Finite Group Databasebeta

← C2×#11565 #480515

Subgroups — C2×#11565 #480515

Order 512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2545C211yesmaximal, characteristic
2C2×#1467C211yesmaximal, characteristic, complemented
3C2×#1461C211yesmaximal, characteristic, complemented
4#11565C218yesmaximal, direct product factor
5C2×#1319C212yesmaximal, complemented
6C2×#1318C212yesmaximal
7C22×C4⋊Q8C411yescharacteristic
8C22×#124C411yescharacteristic
9C22×#124C411yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#432—21no—
12C2×#431—21no—
13#2545—22no—
14#2545—22no—
15#2545C414yes—
16#2545C2214yes—
17C2×#396—21no—
18C2×#396—21no—
19C2×#391C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#388—21no—
22C2×#388—21no—
23C2×C4×C4⋊C4C411yescharacteristic
24#1467—22no—
25#1467C2214yescomplemented
26#1461—22no—
27#1461C2214yescomplemented
28C2×#324—21no—
29C2×#324—21no—
30C2×#324C2212yes—
31C2×#324C2212yes—
32#1319C2218yescomplemented
33#1318C2218yes—
34C23×Q8C2×C411yescharacteristic
35C23×Q8C2×C411yescharacteristic
36C23×D4C2×C411yescharacteristic
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—44no—
39C2×C4⋊Q8C2×C414yes—
40C2×#124—22no—
41C2×#124—22no—
42C2×#124—42no—
43C2×#124—41no—
44C2×#124—41no—
45C2×#124—44no—
46C2×#124C2×C414yes—
47C2×#124C2×C414yes—
48C2×C4×Q8—41no—
49C2×C4×Q8—41no—
50C2×C4×Q8—42no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8D412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8C2312yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72#432—24no—
73#432—42no—
74#431—24no—
75#431—42no—
76C2×C4⋊C8—41no—
77C2×C4⋊C8—41no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×#97—41no—
81C2×C4×C8—41no—
82#396—24no—
83#396—24no—
84#396—24no—
85#396—24no—
86#391—21no—
87#391—21no—
88#391C2×C412yes—
89#391C2312yes—
90#390—21no—
91#390—21no—
92#390C2×C412yes—
93#390C2312yes—
94#388—24no—
95#388—24no—
96#388—24no—
97#388—24no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C414yes—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103#324—24no—
104#324—24no—
105#324—22no—
106#324—22no—
107#324C2314yes—
108#324C2314yes—
109C25—21no—
110C22×Q8—24no—
111C22×Q8—21no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8—41no—
115C22×Q8—41no—
116C22×Q8—41no—
117C22×Q8—41no—
118C22×Q8—42no—
119C22×Q8C22⋊C412yes—
120C22×Q8C22×C412yes—
121C22×Q8C22×C412yes—
122C22×D4—21no—
123C22×D4—24no—
124C22×D4—41no—
125C22×D4—41no—
126C22×D4—42no—
127C22×D4C22×C412yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C22×C8—42no—
139C22×C8—42no—
140C4⋊Q8—48no—
141C4⋊Q8—48no—
142#124—44no—
143#124—44no—
144#124—44no—
145#124—44no—
146#124—48no—
147#124—48no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×Q8—42no—
153C4×Q8—42no—
154C4×Q8—42no—
155C4×Q8—42no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2411yescharacteristic
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—44no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42C22⋊C412yes—
207C2×C42C22⋊C412yes—
208C2×C42C22⋊C412yes—
209C2×C42C22×C412yes—
210C2×C42C22×C412yes—
211C2×C42C22×C412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C4⋊C8—44no—
215C4⋊C8—44no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220#97—44no—
221C4×C8—44no—
222#95—24no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227C24—22no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24—41no—
234C24—41no—
235C24—42no—
236C24C2×C22⋊C411yescharacteristic
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—48no—
245C2×Q8—48no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—84no—
252C2×Q8—84no—
253C2×D4—24no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—84no—
260C2×D4—84no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4#9911yescharacteristic
295C22×C4#9911yescharacteristic
296C22×C4#9912yes—
297C22×C4#10014yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4#13114yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—84no—
312C2×C8—84no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—48no—
322C4⋊C4—48no—
323C4⋊C4—48no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—48no—
329C22⋊C4—48no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—48no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C23—22no—
350C23—22no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—84no—
359C23—84no—
360C23—84no—
361C23—84no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23#40812yes—
369C23C2×#9911yescharacteristic
370C23C2×#9911yescharacteristic
371C23C2×#10012yes—
372C23C22×C22⋊C411yescharacteristic
373Q8—48no—
374Q8—48no—
375Q8—84no—
376Q8—84no—
377D4—84no—
378D4—84no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—48no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—84no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4#40812yes—
428C2×C4C2×#9912yes—
429C2×C4C2×#9912yes—
430C2×C4C2×#10014yes—
431C2×C4C2×#13112yes—
432C8—84no—
433C8—84no—
434C22—24no—
435C22—24no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—164no—
448C22—164no—
449C22#147112yes—
450C22#147112yes—
451C22#212612yes—
452C22#213312yes—
453C22C2×#40811yescharacteristic
454C22C22×#9911yescharacteristic
455C22C22×#10011yescharacteristic
456C4—22no—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—24no—
461C4—44no—
462C4—41no—
463C4—41no—
464C4—42no—
465C4—41no—
466C4—41no—
467C4—42no—
468C4—82no—
469C4—82no—
470C2—21no—
471C2—21no—
472C2—22no—
473C2—162no—
474C2#1156514yesdirect product factor
475C2C2×#147111yescharacteristic
476C2C2×#212611yescharacteristic
477C2C2×#213311yescharacteristic
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×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1467C2semidirect
3. C2×#1461C2semidirect
4. #11565C2direct (N × H)
5. C2×#1319C2semidirect
25. #1467C22semidirect
27. #1461C22semidirect
32. #1319C22semidirect
474. C2#11565direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25452
C2×#14672
C2×#14612
#115652
C2×#13192
C2×#13182