Finite Group Databasebeta

← C2×#21462009 #30616488

Subgroups — C2×#21462009 #30616488

Order 15616.

1698 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840704C211yesmaximal, characteristic
2C2×C4×C22⋊#6923C211yesmaximal, characteristic
3#21462009C218yesmaximal, direct product factor
4C2×#16840640C214yesmaximal
5C22×#14519757C211yesmaximal, characteristic
6C22×C4⋊#6923—21no—
7C22×C22⋊#6923C2212yes—
8C22×C22⋊#6923C2212yes—
9C22×C4×#6923—21no—
10#16840704—22no—
11#16840704C2214yes—
12C4×C22⋊#6923—22no—
13C4×C22⋊#6923C2214yes—
14C2×(C2×#6923)⋊C4—21no—
15C2×(C2×#6923)⋊C4—21no—
16#16840640C22116yes—
17C23×Dic122C411yescharacteristic, complemented
18C23×C4×D61C2211yescharacteristic
19C2×#14519757—21no—
20C2×#14519757—21no—
21C2×#14519757—22no—
22C2×#14519757—21no—
23C2×#14519757—21no—
24C2×#14519757—22no—
25C2×#14519757C414yescomplemented
26C2×#14519757C2214yes—
27C22×C4⋊Dic61C412yescomplemented
28C22×#11758198C2212yes—
29C22×C4⋊C4×C61C411yescharacteristic, complemented
30C23×#6923—22no—
31C23×#6923—22no—
32C2×C4⋊#6923—24no—
33C2×C4⋊#6923—44no—
34C2×C22⋊#6923—22no—
35C2×C22⋊#6923—24no—
36C2×C22⋊#6923—28no—
37C2×C22⋊#6923—22no—
38C2×C22⋊#6923Q818yescomplemented
39C2×C22⋊#6923C2314yes—
40C2×C22⋊#6923C2314yes—
41C2×C4×#6923—24no—
42C2×C4×#6923—44no—
43(C2×#6923)⋊C4—24no—
44(C2×#6923)⋊C4—24no—
45C24×D61C2311yescharacteristic
46C22×Dic122—21no—
47C22×Dic122—22no—
48C22×Dic122—21no—
49C22×Dic122—44no—
50C22×Dic122C2×C414yescomplemented
51C23×Dic61C2×C412yes—
52C23×Dic61C2×C411yescharacteristic
53C22×C4×D61—44no—
54C22×C4×D61C2×C412yes—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D412yes—
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D412yes—
61C22×C4×D61C2312yes—
62#14519757—24no—
63#14519757—24no—
64#14519757—24no—
65#14519757—24no—
66#14519757—24no—
67#14519757—24no—
68#14519757—24no—
69#14519757—24no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—24no—
73C2×C4⋊Dic61C2×C418yescomplemented
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198C2×C414yes—
79C2×#11758198C2314yes—
80C23×C4×C61C2×C412yes—
81C23×C4×C61C2×C411yescharacteristic
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61C2×C414yescomplemented
86C24×C61C22×C411yescharacteristic
87C23×D61—42no—
88C23×D61—42no—
89C23×D61C22×C411yescharacteristic
90C23×D61C2×D411yescharacteristic
91C23×D61C2×D411yescharacteristic
92C23×D61C2×D411yescharacteristic
93C23×D61C2×D411yescharacteristic
94C23×D61C2×Q814yes—
95C23×D61#4114yes—
96C23×D61C2411yescharacteristic
97C22×#6923—28no—
98C22×#6923—24no—
99C22×#6923—24no—
100C22×#6923—44no—
101C22×#6923—44no—
102C22×#6923—44no—
103C22×#6923—44no—
104C22×#6923—44no—
105C22×C4×C61—24no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—44no—
109C22×C4×C61C22⋊C411yescharacteristic
110C22×C4×C61C22⋊C412yes—
111C22×C4×C61C22⋊C411yescharacteristic
112C22×C4×C61C22×C412yes—
113C22×C4×C61C22×C414yes—
114C22×Dic61—22no—
115C22×Dic61—22no—
116C22×Dic61—42no—
117C22×Dic61—44no—
118C22×Dic61C22⋊C411yescharacteristic
