Finite Group Databasebeta

← C2×#31569 #7505088

Subgroups — C2×#31569 #7505088

Order 512.

2312 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2420C212yesmaximal, complemented
2#31569C2116yesmaximal, direct product factor
3C2×#2101C212yesmaximal, complemented
4C2×#2075C211yesmaximal, characteristic, complemented
5C2×#2067C212yesmaximal, complemented
6C2×#1999C211yesmaximal, characteristic
7C2×#1895C212yesmaximal, complemented
8C2×#1879C212yesmaximal, complemented
9C2×#1851C211yesmaximal, characteristic, complemented
10C2×#1804C212yesmaximal, complemented
11C23×D8C2211yescharacteristic
12C23×M16C2211yescharacteristic
13C22×#121C2212yes—
14C22×C4×D4C2212yes—
15C22×#107C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21#2420—28no—
22#2420C22116yescomplemented
23C2×#397—22no—
24C2×#394—22no—
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#386—22no—
28C2×#382C2212yes—
29C2×#381—22no—
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#2101C22116yescomplemented
33#2075C2218yescomplemented
34#2067C22116yescomplemented
35#1999C2218yes—
36#1895C22116yescomplemented
37#1879C22116yescomplemented
38#1851C2218yescomplemented
39#1804C22116yescomplemented
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#340C2212yes—
45C2×#336C2212yes—
46C23×D4—22no—
47C23×D4C2312yes—
48C24×C4—22no—
49C24×C4C2311yescharacteristic
50C22×D8—28no—
51C22×D8—42no—
52C22×D8C2314yes—
53C22×M16—42no—
54C22×M16—44no—
55C22×M16C2314yes—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C2×#121—24no—
60C2×#121—28no—
61C2×#121—44no—
62C2×#121C2318yes—
63C2×C4×D4—28no—
64C2×C4×D4—24no—
65C2×C4×D4—44no—
66C2×C4×D4C2318yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4D412yes—
77C22×C42C2312yes—
78C2×#107—42no—
79C2×#107C2314yes—
80C2×C4⋊C8—44no—
81C2×C4⋊C8C2318yes—
82C2×#102—44no—
83C2×#102—44no—
84C2×#102—44no—
85C2×#102C2318yes—
86C2×#102C2318yes—
87C2×#102C2318yes—
88C2×C22⋊C8—42no—
89C2×C22⋊C8C2314yes—
90#397—216no—
91#394—216no—
92#390—24no—
93#390C2318yes—
94#389—24no—
95#389C2318yes—
96#386—216no—
97#382—24no—
98#382C2318yes—
99#381—216no—
100#380—24no—
101#380C2318yes—
102C4×C22⋊C4D414yes—
103C4×C22⋊C4C2314yes—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95D411yescharacteristic
109C2×#95D411yescharacteristic
110#340C2314yes—
111#340C2318yes—
112#340C2318yes—
113#340C2318yes—
114#340C2314yes—
115#336C2318yes—
116C25—22no—
117C25C2×D411yescharacteristic
118C22×D4—28no—
119C22×D4—22no—
120C22×D4—24no—
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—48no—
126C22×D4—48no—
127C22×D4—84no—
128C22×D4#4118yes—
129C22×D4C2414yes—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—28no—
138C23×C4—48no—
139C23×C4—48no—
140C23×C4—84no—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2412yes—
146C23×C4C2411yescharacteristic
147C2×D8—216no—
148C2×D8—44no—
149C2×D8—416no—
150C2×M16—44no—
151C2×M16—48no—
152C2×M16—416no—
153C22×C8—24no—
154C22×C8—24no—
155C22×C8—24no—
156C22×C8—24no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—24no—
160C22×C8C2414yes—
161C22×C8C2412yes—
162C22×C8C2412yes—
163#121—216no—
164#121—48no—
165#121—416no—
166C4×D4—216no—
167C4×D4—48no—
168C4×D4—416no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—84no—
180C2×C4⋊C4C2414yes—
181C2×C4⋊C4C2414yes—
182C2×C4⋊C4C2414yes—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4—48no—
191C2×C22⋊C4—84no—
192C2×C22⋊C4—84no—
193C2×C22⋊C4—84no—
194C2×C22⋊C4C2×D418yes—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42C2×D414yes—
198C2×C42C2414yes—
199#107—44no—
200C4⋊C8—48no—
201#102—48no—
202#102—48no—
203#102—48no—
