Finite Group Databasebeta

← C4×#2133 #493019

Subgroups — C4×#2133 #493019

Order 512.

1243 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
1C4×#536C211yesmaximal, characteristic
2C2×#2133C212yesmaximal
3C2×C4×#101C212yesmaximal, complemented
4#12632C212yesmaximal, complemented
5#12631C212yesmaximal, complemented
6C4×#356C212yesmaximal, complemented
7C4×#355C212yesmaximal, complemented
8#12586C212yesmaximal, complemented
9C2×#536C2211yescharacteristic
10C2×C4×#41C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×#101C2214yes—
13C2×C4×M16—22no—
14#2327C411yescharacteristic
15#2327C2211yescharacteristic
16#2326C411yescharacteristic
17#2326C2211yescharacteristic
18#2323—22no—
19#2322—22no—
20#2317C411yescharacteristic
21#2311C411yescharacteristic
22#2310C2211yescharacteristic
23C4×C4⋊Q8C411yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#124C411yescharacteristic
26C4×#124C2211yescharacteristic
27C4×C22⋊Q8—22no—
28C4×#123—22no—
29C4×#117C411yescharacteristic
30#2133C4116yesdirect product factor
31C2×#356C2214yes—
32C2×#355C2214yes—
33C4×#101—22no—
34C4×#101—22no—
35C4×#101C2214yes—
36C4×#101C2214yes—
37C2×#343—22no—
38#1394C2214yes—
39#1394C2214yes—
40C22×#41C2×C411yescharacteristic
41C23×Q8C2311yescharacteristic
42C22×M16—24no—
43#536—24no—
44#536C2×C418yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#124C2×C411yescharacteristic
48C2×#124C2311yescharacteristic
49C2×#123—22no—
50C2×C22⋊Q8—22no—
51C4×#41—42no—
52C4×#41—42no—
53C2×C4×Q8—42no—
54C2×C4×Q8—42no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D412yes—
60C2×C4×Q8D412yes—
61C2×C4×Q8C2311yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×#117—21no—
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4D412yes—
73C22×C42—21no—
74C22×C42C2×C411yescharacteristic
75C22×C42D412yes—
76C2×#101—24no—
77C2×#101—28no—
78C2×#101C2×C4116yescomplemented
79C2×#101C2318yes—
80C4×M16—24no—
81C4×M16—24no—
82C4×M16—42no—
83C4×M16—42no—
84C2×#97—42no—
85C2×C4×C8—42no—
86#391—22no—
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#391C2×C411yescharacteristic
90#391C2×C411yescharacteristic
91#391C2311yescharacteristic
92#391C2311yescharacteristic
93#390C2×C411yescharacteristic
94#390C2×C411yescharacteristic
95#389—22no—
96#389—22no—
97#389—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#386—22no—
107#385—22no—
108#385—22no—
109#385—22no—
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#382C2×C411yescharacteristic
113#382C2×C411yescharacteristic
114#381—22no—
115#380—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124#376C2×C411yescharacteristic
125#376C2×C411yescharacteristic
126C43C2×C411yescharacteristic
127C43C2×C411yescharacteristic
128#356C2×C4116yescomplemented
129#355C2×C4116yescomplemented
130#343—24no—
131#343—24no—
132C2×#41—22no—
133C2×#41—42no—
134C2×#41—42no—
135C2×#41C4214yes—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8C22×C411yescharacteristic
139C22×Q8C22×C414yes—
140C22×Q8C22×C411yescharacteristic
141C22×Q8C2×D412yes—
142C22×Q8#4114yes—
143C22×Q8C2411yescharacteristic
144C22×D4—21no—
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D412yes—
154C2×M16—216no—
155C2×M16—28no—
156C2×M16—48no—
157C2×M16—44no—
158C22×C8—44no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8C4214yes—
162C4⋊Q8C22×C414yes—
163#124—22no—
164#124—22no—
165#124C4214yes—
166#124C22×C414yes—
167#123—216no—
168C22⋊Q8—216no—
169C4×Q8—41no—
170C4×Q8—42no—
171C4×Q8—41no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—42no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—42no—
185C4×D4—44no—
186C4×D4—42no—