119C22×Dic61C22⋊C411yescharacteristic
120C22×Dic61C22⋊C412yes—
121C22×Dic61C22×C414yes—
122C22×Dic61C22×C412yes—
123C2×C4×D61—24no—
124C2×C4×D61—24no—
125C2×C4×D61—44no—
126C2×C4×D61—44no—
127C2×C4×D61—44no—
128C2×C4×D61—44no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×C4×D61C22⋊C412yes—
132C2×C4×D61C22⋊C412yes—
133C2×C4×D61C22⋊C412yes—
134C2×C4×D61C22⋊C412yes—
135C2×C4×D61C2×D412yes—
136C2×C4×D61C2×D412yes—
137C2×C4×D61C2×D412yes—
138C2×C4×D61C2×D412yes—
139C2×Dic122—24no—
140C2×Dic122—24no—
141C2×Dic122—44no—
142C2×Dic122—48no—
143C2×Dic122—44no—
144C2×Dic122—48no—
145C22⋊#6923—28no—
146C22⋊#6923—24no—
147C22⋊#6923—28no—
148C22⋊#6923—44no—
149C22⋊#6923C2×Q818yescomplemented
150C4⋊#6923—48no—
151C4×#6923—48no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61—24no—
154#11758198—24no—
155#11758198—24no—
156#11758198—24no—
157#11758198—24no—
158C4⋊Dic61—28no—
159C4⋊Dic61—28no—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61C2×C22⋊C411yescharacteristic
163C23×C61C2×C22⋊C411yescharacteristic
164C23×C61C4×Q814yes—
165C23×C61C23×C411yescharacteristic
166C22×D61—22no—
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—48no—
178C22×D61—48no—
179C22×D61—41no—
180C22×D61C2×C22⋊C411yescharacteristic
181C22×D61C2×C22⋊C411yescharacteristic
182C22×D61C4×Q814yes—
183C22×D61C22⋊Q814yes—
184C22×D61C22⋊Q814yes—
185C22×D61#12412yes—
186C22×D61#12412yes—
187C22×D61C4⋊Q812yes—
188C22×D61C4⋊Q812yes—
189C22×D61C22×D411yescharacteristic
190C22×D61C22×D411yescharacteristic
191C22×D61C22×Q812yes—
192C22×D61C2×#4112yes—
193C2×#6923—28no—
194C2×#6923—44no—
195C2×#6923—44no—
196C2×#6923—416no—
197C2×#6923—44no—
198C2×#6923—44no—
199C2×#6923—44no—
200C2×#6923—88no—
201C2×C4×C61—22no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—44no—
206C2×C4×C61—44no—
207C2×C4×C61—44no—
208C2×C4×C61#9914yes—
209C2×C4×C61C2×C22⋊C412yes—
210C2×C4×C61C2×C22⋊C412yes—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—24no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—44no—
217C2×Dic61—44no—
218C2×Dic61—44no—
219C2×Dic61C2×C22⋊C412yes—
220C2×Dic61C2×C22⋊C412yes—
221C4×D61—24no—
222C4×D61—24no—
223C4×D61—24no—
224C4×D61—24no—
225C4×D61—42no—
226C4×D61—42no—
227C4×D61—42no—
228C4×D61—42no—
229Dic122—48no—
230Dic122—48no—
231C2×#1996—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—28no—
237C22×C61#39112yes—
238C22×C61#39112yes—
239C22×C61C2×#9912yes—
240C22×C61#41012yes—
241C22×C61C22×C22⋊C411yescharacteristic
242C22×C61C2×C4×Q812yes—
243C2×D61—21no—
244C2×D61—21no—
245C2×D61—21no—
246C2×D61—21no—
247C2×D61—21no—
248C2×D61—21no—
249C2×D61—21no—
250C2×D61—21no—
251C2×D61—28no—
252C2×D61—24no—
253C2×D61—24no—
254C2×D61—44no—
255C2×D61—44no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61#39112yes—
261C2×D61#39112yes—
262C2×D61C2×C22⋊Q812yes—
263C2×D61C2×#12411yescharacteristic
264C2×D61C2×C4⋊Q811yescharacteristic
265#6923—48no—
266#6923—84no—
267C4×C61—22no—
268C4×C61—44no—
269C4×C61C2×#9914yes—
270Dic61—22no—
271Dic61—22no—
