28no—
212C22⋊C4—48no—
213C22⋊C4—48no—
214C22⋊C4—44no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C23—22no—
219C23—22no—
220C23—24no—
221C23—24no—
222C23—22no—
223C23—24no—
224C23—22no—
225C23—24no—
226C23—24no—
227C23—24no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—24no—
233C23—24no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—24no—
238C23—24no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23#12114yes—
258C23#12114yes—
259C23#12114yes—
260C23C22⋊Q814yes—
261C23C22⋊Q814yes—
262C23C22⋊Q814yes—
263C23#12614yes—
264C23C22×D411yescharacteristic
265C23C22×D411yescharacteristic
266C23C22×D411yescharacteristic
267C23C22×Q811yescharacteristic
268C23C2×#4111yescharacteristic
269C23C2×#4111yescharacteristic
270C23C2×#4111yescharacteristic
271C23C2×#4112yes—
272C23C2×#4112yes—
273C23H(5,𝔽2)12yes—
274C23H(5,𝔽2)12yes—
275D4—416no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—416no—
296C2×C4—416no—
297C2×C4—416no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
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—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4#12314yes—
334C22—28no—
335C22—24no—
336C22—24no—
337C22—24no—
338C22—24no—
339C22—24no—
340C22—28no—
341C22—24no—
342C22—28no—
343C22—28no—
344C22—44no—
345C22—42no—
346C22—42no—
347C22—44no—
348C22—42no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—82no—
354C22—82no—
355C22#39618yes—
356C22C2×#12111yescharacteristic
357C22C2×#12111yescharacteristic
358C22C2×#12111yescharacteristic
359C22C2×C22⋊Q811yescharacteristic
360C22C2×C22⋊Q811yescharacteristic
361C22C2×C22⋊Q811yescharacteristic
362C22C2×#12311yescharacteristic
363C22#52511yescharacteristic
364C22C2×#12611yescharacteristic
365C22#52912yes—
366C22#53212yes—
367C22#53412yes—
368C22#53411yescharacteristic
369C22#53511yescharacteristic
370C22#53812yes—
371C22#54612yes—
372C22#54612yes—
373C22#55112yes—
374C22#55112yes—
375C22#55312yes—
376C22#55312yes—
377C22#55412yes—
378C22#55412yes—
379C4—24no—
380C4—44no—
381C4—42no—
382C4—44no—
383C4—42no—
384C4—44no—
385C4—81no—
386C4—81no—
387C4—82no—
388C2—28no—
389C2—82no—
390C2C2×#39611yescharacteristic
391C2#241111yescharacteristic
392C2#241211yescharacteristic
393C2#245012yes—
394C2#248711yescharacteristic
395C2#251012yes—
396C2#254312yes—
397C2#256412yes—
398C2#257911yescharacteristic
399C2#261711yescharacteristic
400C2#285411yescharacteristic
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
4. C2×#385C2semidirect
5. C2×#380C2semidirect
6. C22×#95C2semidirect
7. #1443C2semidirect
8. #1441C2semidirect
9. #1439C2semidirect
10. #1439C2semidirect
11. C22⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3862
C2×#3852
C2×#3802
C22×#952
#14432
#14412
#14392
C22⋊#952