Finite Group Databasebeta

← C2×#21456766 #30551203

Subgroups — C2×#21456766 #30551203

Order 15616.

2134 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2C2×C4⋊Q8×D61C211yesmaximal, characteristic
3#21456766C2116yesmaximal, direct product factor
4C2×#16840140C212yesmaximal, complemented
5C2×#16840111C212yesmaximal, complemented
6C2×#16839973C211yesmaximal, characteristic, complemented
7C2×#16839972C211yesmaximal, characteristic, complemented
8C2×#16839969C211yesmaximal, characteristic, complemented
9C2×#16839919C212yesmaximal
10C2×#16839916C212yesmaximal
11C2×#16839814C211yesmaximal, characteristic
12C2×C61×#462C211yesmaximal, characteristic, complemented
13C22×Q8×D61C2212yes—
14C22×Dic244C2212yes—
15C22×#11758202C2212yes—
16C22×#11758202C2212yes—
17#16840316—24no—
18#16840316C2218yes—
19C2×#14519777C2211yescharacteristic
20C2×#14519776—21no—
21C2×#14519776—21no—
22C2×#14519776C2211yescharacteristic
23C2×#14519775C2212yes—
24C2×#14519775C2211yescharacteristic
25C4⋊Q8×D61—24no—
26C4⋊Q8×D61C2218yes—
27C2×#14519767—21no—
28C2×#14519757C2211yescharacteristic
29C2×#14519755—21no—
30C2×#14519753C2211yescharacteristic
31C2×C4×Dic122—21no—
32C2×C4×Dic122C2211yescharacteristic
33C2×#14519734—21no—
34C2×C4⋊C4×D61—21no—
35C2×C4⋊C4×D61—21no—
36C2×C42×D61C2211yescharacteristic
37#16840140C22116yescomplemented
38#16840111C22116yescomplemented
39#16839973C2218yescomplemented
40#16839972C2218yescomplemented
41#16839969C2218yescomplemented
42#16839919C22116yes—
43#16839916C22116yes—
44#16839814C2218yes—
45C2×C4⋊#92418C2211yescharacteristic
46C2×#14519702C2212yes—
47C2×#14519701C2212yes—
48C2×#14519700C2212yes—
49C2×#14519689C2212yes—
50C2×C8×Dic61C2212yes—
51C22×Q16×C61C2212yes—
52C2×C4⋊Q8×C61C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C61×#462C2218yescomplemented
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2212yes—
57C22×Dic122—21no—
58C22×Dic122—22no—
59C22×Dic122C2311yescharacteristic
60C22×Dic122C2312yes—
61C2×Q8×D61—24no—
62C2×Q8×D61—44no—
63C2×Q8×D61—44no—
64C2×Q8×D61C2318yes—
65C22×C4×D61—22no—
66C22×C4×D61D411yescharacteristic
67C22×C4×D61D411yescharacteristic
68C22×C4×D61C2311yescharacteristic
69C2×Dic244—44no—
70C2×Dic244C2318yes—
71C2×#11758202—44no—
72C2×#11758202—44no—
73C2×#11758202C2318yes—
74C2×#11758202C2318yes—
75C22×#92418C2312yes—
76#14519777—22no—
77#14519777C2314yes—
78#14519776—28no—
79#14519776—28no—
80#14519776—22no—
81#14519776C2314yes—
82#14519775—24no—
83#14519775—22no—
84#14519775C2318yes—
85#14519775C2314yes—
86#14519767—28no—
87#14519757—22no—
88#14519757C2314yes—
89#14519755—28no—
90#14519753—22no—
91#14519753C2314yes—
92C4×Dic122—22no—
93C4×Dic122—28no—
94C4×Dic122C2314yes—
95#14519734—28no—
96C2×C4⋊Dic61—21no—
97C2×C4⋊Dic61—21no—
98C2×C4⋊Dic61—21no—
99C2×C4⋊Dic61C2311yescharacteristic
100C2×#11758196—21no—
101C2×#11758196—22no—
102C2×#11758196—22no—
103C2×#11758196—21no—
104C2×#11758196—21no—
105C2×#11758196—21no—
106C2×#11758196—21no—
107C4⋊C4×D61—28no—
108C4⋊C4×D61—28no—
109C2×#11758198—21no—
110C2×#11758198—21no—
111C2×C4×Dic61—21no—
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115C42×D61D414yes—
116C42×D61C2314yes—
117C4⋊#92418C2314yes—
118#14519702C2318yes—