272Dic61—44no—
273C22×C22⋊Q8—611no—
274C2×#380—611no—
275C2×C4×C22⋊C4—611no—
276#1996—618no—
277C2×#342—614no—
278C2×C61—22no—
279C2×C61—22no—
280C2×C61#199614yescomplemented
281C2×C61C2×#39111yescharacteristic
282C2×C61C22×#9911yescharacteristic
283C2×C61C2×#41011yescharacteristic
284D61—22no—
285D61—21no—
286D61—21no—
287D61—22no—
288D61—22no—
289C23×Q8—611no—
290C24×C4—611no—
291C2×C22⋊Q8—614no—
292C2×C22⋊Q8—614no—
293C2×C22⋊Q8—1221no—
294C2×C22⋊Q8—1221no—
295C2×C22⋊Q8—1221no—
296C2×C22⋊Q8—1221no—
297C2×C22⋊Q8—1222no—
298C2×C22⋊Q8—1222no—
299C22×C4⋊C4—612no—
300C22×C4⋊C4—1221no—
301C22×C4⋊C4#692311yescharacteristic, complemented
302C22×C22⋊C4—612no—
303C22×C22⋊C4—612no—
304C22×C22⋊C4—612no—
305C22×C42—1221no—
306#380—614no—
307#380—1222no—
308C4×C22⋊C4—614no—
309C4×C22⋊C4—1222no—
310C2×#95—1221no—
311C2×#95—1221no—
312#342—6116no—
313C61C2×#199611yescharacteristic, complemented, Sylow
314C25—611no—
315C22×Q8—614no—
316C22×Q8—1222no—
317C22×Q8—1221no—
318C22×Q8—1221no—
319C22×Q8—2444no—
320C23×C4—611no—
321C23×C4—611no—
322C23×C4—611no—
323C23×C4—611no—
324C23×C4—611no—
325C23×C4—612no—
326C23×C4—612no—
327C23×C4—612no—
328C23×C4—612no—
329C23×C4—612no—
330C23×C4—1222no—
331C23×C4—1222no—
332C23×C4—2444no—
333C23×C4C2×#692312yes—
334C23×C4C2×#692311yescharacteristic
335C22⋊Q8—1224no—
336C22⋊Q8—1224no—
337C22⋊Q8—1224no—
338C22⋊Q8—1224no—
339C22⋊Q8—1224no—
340C22⋊Q8—1224no—
341C22⋊Q8—1224no—
342C22⋊Q8—1224no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—618no—
347C2×C4⋊C4—1224no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1224no—
351C2×C4⋊C4—2444no—
352C2×C4⋊C4C2×#692314yescomplemented
353C2×C22⋊C4—618no—
354C2×C22⋊C4—614no—
355C2×C22⋊C4—614no—
356C2×C22⋊C4—614no—
357C2×C22⋊C4—614no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4—1224no—
363C2×C22⋊C4—1222no—
364C2×C22⋊C4—1228no—
365C2×C22⋊C4—1222no—
366C2×C42—1224no—
367C2×C42—2444no—
368#95—1224no—
369#95—1224no—
370C24—611no—
371C24—611no—
372C24—611no—
373C24—611no—
374C24—614no—
375C24—614no—
376C24—611no—
377C24—611no—
378C24—2442no—
379C24—2442no—
380C24C22×#692311yescharacteristic
381C2×Q8—1224no—
382C2×Q8—1224no—
383C2×Q8—2448no—
384C2×Q8—2444no—
385C2×Q8—2448no—
386C2×Q8—2444no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—44no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—612no—
394C22×C4—612no—
395C22×C4—612no—
396C22×C4—612no—
397C22×C4—612no—
398C22×C4—612no—
399C22×C4—612no—
400C22×C4—612no—
401C22×C4—612no—
402C22×C4—612no—
403C22×C4—614no—
404C22×C4—1224no—
405C22×C4—1224no—
406C22×C4—1224no—
407C22×C4—1228no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1224no—
411C22×C4—2444no—
412C22×C4—2444no—
413C22×C4—2444no—
414C22×C4—2444no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4—2444no—
419C22×C4—2444no—
420C22×C4—2444no—
421C22×C4—2444no—
422C22×C4—2444no—
423C22×C4—2444no—
424C22×C4C22⋊#692312yes—
425C22×C4C22⋊#692311yescharacteristic
426C22×C4C22⋊#692311yescharacteristic
427C22×C4C22×#692312yes—
