Finite Group Databasebeta

← #24568

Subgroups — #24568

Order 256.

520 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×#408C211yesmaximal, characteristic, complemented
3#2615C211yesmaximal, characteristic
4#2311C211yesmaximal, characteristic, complemented
5#2077C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1893C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic, complemented
12#1854C211yesmaximal, characteristic, complemented
13#1823C211yesmaximal, characteristic, complemented
14#1809C211yesmaximal, characteristic, complemented
15#1807C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#481—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33#408—24no—
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38#393—21no—
39#392—21no—
40#392—21no—
41#392—21no—
42#391—21no—
43#390—21no—
44#386—21no—
45#386—21no—
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8D411yescharacteristic
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4D411yescharacteristic
72C22×D4D411yescharacteristic
73C22×D4D411yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—41no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89#121—22no—
90#121—22no—
91#121—22no—
92#121—22no—
93#121—41no—
94#121—41no—
95#117—41no—
96#117—41no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
118C2×C42—21no—
119#107—41no—
120#106—22no—
121#106—22no—
122#106—41no—
123#102—22no—
124#102—22no—
125#102—41no—
126#102—41no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8D414yescomplemented
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24C2×D411yescharacteristic
140C24C2×D411yescharacteristic
141C24C2×D411yescharacteristic
142#41—44no—
143#41—44no—
144#41—44no—
145#41—44no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—41no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
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—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D412yes—
200C22×C4C2×D411yescharacteristic
201C22×C4#4112yes—
202C22×C4#4112yes—
203C22×C4#4112yes—
204C22×C4C2411yescharacteristic
205M16—44no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—44no—
217C2×C8C2×D412yes—
218C2×C8C2×D412yes—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C22⋊C4—81no—
245C42—42no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#12112yes—
273C23#12312yes—
274C23C22×D411yescharacteristic
275Q8—44no—
276D4—44no—
277D4—44no—
278D4—44no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—41no—
283D4—84no—
284D4—84no—
285D4—84no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
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—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—84no—
327C2×C4#12112yes—
328C2×C4#12312yes—
329C2×C4C22×D411yescharacteristic
330C2×C4C2×#4111yescharacteristic
331C2×C4C2×#4111yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C2×C4H(5,𝔽2)11yescharacteristic
335C8—22no—
336C8—22no—
337C8—41no—
338C8—41no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22C2×#12111yescharacteristic
368C22C2×#12311yescharacteristic
369C22#53411yescharacteristic
370C22#57611yescharacteristic
371C22#57711yescharacteristic
372C22#57711yescharacteristic
373C22#57711yescharacteristic
374C4—22no—
375C4—22no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4#54611yescharacteristic
385C4#54611yescharacteristic
386C4#55111yescharacteristic
387C4#55111yescharacteristic
388C2—42no—
389C2—42no—
390C2—81no—
391C2—161no—
392C2#241211yescharacteristic
393C2#305711yescharacteristic
394C2#306211yescharacteristic
395C2#309611yescharacteristic
396C2#309911yescharacteristic
397C2#319811yescharacteristic
398C2#320411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2615C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#481C2semidirect
2. C2×#408C2semidirect
4. #2311C2semidirect
5. #2077C2semidirect
6. #2077C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1895C2semidirect
10. #1893C2semidirect
11. #1875C2semidirect
12. #1854C2semidirect
13. #1823C2semidirect
14. #1809C2semidirect
15. #1807C2semidirect
130. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4082
#26152
#23112
#20772
#20342
#20332
#18952
#18932
#18752
#18542
#18232
#18092
#18072