Finite Group Databasebeta

← C2×C4×#352 #492986

Subgroups — C2×C4×#352 #492986

Order 512.

1588 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C22×#352C212yesmaximal
3C2×C4×#100C211yesmaximal, characteristic
4C2×C4×#99C211yesmaximal, characteristic
5C4×#352C218yesmaximal, direct product factor
6C2×#1390C212yesmaximal
7C22×#123C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C23×C42C411yescharacteristic, complemented
10C22×#100C2212yes—
11C22×#99C2212yes—
12C2×C4×M16—21no—
13C4×#123—22no—
14C4×#123—22no—
15C4×#123C414yescomplemented
16C4×#123C2214yes—
17C2×#388—21no—
18C2×#387—21no—
19C2×#386C2211yescharacteristic
20C2×#385C411yescharacteristic, complemented
21C2×#381C411yescharacteristic, complemented
22C2×C4×C4⋊C4—21no—
23C2×C4×C22⋊C4—21no—
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4C411yescharacteristic, complemented
26C2×#352C4116yesdirect product factor
27C2×#352C2218yes—
28C4×#100—22no—
29C4×#100C2214yes—
30C4×#99—22no—
31C4×#99C2214yes—
32C2×#343—21no—
33C2×#342—21no—
34#1390C2218yes—
35C23×D4C2311yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4D411yescharacteristic
40C22×M16—22no—
41C2×#123—21no—
42C2×#123—21no—
43C2×#123—24no—
44C2×#123C2×C418yes—
45C2×#123C2×C412yes—
46C2×#123C2312yes—
47C2×C4×D4—41no—
48C2×C4×D4—42no—
49C2×C4×D4—41no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2312yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42—22no—
66C22×C42—21no—
67C22×C42—21no—
68C22×C42—42no—
69C22×C42—41no—
70C22×C42—41no—
71C22×C42—41no—
72C22×C42C2×C414yescomplemented
73C22×C42D411yescharacteristic
74C2×#100—22no—
75C2×#100—24no—
76C2×#100C2×C418yes—
77C2×#100C2314yes—
78C2×#99—22no—
79C2×#99—24no—
80C2×#99C2×C418yes—
81C2×#99C2314yes—
82C4×M16—24no—
83C4×M16—42no—
84C2×#97—41no—
85C2×C4×C8—41no—
86#388—24no—
87#388—24no—
88#387—24no—
89#387—24no—
90#386—21no—
91#386—21no—
92#386C2×C412yes—
93#386C2312yes—
94#385—22no—
95#385C2×C414yescomplemented
96#381—22no—
97#381C2×C414yescomplemented
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C22⋊C4—24no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—42no—
105C4×C22⋊C4C2×C414yescomplemented
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—41no—
112#352C2×C4132yesdirect product factor
113#343—24no—
114#342—24no—
115C25C22×C411yescharacteristic
116C25C2×D411yescharacteristic
117C22×D4—41no—
118C22×D4—41no—
119C22×D4—42no—
120C22×D4C22×C411yescharacteristic
121C22×D4C22×C414yes—
122C22×D4C2×D411yescharacteristic
123C22×D4C2×D411yescharacteristic
124C22×D4#4114yes—
125C22×D4C2411yescharacteristic
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—42no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4—42no—
148C23×C4C4214yescomplemented
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C412yes—
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C2×M16—28no—
161C2×M16—24no—
162C2×M16—44no—
163C2×M16—42no—
164C22×C8—42no—
165#123—24no—
166#123—28no—
167#123—28no—
168#123—24no—
169#123C4218yescomplemented
170#123C22×C418yes—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—42no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4C4214yescomplemented
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22×C412yes—
210C2×C22⋊C4C22×C412yes—
211C2×C22⋊C4C22×C412yes—
212C2×C22⋊C4C2×D412yes—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—44no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42—41no—
225C2×C42—42no—
226C2×C42—41no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—84no—
231C2×C42C22⋊C412yes—
232C2×C42C2×D412yes—
233#100—28no—
234#100—24no—
235#100C22×C418yes—
236#99—28no—
237#99—24no—
238#99C22×C418yes—
239#97—44no—
240C4×C8—44no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—44no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—21no—
253C24—22no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24C2×C4212yes—
259C24C2×C22⋊C411yescharacteristic
260C24C2×C22⋊C411yescharacteristic
261C24C2×C22⋊C411yescharacteristic