119#14519701C2318yes—
120#14519700C2318yes—
121#14519689C2318yes—
122C8×Dic61C2318yes—
123C22×Q8×C61C2312yes—
124C22×Q8×C61C2311yescharacteristic
125C2×Q16×C61—44no—
126C2×Q16×C61C2318yes—
127C22×C8×C61C2312yes—
128C4⋊Q8×C61—22no—
129C4⋊Q8×C61C2314yes—
130C4×Q8×C61—22no—
131C4×Q8×C61C2314yes—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61—21no—
134C2×C4⋊C4×C61—21no—
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C42×C61—21no—
137C2×C42×C61C2311yescharacteristic
138C4⋊C8×C61C2314yes—
139#103×C61C2318yes—
140C23×D61C2×D411yescharacteristic
141C2×Q8×C61—22no—
142C2×Q8×C61—21no—
143C2×Q8×C61—24no—
144C2×Q8×C61—24no—
145C2×Q8×C61—22no—
146C2×Q8×C61—22no—
147C2×Q8×C61C2414yes—
148C2×Q8×C61C2412yes—
149C22×C4×C61—22no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—22no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161Q8×D61—48no—
162Q8×D61—48no—
163Q8×D61—416no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—24no—
167C2×C4×D61—28no—
168C2×C4×D61—48no—
169C2×C4×D61Q1618yescomplemented
170C2×C4×D61C2×D414yes—
171C2×C4×D61C2×D412yes—
172C2×C4×D61C2×D414yes—
173C2×C4×D61C2412yes—
174C2×Dic122—24no—
175C2×Dic122—24no—
176C2×Dic122—22no—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C2×Dic122—21no—
180C2×Dic122—28no—
181C2×Dic122—24no—
182C2×Dic122—48no—
183C2×Dic122—44no—
184C2×Dic122C2414yes—
185C2×Dic122C2412yes—
186Q16×C61—48no—
187C2×C8×C61—24no—
188C2×C8×C61C2414yes—
189C4⋊C4×C61—24no—
190C4⋊C4×C61—24no—
191C4⋊C4×C61—24no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61C2412yes—
194C42×C61—24no—
195C42×C61C2×D412yes—
196C42×C61C2412yes—
197#11758202—48no—
198#11758202—48no—
199#11758198—24no—
200#11758198—24no—
201C4⋊Dic61—24no—
202C4⋊Dic61—24no—
203C4⋊Dic61—21no—
204C4⋊Dic61—24no—
205C4⋊Dic61C2412yes—
206#11758196—24no—
207#11758196—28no—
208#11758196—28no—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—24no—
213C4×Dic61—24no—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D414yes—
216C4×Dic61C2×D414yes—
217C4×Dic61C2412yes—
218C2×#92418—24no—
219C2×#92418C2414yes—
220Dic244—48no—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—22no—
224C22×D61C2×Q1614yes—
225C22×D61C22×D412yes—
226Q8×C61—22no—
227Q8×C61—24no—
228Q8×C61—22no—
229Q8×C61—24no—
230Q8×C61—44no—
231Q8×C61—42no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—44no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61C2×Q1614yes—
240C2×C4×C61C22×D412yes—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D412yes—
243C2×C4×C61H(5,𝔽2)12yes—
244C2×C4×C61H(5,𝔽2)12yes—
245C2×C4×C61C2511yescharacteristic
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—24no—
250C2×Dic61—44no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61C2×Q1614yes—
254C2×Dic61#13714yes—
255C2×Dic61C22×D412yes—
256C2×Dic61C22×D412yes—
257C2×Dic61C22×D412yes—
258C4×D61—24no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—28no—
262C4×D61—416no—
263C4×D61C2×Q16116yescomplemented
264Dic122—22no—
265Dic122—24no—
266Dic122—22no—
267Dic122—24no—
268Dic122—416no—
269Dic122—42no—
270Dic122—44no—
271Dic122—48no—
272C8×C61—24no—
273#92418—24no—
274C2×#3209—611nomaximal, Sylow
275C22×C61#53512yes—
276C22×C61C22×Q1612yes—
