Finite Group Databasebeta

← #24756

Subgroups — #24756

Order 256.

535 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#469C211yesmaximal, characteristic
2C2×#468C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic, complemented
4C2×#423C211yesmaximal, characteristic, complemented
5#2850C211yesmaximal, characteristic, complemented
6C4×#121C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2066C211yesmaximal, characteristic, complemented
9#2055C211yesmaximal, characteristic, complemented
10#2054C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1949C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic, complemented
14#1894C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4C2211yescharacteristic
26C22×C42C2211yescharacteristic
27#469—24no—
28#468—24no—
29#463—24no—
30C2×#107C2211yescharacteristic
31C2×#106C2211yescharacteristic
32#423—24no—
33C2×C4⋊C8C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#389C2211yescharacteristic
49#388—21no—
50#388—21no—
51#388—21no—
52#386—21no—
53#386—21no—
54#386—21no—
55#382C2211yescharacteristic
56#380C2211yescharacteristic
57C4×C4⋊C4C2211yescharacteristic
58C4×C22⋊C4—21no—
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4D411yescharacteristic
70C23×C4D411yescharacteristic
71C23×C4C2311yescharacteristic
72C2×SD16—22no—
73C2×SD16—22no—
74C2×SD16—41no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—41no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—41no—
83C2×M16—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88#124—22no—
89#124—22no—
90C22⋊Q8—22no—
91C22⋊Q8—22no—
92#121—22no—
93#121—22no—
94#121—22no—
95#121D414yes—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—44no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
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×C42—22no—
118C2×C42—22no—
119C2×C42—21no—
120C2×C42D411yescharacteristic
121C2×C42D411yescharacteristic
122C2×C42C2311yescharacteristic
123C2×C42C2311yescharacteristic
124#107—22no—
125#106—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8D414yescomplemented
130#103—22no—
131#103—22no—
132#103—41no—
133#102—22no—
134#102—22no—
135#102—22no—
136#102—22no—
137#102—41no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145C24—21no—
146C24—21no—
147C24C2×D411yescharacteristic
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×D4—24no—
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—21no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4C2×D412yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D412yes—
196C22×C4C2×D411yescharacteristic
197C22×C4#4112yes—
198C22×C4C2411yescharacteristic
199SD16—44no—
200D8—44no—
201M16—44no—
202M16—44no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8C2×D412yes—
214C2×C8C2×D412yes—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4C2×D412yes—
227C22⋊C4—24no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C42—22no—
239C42—22no—
240C42—21no—
241C42—21no—
242C42—42no—
243C42C2×D412yes—
244C42#4112yes—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23#12112yes—
266C23C22×D411yescharacteristic
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271D4—22no—
272D4—22no—
273D4—48no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—84no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—44no—
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—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—84no—
317C2×C4#12112yes—
318C2×C4#12112yes—
319C2×C4#12112yes—
320C2×C4AGL(1,ℤ/8)12yes—
321C2×C4C22×D411yescharacteristic
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C42−1+411yescharacteristic
327C8—22no—
328C8—22no—
329C8—41no—
330C8—41no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22C2×#12111yescharacteristic
353C22C2×#12111yescharacteristic
354C22#53711yescharacteristic
355C22C2×AGL(1,ℤ/8)11yescharacteristic
356C22#57511yescharacteristic
357C22#57511yescharacteristic
358C22#57511yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—42no—
364C4—41no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4D4211yescharacteristic
371C4#54711yescharacteristic
372C4#54811yescharacteristic
373C4#55311yescharacteristic
374C2—42no—
375C2—82no—
376C2—161no—
377C2#241611yescharacteristic
378C2#303611yescharacteristic
379C2#304011yescharacteristic
380C2#305411yescharacteristic
381C2#305511yescharacteristic
382C2#325111yescharacteristic
383C2#325611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#469C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#468C2semidirect
3. C2×#463C2semidirect
4. C2×#423C2semidirect
5. #2850C2semidirect
6. C4×#121C2semidirect
7. #2067C2semidirect
8. #2066C2semidirect
9. #2055C2semidirect
10. #2054C2semidirect
11. #1983C2semidirect
12. #1949C2semidirect
13. #1895C2semidirect
14. #1894C2semidirect
15. #1804C2semidirect
129. 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×#4682
C2×#4632
C2×#4232
#28502
C4×#1212
#20672
#20662
#20552
#20542
#19832
#19492
#18952
#18942
#18042