Finite Group Databasebeta

← #33347

Subgroups — #33347

Order 256.

618 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3449C211yesmaximal, characteristic
2C2×#413C211yesmaximal, characteristic, complemented
3C2×#410C211yesmaximal, characteristic, complemented
4#2123C212yesmaximal, complemented
5#2122C212yesmaximal, complemented
6#2121C214yesmaximal, complemented
7#2120C214yesmaximal, complemented
8C22×#41C411yescharacteristic, complemented
9C22×#41C2211yescharacteristic
10C22×M16—21no—
11#537—21no—
12#537—21no—
13#536—21no—
14#536C411yescharacteristic, complemented
15#536C2211yescharacteristic
16#535—21no—
17#535C411yescharacteristic, complemented
18#535C2211yescharacteristic
19#534C412yescomplemented
20#534C2212yes—
21#532—21no—
22#532—21no—
23#525—21no—
24#525—21no—
25#525C412yescomplemented
26#525C2212yes—
27C2×#117—21no—
28C2×#117C411yescharacteristic, complemented
29#413—21no—
30#413—21no—
31#413C2212yes—
32#413C2212yes—
33C2×#101—21no—
34C2×#100C2212yes—
35#410—21no—
36#410—21no—
37#410C2212yes—
38#410C2212yes—
39C2×#99—21no—
40C2×#99C2212yes—
41#352C2218yescomplemented
42AΣL(1,𝔽16)C2218yescomplemented
43C2×#41—21no—
44C2×#41—21no—
45C2×#41—21no—
46C2×#41—21no—
47C2×#41—41no—
48C2×#41—41no—
49C2×#41—41no—
50C2×#41—41no—
51C2×#41—41no—
52C2×#41—41no—
53C2×#41—41no—
54C2×#41—41no—
55C2×#41C2×C411yescharacteristic
56C2×#41C2×C411yescharacteristic
57C2×#41D411yescharacteristic
58C2×#41D411yescharacteristic
59C2×#41D411yescharacteristic
60C2×#41D411yescharacteristic
61C2×#41C2311yescharacteristic
62C2×#41C2311yescharacteristic
63C22×Q8C2×C411yescharacteristic, complemented
64C22×Q8D411yescharacteristic
65C22×D4C2×C411yescharacteristic
66C22×D4C2×C412yes—
67C22×D4C2×C411yescharacteristic, complemented
68C22×D4D411yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4C2×C411yescharacteristic
71C23×C4C2×C412yescomplemented
72C23×C4D411yescharacteristic
73C23×C4D411yescharacteristic
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16—41no—
80C22×C8—41no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83#127—21no—
84#127—21no—
85#124—21no—
86#124—21no—
87#124—21no—
88#124—21no—
89#123—22no—
90#123—22no—
91#123C2×C412yes—
92#123C2×C412yescomplemented
93#123C2×C412yescomplemented
94#123C2312yes—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C22⋊Q8—21no—
98C22⋊Q8—21no—
99C22⋊Q8—21no—
100C22⋊Q8—21no—
101#121—22no—
102#121—22no—
103#121—22no—
104#121—22no—
105#121—22no—
106#121—22no—
107#121—21no—
108#121—21no—
109#121—21no—
110#121—21no—
111#120—22no—
112#120—22no—
113#120C2×C412yes—
114#120C2×C412yescomplemented
115#120C2×C412yescomplemented
116#120C2312yes—
117C4×D4—22no—
118C4×D4—22no—
119C4×D4—22no—
120C4×D4—22no—
121#117—21no—
122#117—21no—
123#117—22no—
124#117—22no—
125#117—41no—
126#117—41no—
127#117C2×C412yescomplemented
128#117C2×C411yescharacteristic, complemented
129#117C2×C411yescharacteristic, complemented
130C2×C4⋊C4—21no—
131C2×C4⋊C4—41no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4C2×C412yes—
134C2×C42—21no—
135C2×C42—41no—
136#101—22no—
137#101—22no—
138#100—22no—
139#100C2314yes—
140#99—22no—
141#99—22no—
142#99—22no—
143#99C2314yes—
144C24C22⋊C411yescharacteristic
145C24C22×C411yescharacteristic
146C24C22×C412yes—
147C24C2×D411yescharacteristic
148#41—41no—
149#41—41no—
150#41—41no—
151#41—41no—
152#41—41no—
153#41—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159#41—41no—
160#41—41no—
161#41—41no—
162#41—41no—
163#41—41no—
164#41—42no—
165#41—42no—
166#41—42no—
167#41—42no—
168#41—42no—
169#41—42no—
170#41—41no—
171#41—41no—
172#41—41no—
173#41—41no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C22⋊C411yescharacteristic
181C2×Q8C22⋊C411yescharacteristic
182C2×Q8C2×D411yescharacteristic
183C2×Q8C2×D411yescharacteristic