277C22×C61C2×#13712yes—
278C22×C61C23×D411yescharacteristic
279C2×D61—22no—
280C2×D61—28no—
281C2×D61—22no—
282C2×D61C22×Q1614yes—
283C4×C61—22no—
284C4×C61—44no—
285C4×C61—42no—
286C4×C61—42no—
287C4×C61#53512yes—
288C4×C61C22×Q1614yes—
289C4×C61C2×H(5,𝔽2)11yescharacteristic
290C4×C61C2×H(5,𝔽2)11yescharacteristic
291Dic61—22no—
292Dic61—44no—
293Dic61—42no—
294Dic61—42no—
295Dic61C22×Q1614yes—
296Dic61C2×#13714yes—
297C2×#533—611no—
298C22×C4⋊Q8—611no—
299#3209—6116no—
300C2×#494—612no—
301C2×#487—612no—
302C2×#462—611no—
303C2×#462—611no—
304C2×#462—611no—
305C2×#462D6111yescharacteristic, complemented
306C2×#451—612no—
307C2×#451—612no—
308C2×#431—611no—
309C2×C61#320914yescomplemented
310C2×C61C2×#53511yescharacteristic
311C2×C61C23×Q1611yescharacteristic
312C2×C61C22×#13711yescharacteristic
313D61—24no—
314C23×Q8—612no—
315C22×Q16—612no—
316C22×Q16—612no—
317C22×Q16—612no—
318C22×Q16C2×D6112yes—
319#533—618no—
320#533—1224no—
321C2×C4⋊Q8—611no—
322C2×C4⋊Q8—612no—
323C2×C4⋊Q8—611no—
324C2×C4⋊Q8—618no—
325C2×C4⋊Q8—611no—
326C2×C4⋊Q8—1221no—
327C2×C4⋊Q8—1221no—
328C2×C4⋊Q8—1224no—
329C2×C4⋊Q8C2×D6111yescharacteristic
330C2×#125—1221no—
331C2×C22⋊Q8—611no—
332C2×C22⋊Q8—611no—
333C2×C22⋊Q8—1221no—
334C2×C4×Q8—611no—
335C2×C4×Q8—1221no—
336C2×C4×Q8C2×D6111yescharacteristic
337C2×#117—1221no—
338C22×C4⋊C4—1221no—
339C22×C4⋊C4—1221no—
340C22×C42—611no—
341#494—6116no—
342#487—6116no—
343#462—618no—
344#462—618no—
345#462—618no—
346#462C2×D6118yescomplemented
347#451—6116no—
348#451—6116no—
349#431—618no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8C2×D6111yescharacteristic
352C2×#103—612no—
353C2×#103—612no—
354C2×#103—612no—
355C2×#103C2×D6112yes—
356C2×C22⋊C8—612no—
357C2×C4×C8—612no—
358C61C2×#320911yescharacteristic, complemented, Sylow
359C22×Q8—618no—
360C22×Q8—611no—
361C22×Q8—612no—
362C22×Q8—1221no—
363C22×Q8—1222no—
364C22×Q8—1224no—
365C22×Q8—2444no—
366C22×Q8—2444no—
367C22×Q8C22×D6111yescharacteristic
368C22×Q8C22×D6112yes—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—611no—
372C23×C4—1222no—
373C2×Q16—44no—
374C2×Q16—618no—
375C2×Q16—618no—
376C2×Q16—618no—
377C2×Q16—2444no—
378C2×Q16—2444no—
379C2×Q16—2444no—
380C2×Q16C22×D6118yes—
381C22×C8—612no—
382C22×C8C22×D6112yes—
383C4⋊Q8—22no—
384C4⋊Q8—614no—
385C4⋊Q8—618no—
386C4⋊Q8—614no—
387C4⋊Q8—614no—
388C4⋊Q8—1228no—
389C4⋊Q8—1228no—
390C4⋊Q8—1222no—
391C4⋊Q8—1224no—
392C4⋊Q8—1222no—
393C4⋊Q8—1222no—
394C4⋊Q8C22×D6114yes—
395#125—1228no—
396C22⋊Q8—614no—
397C22⋊Q8—614no—
398C22⋊Q8—1222no—
399C22⋊Q8—1228no—
400C22⋊Q8—1222no—
401C4×Q8—22no—
402C4×Q8—614no—
403C4×Q8—1228no—
404C4×Q8—1222no—
405C4×Q8C22×D6114yes—
406#117—1228no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—1228no—
412C2×C4⋊C4—1228no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C42—21no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—614no—
431C2×C42—614no—
432C2×C42—1221no—
433C2×C42C22×D6111yescharacteristic
434C4⋊C8—614no—
435C4⋊C8C22×D6114yes—
436#103—618no—
437#103—618no—
438#103—618no—
439#103C22×D6118yes—
440C22⋊C8—618no—
441C4×C8—618no—
442C24—611no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—22no—
448C2×Q8—24no—
449C2×Q8—614no—