204C22⋊C8—44no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—28no—
209#95C2×D414yes—
210#95C2×D414yes—
211C24—22no—
212C24—28no—
213C24—24no—
214C24—24no—
215C24—22no—
216C24—22no—
217C24—24no—
218C24—44no—
219C24—44no—
220C24—44no—
221C24—44no—
222C24—44no—
223C24—44no—
224C24C22×D412yes—
225C24C22×D411yescharacteristic
226C2×D4—216no—
227C2×D4—28no—
228C2×D4—216no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—416no—
235C2×D4—416no—
236C2×D4—44no—
237C2×D4—48no—
238C2×D4—44no—
239C2×D4—48no—
240C2×D4—88no—
241C2×D4—832no—
242C2×D4C2×#4118yes—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—416no—
261C22×C4—416no—
262C22×C4—48no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—88no—
275C22×C4—832no—
276C22×C4C22×D412yes—
277C22×C4C22×D414yes—
278C22×C4C22×D414yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C2×#4114yes—
281C22×C4H(5,𝔽2)12yes—
282C22×C4H(5,𝔽2)14yes—
283C22×C42−1+412yes—
284C22×C4C2511yescharacteristic
285D8—416no—
286M16—416no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—48no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—88no—
307C22⋊C4—48no—
308C22⋊C4—416no—
309C22⋊C4—416no—
310C22⋊C4—88no—
311C22⋊C4—88no—
312C22⋊C4—88no—
313C42—28no—
314C42—24no—
315C42—24no—
316C23—22no—
317C23—28no—
318C23—24no—
319C23—24no—
320C23—22no—
321C23—22no—
322C23—44no—
323C23—44no—
324C23—416no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—48no—
329C23—416no—
330C23—416no—
331C23—48no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—88no—
337C23—88no—
338C23—88no—
339C23—88no—
340C23#53712yes—
341C23#54614yes—
342C23#57514yes—
343C23#57612yes—
344C23#57612yes—
345C23C23×D411yescharacteristic
346D4—28no—
347D4—28no—
348D4—44no—
349D4—416no—
350D4—44no—
351D4—832no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—44no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—44no—
364C2×C4—48no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—832no—
377C2×C4—88no—
378C2×C4—88no—
379C2×C4—88no—
380C2×C4#53712yes—
381C2×C4#53918yes—
382C2×C4#54614yes—
383C2×C4#55114yes—
384C2×C4#56014yes—
385C2×C4C22×#4112yes—
386C2×C4C2×H(5,𝔽2)12yes—
387C2×C4C2×H(5,𝔽2)11yescharacteristic
388C2×C4C2×2−1+411yescharacteristic
389C8—42no—
390C8—42no—
391C8—44no—
392C22—42no—
393C22—48no—
394C22—44no—
395C22—48no—
396C22—48no—
397C22—44no—
398C22—42no—
399C22—42no—
400C22—88no—
401C22—832no—
402C22—84no—
403C22—84no—
404C22—84no—
405C22—84no—
406C22#269314yes—
407C22#325614yes—
408C22#325914yes—
409C22#331918yes—
410C22#332218yes—
411C22C2×#53711yescharacteristic
412C22C2×#54612yes—
413C22C2×#57512yes—
414C22C2×#57611yescharacteristic
415C22C2×#57611yescharacteristic
416C4—42no—
417C4—44no—
418C4—44no—
419C4—84no—
420C4—84no—
421C4—84no—
422C4#269314yes—
423C4C2×#53912yes—
424C4C2×#55111yescharacteristic
425C4C2×#56011yescharacteristic
426C2—82no—
427C2—88no—
428C2#3156918yesdirect product factor
429C2C2×#269311yescharacteristic
430C2C2×#325611yescharacteristic
431C2C2×#325911yescharacteristic
432C2C2×#331912yes—
433C2C2×#332212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2420C2semidirect
2. #31569C2direct (N × H)
3. C2×#2101C2semidirect
4. C2×#2075C2semidirect
5. C2×#2067C2semidirect
7. C2×#1895C2semidirect
8. C2×#1879C2semidirect
9. C2×#1851C2semidirect
10. C2×#1804C2semidirect
22. #2420C22semidirect
32. #2101C22semidirect
33. #2075C22semidirect
34. #2067C22semidirect
36. #1895C22semidirect
37. #1879C22semidirect
38. #1851C22semidirect
39. #1804C22semidirect
428. C2#31569direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24202
#315692
C2×#21012
C2×#20752
C2×#20672
C2×#19992
C2×#18952
C2×#18792
C2×#18512
C2×#18042