Finite Group Databasebeta

← #31061

Subgroups — #31061

Order 256.

593 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#452C211yesmaximal, characteristic, complemented
2C2×#452C211yesmaximal, characteristic, complemented
3C2×#451C211yesmaximal, characteristic, complemented
4C2×#448C211yesmaximal, characteristic, complemented
5#2409C211yesmaximal, characteristic, complemented
6#2407C211yesmaximal, characteristic
7#2056C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2036C211yesmaximal, characteristic, complemented
10#2035C211yesmaximal, characteristic, complemented
11#2035C211yesmaximal, characteristic, complemented
12#2033C211yesmaximal, characteristic, complemented
13#1969C211yesmaximal, characteristic, complemented
14#1968C211yesmaximal, characteristic, complemented
15#1954C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27#452—24no—
28#452—24no—
29#451—24no—
30#448—24no—
31C2×#107C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C4×C8C2211yescharacteristic
47#395—21no—
48#394—21no—
49#393—21no—
50#392—21no—
51#391C2211yescharacteristic
52#391C2211yescharacteristic
53#391C2211yescharacteristic
54#390C2211yescharacteristic
55#386—21no—
56#386—21no—
57#385—21no—
58#385—21no—
59#380C2211yescharacteristic
60#380C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41—41no—
68C2×#41—41no—
69C22×Q8—21no—
70C22×Q8—22no—
71C22×Q8—22no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8C2311yescharacteristic
77C22×Q8C2311yescharacteristic
78C22×Q8C2311yescharacteristic
79C22×D4—21no—
80C22×D4—21no—
81C22×D4D411yescharacteristic
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C2×Q16—22no—
88C2×Q16—22no—
89C2×Q16—41no—
90C2×Q16—41no—
91C2×Q16—41no—
92C2×Q16D414yescomplemented
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—41no—
96C2×SD16—41no—
97C2×SD16—41no—
98C2×SD16D414yescomplemented
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109#121—22no—
110#121—41no—
111C4×D4—41no—
112C4×D4—41no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4D411yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4D411yescharacteristic
130C2×C22⋊C4D411yescharacteristic
131C2×C42C2311yescharacteristic
132#107—41no—
133#106—41no—
134C4⋊C8—41no—
135#103—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#103—22no—
141#102—22no—
142#102—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C4×C8—41no—
148#95—21no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156C24—21no—
157C24C2×D411yescharacteristic
158C24C2×D411yescharacteristic
159#41—44no—
160#41—44no—
161#41—44no—
162#41—44no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—44no—
184C2×Q8—44no—
185C2×Q8C2×D412yes—
186C2×Q8C2×D412yes—
187C2×Q8C2×D412yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—81no—
202C2×D4C2×D412yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2×D411yescharacteristic
232C22×C4C2411yescharacteristic
233Q16—44no—
234Q16—44no—
235SD16—44no—
236SD16—44no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8C2×D412yes—
248C2×C8C2×D412yes—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—81no—
270C22⋊C4—81no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—22no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#13512yes—
294C23C22×D411yescharacteristic
295Q8—22no—
296Q8—22no—
297Q8—22no—
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8—48no—
302Q8—41no—
303Q8—41no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8—41no—
308D4—22no—
309D4—22no—
310D4—44no—
311D4—48no—
312D4—41no—
313D4—41no—
314D4—84no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4#13712yes—
358C2×C4C22×D411yescharacteristic
359C2×C4C22×D411yescharacteristic
360C2×C4H(5,𝔽2)11yescharacteristic
361C2×C4H(5,𝔽2)11yescharacteristic
362C2×C4H(5,𝔽2)11yescharacteristic
363C2×C4H(5,𝔽2)11yescharacteristic
364C8—22no—
365C8—22no—
366C8—41no—
367C8—41no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—41no—
382C22—42no—
383C22—41no—
384C22—41no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22#53411yescharacteristic
390C22D4211yescharacteristic
391C22D4211yescharacteristic
392C22C2×#13511yescharacteristic
393C22C2×#13711yescharacteristic
394C22#57711yescharacteristic
395C22#57811yescharacteristic
396C4—22no—
397C4—41no—
398C4—41no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—41no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4#53411yescharacteristic
408C4#53411yescharacteristic
409C4D4211yescharacteristic
410C4H(3,𝔽4)11yescharacteristic
411C2—42no—
412C2—81no—
413C2—82no—
414C2#267911yescharacteristic
415C2#319011yescharacteristic
416C2#320111yescharacteristic
417C2#327611yescharacteristic
418C2#327811yescharacteristic
419C2#328211yescharacteristic
420C2#328511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2407C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#452C2semidirect
2. C2×#452C2semidirect
3. C2×#451C2semidirect
4. C2×#448C2semidirect
5. #2409C2semidirect
7. #2056C2semidirect
8. #2055C2semidirect
9. #2036C2semidirect
10. #2035C2semidirect
11. #2035C2semidirect
12. #2033C2semidirect
13. #1969C2semidirect
14. #1968C2semidirect
15. #1954C2semidirect
92. C2×Q16D4semidirect
98. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4522
C2×#4512
C2×#4482
#24092
#24072
#20562
#20552
#20362
#20352
#20332
#19692
#19682
#19542