Finite Group Databasebeta

← #14623

Subgroups — #14623

Order 256.

955 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#392C211yesmaximal, characteristic
2C2×#390C211yesmaximal, characteristic, complemented
3C2×#388C211yesmaximal, characteristic, complemented
4#1442C211yesmaximal, characteristic, complemented
5#1440C212yesmaximal, complemented
6#1439C211yesmaximal, characteristic, complemented
7#1439C212yesmaximal, complemented
8#1438C212yesmaximal, complemented
9#1438C212yesmaximal, complemented
10#1438C211yesmaximal, characteristic, complemented
11#1434C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C23×D4C2212yes—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C42C2211yescharacteristic
26#392C418yescomplemented
27#390—24no—
28#388—24no—
29C2×#95C2211yescharacteristic
30C2×#95C2211yescharacteristic
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C25C2311yescharacteristic
39C25C2312yes—
40C22×D4—22no—
41C22×D4—22no—
42C22×D4—42no—
43C22×D4—42no—
44C22×D4—42no—
45C22×D4—42no—
46C22×D4—42no—
47C22×D4—42no—
48C22×D4—42no—
49C22×D4—42no—
50C22×D4C2×C414yes—
51C22×D4C2×C418yes—
52C23×C4C2311yescharacteristic
53C23×C4C2312yes—
54C23×C4C2311yescharacteristic
55C23×C4C2312yes—
56C23×C4C2311yescharacteristic
57C23×C4C2312yes—
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—42no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
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⋊C4C2×C414yes—
89C2×C22⋊C4C2×C418yes—
90C2×C42—42no—
91C2×C42D414yescomplemented
92#95—22no—
93#95—22no—
94#95—22no—
95#95—42no—
96#95—42no—
97#95—42no—
98#95—42no—
99#95—42no—
100#95C2×C414yes—
101C24—21no—
102C24—22no—
103C24—21no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—21no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—22no—
125C24C22×C412yes—
126C24C22×C414yes—
127C24C2411yescharacteristic
128C2×D4—28no—
129C2×D4—48no—
130C2×D4—48no—
131C2×D4—48no—
132C2×D4—48no—
133C2×D4—44no—
134C2×D4—44no—
135C2×D4—48no—
136C2×D4—48no—
137C2×D4—48no—
138C2×D4—48no—
139C2×D4—48no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
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—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4C22×C414yes—
185C22×C4C22×C412yes—
186C22×C4C22×C412yes—
187C22×C4C2×D412yes—
188C22×C4C2×D412yes—
189C22×C4C2×D412yes—
190C22×C4#4112yes—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C22⋊C4—28no—
194C22⋊C4—48no—
195C22⋊C4—44no—
196C22⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—44no—
199C22⋊C4—48no—
200C22⋊C4—48no—
201C22⋊C4—48no—
202C42—44no—
203C23—22no—
204C23—24no—
205C23—24no—
206C23—24no—
207C23—22no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—24no—
213C23—22no—
214C23—24no—
215C23—41no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—41no—
220C23—42no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—41no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—41no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23C23×C411yescharacteristic
262C23C22×D411yescharacteristic
263C23C2×#4111yescharacteristic
264C23H(5,𝔽2)12yes—
265C23H(5,𝔽2)12yes—
266C23H(5,𝔽2)12yes—
267C23H(5,𝔽2)12yes—
268C23H(5,𝔽2)11yescharacteristic
269C23H(5,𝔽2)11yescharacteristic
270C23H(5,𝔽2)11yescharacteristic
271C232−1+411yescharacteristic
272D4—416no—
273D4—432no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—41no—
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—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4C4×D414yes—
325C22—44no—
326C22—42no—
327C22—42no—
328C22—44no—
329C22—42no—
330C22—44no—
331C22—44no—
332C22—44no—
333C22—44no—
334C22—44no—
335C22—44no—
336C22—44no—
337C22—82no—
338C22—81no—
339C22—81no—
340C22—82no—
341C22—82no—
342C22—82no—
343C22—82no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22—81no—
348C22—82no—
349C22—81no—
350C22—82no—
351C22C2×C4×D411yescharacteristic
352C22#51812yes—
353C22#51811yescharacteristic
354C22#51812yes—
355C22#52011yescharacteristic
356C22#53411yescharacteristic
357C22#53511yescharacteristic
358C22#53511yescharacteristic
359C22#53511yescharacteristic
360C22#53511yescharacteristic
361C22#53711yescharacteristic
362C22#53812yes—
363C22#54011yescharacteristic
364C22#54012yes—
365C22#54211yescharacteristic
366C22#56012yes—
367C22#56012yes—
368C22#56012yes—
369C22#56012yes—
370C22#56012yes—
371C22H(3,𝔽4)12yes—
372C22#56212yes—
373C22#56212yes—
374C4—24no—
375C4—42no—
376C4—44no—
377C4—42no—
378C4—42no—
379C4—82no—
380C4—81no—
381C4—82no—
382C4—81no—
383C2—84no—
384C2—82no—
385C2#232111yescharacteristic
386C2#232411yescharacteristic
387C2#232811yescharacteristic
388C2#237512yes—
389C2#237712yes—
390C2#281311yescharacteristic
391C2#281511yescharacteristic
392C2#281811yescharacteristic
393C2#281911yescharacteristic
394C2#282812yes—
395C2#282912yes—
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC2×#392C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#390C2semidirect
3. C2×#388C2semidirect
4. #1442C2semidirect
5. #1440C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
8. #1438C2semidirect
9. #1438C2semidirect
10. #1438C2semidirect
11. #1434C2semidirect
26. #392C4semidirect
91. C2×C42D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3922
C2×#3902
C2×#3882
#14422
#14402
#14392
#14382
#14342