Finite Group Databasebeta

← C2×#11564 #480514

Subgroups — C2×#11564 #480514

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#1465C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#11564C218yesmaximal, direct product factor
5C2×#1319C214yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic
7C22×#124C412yes—
8C22×C4×D4C2211yescharacteristic
9C2×#407—21no—
10C22×C22⋊C8—21no—
11#2547—22no—
12#2547—22no—
13#2547C414yes—
14#2547C2214yes—
15C2×#397—21no—
16C2×#397—21no—
17C2×#391C2212yes—
18C2×#380—21no—
19C2×#380—21no—
20C2×C4×C4⋊C4C411yescharacteristic
21#1465—22no—
22#1465C2214yes—
23#1459—22no—
24#1459C2214yes—
25C2×#324C2212yes—
26C2×#324C2212yes—
27C2×#323—21no—
28C2×#323—21no—
29#1319C22116yescomplemented
30C23×Q8C2×C412yes—
31C23×D4C2×C411yescharacteristic
32C24×C4D411yescharacteristic
33C24×C4D411yescharacteristic
34C22×M16—41no—
35C23×C8—41no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—42no—
38C2×C4⋊Q8—41no—
39C2×C4⋊Q8—41no—
40C2×C4⋊Q8C2×C414yes—
41C2×#124—24no—
42C2×#124—48no—
43C2×#124C2×C418yes—
44C2×C4×D4—41no—
45C2×C4×D4—41no—
46C2×C4×D4—42no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C2×C4×D4C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4D411yescharacteristic
58C22×C22⋊C4D411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C412yes—
61#407—24no—
62#407—42no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—42no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69#397—24no—
70#397—24no—
71#397—24no—
72#397—24no—
73#391—22no—
74#391—22no—
75#391C2×C414yes—
76#391C2314yes—
77#380—24no—
78#380—24no—
79#380—24no—
80#380—24no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4C2×C414yes—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86#324—22no—
87#324—22no—
88#324C2314yes—
89#324C2314yes—
90#323—24no—
91#323—24no—
92C25C22⋊C411yescharacteristic
93C25C22⋊C411yescharacteristic
94C22×Q8—22no—
95C22×Q8—28no—
96C22×Q8—44no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8C22×C414yes—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4—41no—
103C22×D4—41no—
104C22×D4C22⋊C412yes—
105C22×D4C22×C412yes—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—44no—
115C23×C4—44no—
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C2×M16—44no—
126C2×M16—84no—
127C22×C8—42no—
128C22×C8—44no—
129C22×C8—42no—
130C22×C8—84no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135#124—416no—
136#124—416no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141C4×D4—42no—
142C4×D4—42no—
143C4×D4—42no—
144C4×D4—42no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2411yescharacteristic
160C2×C22⋊C4—28no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—48no—
168C2×C22⋊C4C22⋊C412yes—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C2×D412yes—
171C2×C22⋊C4C2×D412yes—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—44no—
177C2×C42C22⋊C412yes—
178C2×C42C22×C412yes—
179C2×C42C22×C414yes—
180C22⋊C8—24no—
181C22⋊C8—24no—
182C22⋊C8—42no—
183C22⋊C8—44no—
184C22⋊C8—42no—
185C22⋊C8—44no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191C24—21no—
192C24—22no—
193C24—21no—
194C24—21no—
195C24—22no—
196C24—22no—
197C24—21no—
198C24—42no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24#10014yes—
203C24C2×C22⋊C411yescharacteristic
204C24C2×C22⋊C412yes—
205C24C2×C22⋊C412yes—
206C2×Q8—28no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—88no—
213C2×Q8—88no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—48no—
221C2×D4—48no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—84no—
259C22×C4—84no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4#9912yes—
263C22×C4#9911yescharacteristic
264C22×C4#9911yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4#13114yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274M16—88no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—84no—
278C2×C8—88no—
279C2×C8—84no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—48no—
285C4⋊C4—48no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—416no—
293C22⋊C4—416no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—44no—
299C42—44no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—24no—
304C23—22no—
305C23—24no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—42no—
310C23—42no—
311C23—48no—
312C23—42no—
313C23—42no—
314C23—48no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—84no—
324C23—81no—
325C23—81no—
326C23—84no—
327C23#40812yes—
328C23C2×#9911yescharacteristic
329C23C2×#9911yescharacteristic
330C23C2×#10012yes—
331C23C2×#10014yes—
332C23C22×C22⋊C411yescharacteristic
333Q8—88no—
334Q8—88no—
335D4—48no—
336D4—48no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—88no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4#35414yes—
380C2×C4#40812yes—
381C2×C4C2×#9912yes—
382C2×C4C2×#9912yes—
383C2×C4C2×#13112yes—
384C8—84no—
385C8—84no—
386C22—24no—
387C22—24no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—24no—
393C22—44no—
394C22—44no—
395C22—48no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—82no—
402C22—84no—
403C22—82no—
404C22—84no—
405C22—84no—
406C22#147112yes—
407C22#147112yes—
408C22C2×#35412yes—
409C22#213512yes—
410C22C2×#40811yescharacteristic
411C22C22×#9911yescharacteristic
412C22C22×#10011yescharacteristic
413C4—22no—
414C4—22no—
415C4—22no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—42no—
421C4—84no—
422C4—84no—
423C4C2×#35414yes—
424C2—21no—
425C2—21no—
426C2—22no—
427C2—44no—
428C2—82no—
429C2#1156414yesdirect product factor
430C2C2×#147111yescharacteristic
431C2C22×#35411yescharacteristic
432C2C2×#213511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #11564C2direct (N × H)
5. C2×#1319C2semidirect
29. #1319C22semidirect
429. C2#11564direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#14652
C2×#14592
#115642
C2×#13192