262C24#11712yes—
263C24C4×D414yes—
264C24C23×C411yescharacteristic
265C24C22×D411yescharacteristic
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—41no—
278C2×D4C2×C4212yes—
279C2×D4#11712yes—
280C2×D4C4×D414yes—
281C2×D4C4×D414yes—
282C2×D4C23×C412yes—
283C2×D4C2×#4112yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—44no—
312C22×C4—48no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—48no—
328C22×C4—44no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—48no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—84no—
351C22×C4—82no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4#9911yescharacteristic
355C22×C4#9912yes—
356C22×C4#9911yescharacteristic
357C22×C4C2×C4214yescomplemented
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C4×D414yes—
364C22×C4C22×D411yescharacteristic
365M16—28no—
366M16—44no—
367M16—48no—
368C2×C8—44no—
369C2×C8—48no—
370C4⋊C4—28no—
371C4⋊C4—28no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—48no—
375C22⋊C4—22no—
376C22⋊C4—28no—
377C22⋊C4—28no—
378C22⋊C4—24no—
379C22⋊C4—28no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—48no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4C2×C4214yescomplemented
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—48no—
396C42—48no—
397C42—44no—
398C42—44no—
399C42—88no—
400C23—22no—
401C23—22no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—21no—
414C23—21no—
415C23—24no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—44no—
421C23—44no—
422C23—42no—
423C23—42no—
424C23—48no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—48no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—81no—
435C23—81no—
436C23—84no—
437C23C4×C22⋊C412yes—
438C23C4×C22⋊C412yes—
439C23C4×C22⋊C412yes—
440C23C2×#9911yescharacteristic
441C23C2×#9911yescharacteristic
442C23#41012yes—
443C23C22×C4211yescharacteristic
444C23C22×C22⋊C411yescharacteristic
445C23C2×#11711yescharacteristic
446C23C2×C4×D412yes—
447D4—42no—
448D4—42no—
449D4—44no—
450D4—44no—
451D4—42no—
452D4—42no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4—816no—
510C2×C4#35214yesdirect product factor
511C2×C4C4×C22⋊C412yes—
512C2×C4C2×#9912yes—
513C2×C4C2×#9912yes—
514C2×C4C2×C4×D412yes—
515C8—48no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—24no—
521C22—24no—
522C22—21no—
523C22—22no—
524C22—22no—
525C22—21no—
526C22—22no—
527C22—41no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—41no—
532C22—42no—
533C22—42no—
534C22—44no—
535C22—44no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—48no—
545C22—44no—
546C22—44no—
547C22—81no—
548C22—81no—
549C22—84no—
550C22C4×#9912yes—
551C22C4×#9912yes—
552C22C2×#35212yes—
553C22#212012yes—
554C22C2×C4×C22⋊C411yescharacteristic
555C22C22×#9911yescharacteristic
556C22C2×#41011yescharacteristic
557C4—22no—
558C4—22no—
559C4—22no—
560C4—44no—
561C4—42no—
562C4—42no—
563C4—44no—
564C4—42no—
565C4—44no—
566C4—82no—
567C4—88no—
568C4C2×#35214yesdirect product factor
569C2—21no—
570C2—21no—
571C2—22no—
572C2—42no—
573C2—41no—
574C2—42no—
575C2—41no—
576C2—42no—
577C2C4×#35214yesdirect product factor
578C2C2×C4×#9911yescharacteristic
579C2C22×#35211yescharacteristic
580C2C2×#212011yescharacteristic
subgroupG/H
centerC22×C4#99
commutatorC23C22×C42
Frattini, cosocleC22×D4C24
socleC23C2×#99
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C4×#352C2direct (N × H)
9. C23×C42C4semidirect
15. C4×#123C4semidirect
20. C2×#385C4semidirect
21. C2×#381C4semidirect
25. C2×C4×C22⋊C4C4semidirect
26. C2×#352C4direct (N × H)
72. C22×C42C2×C4semidirect
95. #385C2×C4semidirect
97. #381C2×C4semidirect
105. C4×C22⋊C4C2×C4semidirect
112. #352C2×C4direct (N × H)
148. C23×C4C42semidirect
169. #123C42semidirect
206. C2×C22⋊C4C42semidirect
357. C22×C4C2×C42semidirect
390. C22⋊C4C2×C42semidirect
510. C2×C4#352direct (N × H)
568. C4C2×#352direct (N × H)
577. C2C4×#352direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C22×#3522
C2×C4×#1002
C2×C4×#992
C4×#3522
C2×#13902