187C4×D4—42no—
188#117—22no—
189#117—44no—
190#117—42no—
191#117—42no—
192#117C4214yes—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C412yes—
239C2×C42C22⋊C412yes—
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249#101—216no—
250#101—28no—
251#101C22×C4116yes—
252#97—44no—
253#97—44no—
254C4×C8—44no—
255C4×C8—44no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268C24—21no—
269C24C2×C22⋊C411yescharacteristic
270#41—48no—
271#41—48no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—48no—
278C2×Q8—42no—
279C2×Q8—48no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8C2×C4212yes—
284C2×Q8C2×C4212yes—
285C2×Q8#11712yes—
286C2×Q8C4×D418yes—
287C2×Q8C23×C412yes—
288C2×Q8C2×#4112yes—
289C2×D4—24no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4C2×C4212yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—81no—
322C22×C4#9912yes—
323C22×C4#9912yes—
324C22×C4C2×C4212yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4#11712yes—
334C22×C4C4×D418yes—
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D412yes—
337M16—216no—
338M16—48no—
339M16—416no—
340C2×C8—48no—
341C2×C8—416no—
342C4⋊C4—28no—
343C4⋊C4—24no—
344C4⋊C4—28no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C22⋊C4—24no—
360C22⋊C4—28no—
361C22⋊C4—28no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C42—21no—
369C42—21no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—21no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—21no—
380C42—22no—
381C42—21no—
382C42—22no—
383C42—21no—
384C42—41no—
385C42—41no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—41no—
391C42—41no—
392C42—81no—
393C42—81no—
394C42C2×C4212yes—
395C42C2×C4212yes—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23C4×C22⋊C412yes—
404C23C2×#9912yes—
405C23#41012yes—
406C23C22×C22⋊C411yescharacteristic
407Q8—42no—
408Q8—42no—
409Q8—48no—
410Q8—42no—
411Q8—42no—
412Q8—42no—
413Q8—42no—
414D4—42no—
415D4—42no—
416D4—48no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—28no—
426C2×C4—28no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—48no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—84no—
466C2×C4—82no—
467C2×C4C4×C22⋊C414yes—
468C2×C4C4×C22⋊C412yes—
469C2×C4C2×#9911yescharacteristic
470C2×C4C2×#9912yes—
471C2×C4C2×#9911yescharacteristic
472C2×C4C22×C4211yescharacteristic
473C2×C4C2×#11711yescharacteristic
474C2×C4C2×C4×D414yes—
475C8—416no—
476C22—21no—
477C22—22no—
478C22—21no—
479C22—22no—
480C22—42no—
481C22—42no—
482C22—44no—
483C22—42no—
484C22—82no—
485C22—81no—
486C22—81no—
487C22C4×#9914yes—
488C22C2×C4×C22⋊C411yescharacteristic
489C22C22×#9911yescharacteristic
490C22C2×#41011yescharacteristic
491C4—21no—
492C4—22no—
493C4—24no—
494C4—22no—
495C4—21no—
496C4—24no—
497C4—21no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—41no—
503C4—41no—
504C4—42no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4#213312yesdirect product factor
510C2—21no—
511C2—21no—
512C2—24no—
513C2—82no—
514C2C2×C4×#9911yescharacteristic
515C2C2×#213311yescharacteristic
516C2#3336911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC2×C4C22×C42
Frattini, cosocleC22×Q8C24
socleC22C22×#99
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×#101C2semidirect
4. #12632C2semidirect
5. #12631C2semidirect
6. C4×#356C2semidirect
7. C4×#355C2semidirect
8. #12586C2semidirect
30. #2133C4direct (N × H)
78. C2×#101C2×C4semidirect
128. #356C2×C4semidirect
129. #355C2×C4semidirect
509. C4#2133direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#5362
C2×#21332
C2×C4×#1012
#126322
#126312
C4×#3562
C4×#3552
#125862