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4C22⋊C411yescharacteristic
203C2×D4C22⋊C411yescharacteristic
204C2×D4C22⋊C411yescharacteristic
205C2×D4C22⋊C411yescharacteristic
206C2×D4C22×C411yescharacteristic
207C2×D4C22×C411yescharacteristic
208C2×D4C22×C411yescharacteristic
209C2×D4C2×D411yescharacteristic
210C2×D4C2×D411yescharacteristic
211C2×D4C2×D411yescharacteristic
212C2×D4C2×D411yescharacteristic
213C2×D4C2411yescharacteristic
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—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4C22⋊C411yescharacteristic
233C22×C4C22⋊C411yescharacteristic
234C22×C4C22⋊C411yescharacteristic
235C22×C4C22⋊C411yescharacteristic
236C22×C4C22⋊C411yescharacteristic
237C22×C4C22⋊C411yescharacteristic
238C22×C4C22⋊C411yescharacteristic
239C22×C4C22⋊C411yescharacteristic
240C22×C4C22⋊C411yescharacteristic
241C22×C4C22×C411yescharacteristic
242C22×C4C22×C411yescharacteristic
243C22×C4C22×C412yes—
244C22×C4C2×D411yescharacteristic
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249M16—24no—
250M16—42no—
251M16—42no—
252M16—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—22no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—24no—
275C22⋊C4—42no—
276C22⋊C4C22×C412yes—
277C22⋊C4C22×C412yes—
278C42—22no—
279C42—21no—
280C42—21no—
281C42—42no—
282C42—42no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—22no—
290C23—42no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—82no—
295C23C2×C22⋊C411yescharacteristic
296C23C2×C22⋊C411yescharacteristic
297C23C2×C22⋊C411yescharacteristic
298C23C2×C22⋊C411yescharacteristic
299C23C2×C22⋊C411yescharacteristic
300C23C23×C411yescharacteristic
301C23C22×D411yescharacteristic
302Q8—41no—
303Q8—41no—
304Q8—41no—
305Q8—41no—
306Q8—42no—
307Q8—41no—
308Q8—41no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—42no—
322D4—44no—
323D4—42no—
324D4—42no—
325D4—41no—
326D4—41no—
327D4—82no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—82no—
370C2×C4—84no—
371C2×C4#9911yescharacteristic
372C2×C4#9911yescharacteristic
373C2×C4#9911yescharacteristic
374C2×C4#9911yescharacteristic
375C2×C4C2×C22⋊C411yescharacteristic
376C2×C4C2×C22⋊C411yescharacteristic
377C2×C4C2×C22⋊C411yescharacteristic
378C2×C4C2×C22⋊C411yescharacteristic
379C2×C4C2×C22⋊C411yescharacteristic
380C2×C4C2×C22⋊C411yescharacteristic
381C2×C4C2×C22⋊C411yescharacteristic
382C2×C4C22×D411yescharacteristic
383C8—44no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—84no—
407C22C2×#9911yescharacteristic
408C22C2×#9911yescharacteristic
409C22C22×C22⋊C411yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—84no—
420C4C2×#9911yescharacteristic
421C4C2×#9911yescharacteristic
422C4C2×#9911yescharacteristic
423C4C2×#9911yescharacteristic
424C2—21no—
425C2—21no—
426C2—21no—
427C2—41no—
428C2—41no—
429C2—41no—
430C2—41no—
431C2—41no—
432C2—41no—
433C2—82no—
434C2C22×#9911yescharacteristic
subgroupG/H
centerC4C2×#99
commutatorC23C23×C4
Frattini, cosocleC2×D4C24
socleC2C22×#99
prime radical#3449C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G
LECSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#413C2semidirect
3. C2×#410C2semidirect
4. #2123C2semidirect
5. #2122C2semidirect
6. #2121C2semidirect
7. #2120C2semidirect
8. C22×#41C4semidirect
14. #536C4semidirect
17. #535C4semidirect
19. #534C4semidirect
25. #525C4semidirect
28. C2×#117C4semidirect
41. #352C22semidirect
42. AΣL(1,𝔽16)C22semidirect
63. C22×Q8C2×C4semidirect
67. C22×D4C2×C4semidirect
71. C23×C4C2×C4semidirect
92. #123C2×C4semidirect
93. #123C2×C4semidirect
114. #120C2×C4semidirect
115. #120C2×C4semidirect
127. #117C2×C4semidirect
128. #117C2×C4semidirect
129. #117C2×C4semidirect

Maximal subgroups

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

subgroupindex
#34492
C2×#4132
C2×#4102
#21232
#21222
#21212
#21202