Finite Group Databasebeta

← #16398

Subgroups — #16398

Order 256.

879 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#399C211yesmaximal, characteristic, complemented
2C2×#386C211yesmaximal, characteristic
3C2×#380C211yesmaximal, characteristic, complemented
4C2×#380C211yesmaximal, characteristic, complemented
5#1445C211yesmaximal, characteristic, complemented
6#1442C211yesmaximal, characteristic, complemented
7#1441C212yesmaximal, complemented
8#1439C211yesmaximal, characteristic, complemented
9#1439C212yesmaximal, complemented
10#1439C212yesmaximal, complemented
11#1438C212yesmaximal
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25#399—24no—
26#386—24no—
27#380—24no—
28#380—24no—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2211yescharacteristic
39C25C2311yescharacteristic
40C25C2311yescharacteristic
41C22×D4—22no—
42C22×D4—22no—
43C22×D4—42no—
44C22×D4—42no—
45C22×D4Q814yes—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—24no—
49C23×C4—42no—
50C23×C4D414yes—
51C23×C4C2311yescharacteristic
52C23×C4C2312yes—
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2312yes—
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C2×C4⋊C4—24no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90#95—22no—
91#95—24no—
92#95—22no—
93#95—22no—
94#95—42no—
95#95—42no—
96#95—42no—
97#95—42no—
98#95—42no—
99#95D414yescomplemented
100C24—21no—
101C24—21no—
102C24—22no—
103C24—22no—
104C24—21no—
105C24—22no—
106C24—22no—
107C24—21no—
108C24—21no—
109C24—21no—
110C24—21no—
111C24—22no—
112C24—21no—
113C24—22no—
114C24—21no—
115C24—22no—
116C24—22no—
117C24—21no—
118C24C2×D412yes—
119C24C2×Q812yes—
120C24C2×Q812yes—
121C24#4112yes—
122C24C2411yescharacteristic
123C2×D4—28no—
124C2×D4—48no—
125C2×D4—44no—
126C2×D4—44no—
127C2×D4—48no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—28no—
144C22×C4—28no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—21no—
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—22no—
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—21no—
175C22×C4—22no—
176C22×C4—44no—
177C22×C4—48no—
178C22×C4—48no—
179C22×C4C2×D414yes—
180C22×C4C2×D412yes—
181C22×C4C2×D412yes—
182C22×C4C2×D412yes—
183C22×C4C2×Q812yes—
184C22×C4#4112yes—
185C22×C4#4114yes—
186C4⋊C4—48no—
187C4⋊C4—44no—
188C4⋊C4—48no—
189C4⋊C4—44no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—48no—
193C22⋊C4—44no—
194C22⋊C4—48no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—24no—
204C23—24no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—22no—
210C23—24no—
211C23—22no—
212C23—24no—
213C23—22no—
214C23—22no—
215C23—24no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—42no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—41no—
236C23—42no—
237C23—42no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23C22⋊Q814yes—
243C23C22×D411yescharacteristic
244C23C22×D411yescharacteristic
245C23C22×Q811yescharacteristic
246C23C2×#4111yescharacteristic
247C23C2×#4111yescharacteristic
248C23C2×#4112yes—
249C23H(5,𝔽2)12yes—
250C23H(5,𝔽2)12yes—
251C23H(5,𝔽2)12yes—
252C23H(5,𝔽2)11yescharacteristic
253C232−1+411yescharacteristic
254D4—416no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—416no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4#12114yes—
308C2×C4#12314yes—
309C22—24no—
310C22—24no—
311C22—24no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—44no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—44no—
323C22—42no—
324C22—44no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—82no—
329C22—82no—
330C22—81no—
331C22C2×#12111yescharacteristic
332C22C2×C22⋊Q811yescharacteristic
333C22C2×#12311yescharacteristic
334C22#53411yescharacteristic
335C22#53511yescharacteristic
336C22#53512yes—
337C22#53511yescharacteristic
338C22#53711yescharacteristic
339C22#53812yes—
340C22#54012yes—
341C22#54012yes—
342C22#54212yes—
343C22#54312yes—
344C22#54612yes—
345C22#54611yescharacteristic
346C22D4×Q811yescharacteristic
347C22#55112yes—
348C22#55211yescharacteristic
349C22#55312yes—
350C22#55411yescharacteristic
351C22#55412yes—
352C22#56012yes—
353C22#56212yes—
354C4—24no—
355C4—24no—
356C4—42no—
357C4—44no—
358C4—81no—
359C4—82no—
360C4—81no—
361C4—82no—
362C4—82no—
363C2—44no—
364C2—82no—
365C2#241211yescharacteristic
366C2#241711yescharacteristic
367C2#248711yescharacteristic
368C2#261911yescharacteristic
369C2#262411yescharacteristic
370C2#262611yescharacteristic
371C2#269212yes—
372C2#270312yes—
373C2#274112yes—
374C2#274812yes—
375C2#281311yescharacteristic
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. C2×#399C2semidirect
3. C2×#380C2semidirect
4. C2×#380C2semidirect
5. #1445C2semidirect
6. #1442C2semidirect
7. #1441C2semidirect
8. #1439C2semidirect
9. #1439C2semidirect
10. #1439C2semidirect
99. #95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3992
C2×#3862
C2×#3802
#14452
#14422
#14412
#14392
#14382