Finite Group Databasebeta

← C2×#11780 #481607

Subgroups — C2×#11780 #481607

Order 512.

1280 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
1C2×#2486C211yesmaximal, characteristic
2C2×C4×#100C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#11780C218yesmaximal, direct product factor
5C2×#1346C214yesmaximal, complemented
6C22×#123—21no—
7C22×C4×D4C2211yescharacteristic
8C23×C4⋊C4C411yescharacteristic
9C22×#100—21no—
10C2×#407—21no—
11C2×C4×M16—21no—
12#2486—22no—
13#2486—22no—
14#2486C414yes—
15#2486C2214yes—
16C2×#398C2212yes—
17C2×#397C412yes—
18C2×#389—21no—
19C2×#381—21no—
20C2×#380—21no—
21C2×#380C411yescharacteristic
22C4×#100—22no—
23C4×#100C2214yes—
24C2×#343C2212yes—
25#1465—22no—
26#1465C2214yes—
27C2×#324C2212yes—
28C2×#323—21no—
29#1346C22116yescomplemented
30C23×D4D411yescharacteristic
31C24×C4—21no—
32C24×C4C2×C411yescharacteristic
33C24×C4D411yescharacteristic
34C22×M16—22no—
35C22×M16—41no—
36C2×#123—24no—
37C2×#123—24no—
38C2×#123—48no—
39C2×C4×D4—41no—
40C2×C4×D4—41no—
41C2×C4×D4—42no—
42C2×C4×D4C2×C412yes—
43C2×C4×D4D412yes—
44C2×C4×D4D412yes—
45C2×C4×D4C2312yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—42no—
51C22×C4⋊C4—41no—
52C22×C4⋊C4—41no—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C414yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42D411yescharacteristic
61C2×#100—28no—
62C2×#100—24no—
63#407—24no—
64#407—42no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—41no—
67C4×M16—24no—
68C4×M16—42no—
69C2×#97—41no—
70C2×C4×C8—41no—
71#398—22no—
72#398—22no—
73#398C2×C414yes—
74#398C2314yes—
75#397—24no—
76#397C2×C418yes—
77#389—24no—
78#389—24no—
79#381—24no—
80#381—24no—
81#380—22no—
82#380—24no—
83#380—24no—
84#380C2×C414yes—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95C2×C412yes—
88#343—22no—
89#343C2314yes—
90#324—22no—
91#324C2314yes—
92#323—24no—
93C25C22⋊C411yescharacteristic
94C25C22⋊C411yescharacteristic
95C22×D4—24no—
96C22×D4—24no—
97C22×D4—41no—
98C22×D4—41no—
99C22×D4C22⋊C412yes—
100C22×D4C2×D412yes—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—44no—
111C23×C4—44no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—44no—
115C23×C4—44no—
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C2×M16—28no—
126C2×M16—24no—
127C2×M16—42no—
128C2×M16—44no—
129C2×M16—44no—
130C2×M16—84no—
131C22×C8—42no—
132C22×C8—42no—
133#123—416no—
134#123—416no—
135C4×D4—42no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141C4×D4—42no—
142C4×D4—42no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4C22×C414yes—
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C4⋊C4C2411yescharacteristic
166C2×C22⋊C4—28no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—48no—
174C2×C22⋊C4C22⋊C412yes—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22×C412yes—
177C2×C22⋊C4C2×D412yes—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42C22⋊C412yes—
182C2×C42C2×D412yes—
183#100—216no—
184C22⋊C8—24no—
185C22⋊C8—42no—
186C22⋊C8—44no—
187#97—44no—
188C4×C8—44no—
189#95—24no—
190#95—28no—
191#95—22no—
192#95—24no—
193#95C22×C414yes—
194C24—21no—
195C24—21no—
196C24—22no—
197C24—21no—
198C24—22no—
199C24—21no—
200C24—22no—
201C24—42no—
202C24—42no—
203C24—42no—
204C24—42no—
205C24#10414yes—
206C24C2×C22⋊C411yescharacteristic
207C24C2×C22⋊C412yes—
208C24C2×C22⋊C412yes—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—48no—
216C2×D4—48no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—48no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—48no—
257C22×C4—44no—
258C22×C4—84no—
259C22×C4—82no—
260C22×C4—88no—
261C22×C4—88no—
262C22×C4#9912yes—
263C22×C4#9911yescharacteristic
264C22×C4#9911yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4AGL(1,ℤ/8)14yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274M16—28no—
275M16—48no—
276M16—44no—
277M16—88no—
278C2×C8—44no—
279C2×C8—48no—
280C2×C8—44no—
281C2×C8—84no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—44no—
287C4⋊C4—48no—
288C4⋊C4—44no—
289C4⋊C4—48no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—48no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—48no—
298C22⋊C4—416no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—24no—
308C23—24no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—42no—
313C23—48no—
314C23—42no—
315C23—42no—
316C23—48no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—48no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—81no—
328C23—81no—
329C23—84no—
330C23C2×#9911yescharacteristic
331C23C2×#9911yescharacteristic
332C23#41812yes—
333C23C2×#10412yes—
334C23C2×#10414yes—
335C23C22×C22⋊C411yescharacteristic
336D4—48no—
337D4—48no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—88no—
378C2×C4—816no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4#35214yes—
382C2×C4C2×#9912yes—
383C2×C4C2×#9912yes—
384C2×C4#41812yes—
385C2×C4C2×AGL(1,ℤ/8)12yes—
386C8—48no—
387C8—84no—
388C22—24no—
389C22—24no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—24no—
395C22—44no—
396C22—44no—
397C22—48no—
398C22—44no—
399C22—48no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—84no—
407C22—84no—
408C22#150712yes—
409C22#150712yes—
410C22C2×#35212yes—
411C22#213512yes—
412C22C22×#9911yescharacteristic
413C22C2×#41811yescharacteristic
414C22C22×#10411yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—42no—
423C4—84no—
424C4—84no—
425C4C2×#35214yes—
426C2—21no—
427C2—21no—
428C2—22no—
429C2—44no—
430C2—82no—
431C2#1178014yesdirect product factor
432C2C2×#150711yescharacteristic
433C2C22×#35211yescharacteristic
434C2C2×#213511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11780C2direct (N × H)
5. C2×#1346C2semidirect
29. #1346C22semidirect
431. C2#11780direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24862
C2×C4×#1002
C2×#14652
#117802
C2×#13462