Finite Group Databasebeta

← #24767

Subgroups — #24767

Order 256.

501 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#469C211yesmaximal, characteristic
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4C2×#424C211yesmaximal, characteristic, complemented
5#2620C211yesmaximal, characteristic, complemented
6#2324C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#2055C211yesmaximal, characteristic, complemented
10#2054C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1948C211yesmaximal, characteristic, complemented
13#1894C211yesmaximal, characteristic, complemented
14#1893C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#469—24no—
27#465—24no—
28#463—24no—
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31#424—24no—
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44#396—21no—
45#396—21no—
46#394—21no—
47#394—21no—
48#391C2211yescharacteristic
49#390—21no—
50#390C2211yescharacteristic
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#388—21no—
54#386—21no—
55#385—21no—
56#384C2211yescharacteristic
57#380C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—41no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—41no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—41no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—41no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86#124—22no—
87#124—22no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90#121—22no—
91#121—22no—
92#121—22no—
93#121D414yes—
94C4×D4—22no—
95C4×D4—22no—
96#117—22no—
97#117—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4D411yescharacteristic
121C2×C22⋊C4D411yescharacteristic
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124C2×C42C2311yescharacteristic
125#106—22no—
126#106—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8D414yescomplemented
131#103—22no—
132#103—22no—
133#103—41no—
134#102—22no—
135#102—22no—
136#102—22no—
137#102—22no—
138#102—41no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24—21no—
145C24—21no—
146C24C2×D411yescharacteristic
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×D4—24no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4C2×D412yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D412yes—
190C22×C4#4112yes—
191C22×C4C2411yescharacteristic
192SD16—44no—
193D8—44no—
194M16—44no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—44no—
206C2×C8C2×D412yes—
207C2×C8C2×D412yes—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4C2×D412yes—
224C22⋊C4—24no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C42—21no—
238C42—21no—
239C42—81no—
240C42C2×D412yes—
241C42#4112yes—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23#12112yes—
263C23C22×D411yescharacteristic
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268D4—22no—
269D4—22no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—84no—
277D4—84no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—84no—
313C2×C4#12112yes—
314C2×C4#12112yes—
315C2×C4#12112yes—
316C2×C4C22×D411yescharacteristic
317C2×C4C22×D411yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C2×C4H(5,𝔽2)11yescharacteristic
321C2×C42−1+411yescharacteristic
322C8—22no—
323C8—22no—
324C8—41no—
325C8—41no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22C2×#12111yescharacteristic
348C22C2×#12111yescharacteristic
349C22#53711yescharacteristic
350C22#57611yescharacteristic
351C22#57711yescharacteristic
352C22#57711yescharacteristic
353C22#57711yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—41no—
359C4—41no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4D4211yescharacteristic
366C4#54711yescharacteristic
367C4#54811yescharacteristic
368C4#55311yescharacteristic
369C2—42no—
370C2—82no—
371C2—161no—
372C2#241611yescharacteristic
373C2#304211yescharacteristic
374C2#304511yescharacteristic
375C2#306011yescharacteristic
376C2#306211yescharacteristic
377C2#325711yescharacteristic
378C2#326311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#469C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#465C2semidirect
3. C2×#463C2semidirect
4. C2×#424C2semidirect
5. #2620C2semidirect
6. #2324C2semidirect
7. #2066C2semidirect
8. #2065C2semidirect
9. #2055C2semidirect
10. #2054C2semidirect
11. #1983C2semidirect
12. #1948C2semidirect
13. #1894C2semidirect
14. #1893C2semidirect
15. #1802C2semidirect
130. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4692
C2×#4652
C2×#4632
C2×#4242
#26202
#23242
#20662
#20652
#20552
#20542
#19832
#19482
#18942
#18932
#18022