428C22×C4C22×#692314yes—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—1228no—
432C4⋊C4—1228no—
433C4⋊C4—2448no—
434C22⋊C4—618no—
435C22⋊C4—1224no—
436C22⋊C4—1224no—
437C22⋊C4—1224no—
438C22⋊C4—1224no—
439C22⋊C4—1228no—
440C22⋊C4—1224no—
441C22⋊C4—1228no—
442C22⋊C4—2444no—
443C42—2448no—
444C23—22no—
445C23—22no—
446C23—614no—
447C23—612no—
448C23—612no—
449C23—611no—
450C23—611no—
451C23—611no—
452C23—611no—
453C23—614no—
454C23—612no—
455C23—612no—
456C23—614no—
457C23—612no—
458C23—612no—
459C23—1222no—
460C23—1222no—
461C23—1222no—
462C23—1222no—
463C23—2441no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23—2441no—
470C23—2441no—
471C23—2448no—
472C23—2448no—
473C23C2×C22⋊#692311yescharacteristic
474C23C2×C22⋊#692311yescharacteristic
475C23Q8×#692314yes—
476C23C23×#692311yescharacteristic
477Q8—2448no—
478Q8—2448no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—612no—
487C2×C4—612no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1224no—
491C2×C4—1224no—
492C2×C4—1224no—
493C2×C4—1224no—
494C2×C4—1228no—
495C2×C4—1224no—
496C2×C4—4888no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2444no—
504C2×C4—2444no—
505C2×C4—2444no—
506C2×C4—2444no—
507C2×C4—2444no—
508C2×C4—2444no—
509C2×C4—24416no—
510C2×C4—2444no—
511C2×C4—2444no—
512C2×C4#1451983614yes—
513C2×C4C2×C22⋊#692312yes—
514C2×C4C2×C22⋊#692312yes—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—28no—
520C22—612no—
521C22—611no—
522C22—611no—
523C22—612no—
524C22—612no—
525C22—1221no—
526C22—1221no—
527C22—1221no—
528C22—1221no—
529C22—1221no—
530C22—1221no—
531C22—1221no—
532C22—1221no—
533C22—1224no—
534C22—1228no—
535C22—1224no—
536C22—2442no—
537C22—2444no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22—2444no—
542C22(C2×#6923)⋊Q812yes—
543C22(C2×#6923)⋊Q812yes—
544C22C2×#1451983612yes—
545C22#1684074212yes—
546C22C22×C22⋊#692311yescharacteristic
547C22C2×Q8×#692312yes—
548C4—22no—
549C4—44no—
550C4—1222no—
551C4—1222no—
552C4—4884no—
553C4—2448no—
554C4—2444no—
555C4C2×#1451983614yes—
556C2—22no—
557C2—22no—
558C2—1221no—
559C2—1222no—
560C2—1221no—
561C2—1222no—
562C2—1222no—
563C2#2146200914yesdirect product factor
564C2C2×(C2×#6923)⋊Q811yescharacteristic
565C2C22×#1451983611yescharacteristic
566C2C2×#1684074211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61C23
cosocleC23×D61C24
hypercenterC22×C4⋊C4#6923
hypocenterC61C2×#1996
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21462009C2direct (N × H)
17. C23×Dic122C4semidirect
25. C2×#14519757C4semidirect
27. C22×C4⋊Dic61C4semidirect
29. C22×C4⋊C4×C61C4semidirect
38. C2×C22⋊#6923Q8semidirect
50. C22×Dic122C2×C4semidirect
73. C2×C4⋊Dic61C2×C4semidirect
85. C2×C4⋊C4×C61C2×C4semidirect
149. C22⋊#6923C2×Q8semidirect
280. C2×C61#1996semidirect
301. C22×C4⋊C4#6923semidirect
313. C61C2×#1996semidirect
352. C2×C4⋊C4C2×#6923semidirect
563. C2#21462009direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168407042
C2×C4×C22⋊#69232
#214620092
C2×#168406402
C22×#145197572
C2×#199661