Finite Group Databasebeta

← C2×#11553 #480471

Subgroups — C2×#11553 #480471

Order 512.

1192 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2C2×#1463C211yesmaximal, characteristic, complemented
3C22×#324C211yesmaximal, characteristic, complemented
4#11553C218yesmaximal, direct product factor
5C2×#1324C212yesmaximal, complemented
6C2×#1318C212yesmaximal, complemented
7C22×#121C411yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407—21no—
10C22×C22⋊C8—21no—
11#2504—22no—
12#2504—22no—
13#2504C414yes—
14#2504C2214yes—
15C2×#397—21no—
16C2×#397—21no—
17C2×#390C411yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#386—21no—
20C2×#386—21no—
21C2×#382C411yescharacteristic
22C2×#382C2211yescharacteristic
23C2×C4×C22⋊C4C411yescharacteristic
24#1463—22no—
25#1463C2214yescomplemented
26C2×#324—21no—
27C2×#324—22no—
28C2×#324—21no—
29C2×#324C2212yes—
30C2×#324C2212yes—
31C2×#324C2214yescomplemented
32#1324C2218yescomplemented
33#1318C2218yescomplemented
34C23×D4—21no—
35C23×D4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4D411yescharacteristic
38C24×C4D411yescharacteristic
39C22×M16—41no—
40C23×C8—41no—
41C2×#121—22no—
42C2×#121—42no—
43C2×#121—41no—
44C2×#121—41no—
45C2×#121C2×C414yes—
46C22×C4⋊C4—41no—
47C22×C4⋊C4—41no—
48C22×C4⋊C4—42no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66#407—24no—
67#407—42no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8—41no—
74#397—24no—
75#397—24no—
76#397—24no—
77#397—24no—
78#390—22no—
79#390—21no—
80#390—21no—
81#390C2×C412yes—
82#390C2×C414yes—
83#390C2312yes—
84#386—24no—
85#386—24no—
86#386—24no—
87#386—24no—
88#382—22no—
89#382—21no—
90#382—21no—
91#382C2×C412yes—
92#382C2×C414yes—
93#382C2312yes—
94C4×C22⋊C4—22no—
95C4×C22⋊C4C2×C414yes—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101#324—22no—
102#324—24no—
103#324—24no—
104#324—22no—
105#324C2314yes—
106#324C2314yes—
107C25—21no—
108C25C22⋊C411yescharacteristic
109C22×D4—24no—
110C22×D4—21no—
111C22×D4—24no—
112C22×D4—41no—
113C22×D4—41no—
114C22×D4—42no—
115C22×D4—44no—
116C22×D4—44no—
117C22×D4—42no—
118C22×D4—41no—
119C22×D4—41no—
120C22×D4C22×C412yes—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—42no—
135C23×C4—82no—
136C23×C4—82no—
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C2×M16—44no—
150C2×M16—84no—
151C22×C8—44no—
152C22×C8—42no—
153C22×C8—42no—
154C22×C8—84no—
155#121—44no—
156#121—44no—
157#121—44no—
158#121—44no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2411yescharacteristic
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C42—24no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—42no—
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C22⋊C8—24no—
211C22⋊C8—24no—
212C22⋊C8—42no—
213C22⋊C8—44no—
214C22⋊C8—42no—
215C22⋊C8—44no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223C24—24no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24—41no—
234C24—41no—
235C24—42no—
236C24#9911yescharacteristic
237C24#9912yes—
238C24#9911yescharacteristic
239C24C2×C22⋊C412yes—
240C24C2×C22⋊C411yescharacteristic
241C2×D4—24no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—48no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—42no—
251C2×D4—84no—
252C2×D4—84no—
253C2×D4—88no—
254C2×D4—84no—
255C2×D4—84no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—48no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4#13114yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327M16—88no—
328C2×C8—44no—
329C2×C8—44no—
330C2×C8—88no—
331C2×C8—84no—
332C2×C8—84no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C22⋊C4—48no—
346C22⋊C4—48no—
347C22⋊C4—84no—
348C22⋊C4—84no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C42—24no—
354C42—42no—
355C42—42no—
356C42—42no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—22no—
367C23—24no—
368C23—22no—
369C23—42no—
370C23—42no—
371C23—48no—
372C23—48no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—82no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23—84no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23#35514yes—
391C23#40812yes—
392C23C2×#9912yes—
393C23C2×#9912yes—
394C23C2×#9911yescharacteristic
395C23C2×#9911yescharacteristic
396C23#41312yes—
397C23C22×C22⋊C411yescharacteristic
398D4—84no—
399D4—84no—
400D4—88no—
401D4—88no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4#40812yes—
459C2×C4C2×#13112yes—
460C8—84no—
461C8—84no—
462C22—24no—
463C22—24no—
464C22—28no—
465C22—21no—
466C22—22no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—44no—
475C22—44no—
476C22—48no—
477C22—44no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—164no—
486C22—164no—
487C22#146812yes—
488C22#146812yes—
489C22#212712yes—
490C22C2×#35514yes—
491C22C2×#35512yes—
492C22C2×#40811yescharacteristic
493C22C22×#9911yescharacteristic
494C22C2×#41311yescharacteristic
495C4—22no—
496C4—22no—
497C4—41no—
498C4—41no—
499C4—42no—
500C4—42no—
501C4—41no—
502C4—41no—
503C4—42no—
504C4—82no—
505C4—82no—
506C4—82no—
507C2—21no—
508C2—21no—
509C2—22no—
510C2—24no—
511C2—42no—
512C2—162no—
513C2#1155314yesdirect product factor
514C2C2×#146811yescharacteristic
515C2C2×#212711yescharacteristic
516C2C22×#35511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1463C2semidirect
3. C22×#324C2semidirect
4. #11553C2direct (N × H)
5. C2×#1324C2semidirect
6. C2×#1318C2semidirect
25. #1463C22semidirect
31. C2×#324C22semidirect
32. #1324C22semidirect
33. #1318C22semidirect
513. C2#11553direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#14632
C22×#3242
#115532
C2×#13242
C2×#13182