450C2×Q8—612no—
451C2×Q8—1224no—
452C2×Q8—1228no—
453C2×Q8—1221no—
454C2×Q8—1222no—
455C2×Q8—1222no—
456C2×Q8—1224no—
457C2×Q8—1222no—
458C2×Q8—1224no—
459C2×Q8—2444no—
460C2×Q8—2448no—
461C2×Q8—2448no—
462C2×Q8—2448no—
463C2×Q8—24416no—
464C2×Q8C23×D6114yes—
465C2×Q8C23×D6112yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—614no—
472C22×C4—618no—
473C22×C4—611no—
474C22×C4—614no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—1224no—
478C22×C4—1228no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1222no—
484C22×C4—2448no—
485C22×C4D4×D6111yescharacteristic
486C22×C4D4×D6111yescharacteristic
487C22×C4C23×D6111yescharacteristic
488Q16—48no—
489Q16—2448no—
490Q16—2448no—
491Q16—2448no—
492C2×C8—24no—
493C2×C8—614no—
494C2×C8—1224no—
495C2×C8C23×D6114yes—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—612no—
501C4⋊C4—1228no—
502C4⋊C4—1228no—
503C4⋊C4—1224no—
504C4⋊C4—1224no—
505C4⋊C4—1224no—
506C4⋊C4—1224no—
507C4⋊C4—1224no—
508C4⋊C4—1221no—
509C4⋊C4—1224no—
510C4⋊C4—1224no—
511C4⋊C4—1224no—
512C4⋊C4C23×D6112yes—
513C22⋊C4—1224no—
514C22⋊C4—1224no—
515C42—24no—
516C42—614no—
517C42—614no—
518C42—612no—
519C42—612no—
520C42—1224no—
521C42D4×D6112yes—
522C42C23×D6112yes—
523C23—614no—
524C23—612no—
525C23—1222no—
526C23—1222no—
527C23C2×D4×D6111yescharacteristic
528Q8—22no—
529Q8—24no—
530Q8—22no—
531Q8—24no—
532Q8—42no—
533Q8—44no—
534Q8—1222no—
535Q8—1224no—
536Q8—1222no—
537Q8—1224no—
538Q8—2448no—
539Q8—24416no—
540Q8—2442no—
541Q8—2444no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—614no—
550C2×C4—614no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—6116no—
554C2×C4—612no—
555C2×C4—1228no—
556C2×C4—1222no—
557C2×C4—1224no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1224no—
563C2×C4—24416no—
564C2×C4—2442no—
565C2×C4—2444no—
566C2×C4—2442no—
567C2×C4Q16×D6114yes—
568C2×C4C2×D4×D6112yes—
569C2×C4C2×D4×D6112yes—
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4#1451982212yes—
572C2×C4#1451982312yes—
573C2×C4C24×D6111yescharacteristic
574C8—24no—
575C8—1224no—
576C22—614no—
577C22—1222no—
578C22—1228no—
579C22—1222no—
580C22#1684032812yes—
581C22C2×Q16×D6112yes—
582C22#137×D6112yes—
583C22C22×D4×D6111yescharacteristic
584C4—22no—
585C4—42no—
586C4—44no—
587C4—42no—
588C4—614no—
589C4—614no—
590C4—1222no—
591C4—2442no—
592C4—2444no—
593C4—2442no—
594C4#1684032812yes—
595C4C2×Q16×D6114yes—
596C4C2×#1451982211yescharacteristic
597C4C2×#1451982311yescharacteristic
598C2—1224no—
599C2#2145676614yesdirect product factor
600C2C2×#1684032811yescharacteristic
601C2C22×Q16×D6111yescharacteristic
602C2C2×#137×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#462C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#462D61
hypocenterC61C2×#3209
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
3. #21456766C2direct (N × H)
4. C2×#16840140C2semidirect
5. C2×#16840111C2semidirect
6. C2×#16839973C2semidirect
7. C2×#16839972C2semidirect
8. C2×#16839969C2semidirect
12. C2×C61×#462C2semidirect
37. #16840140C22semidirect
38. #16840111C22semidirect
39. #16839973C22semidirect
40. #16839972C22semidirect
41. #16839969C22semidirect
54. C61×#462C22semidirect
169. C2×C4×D61Q16semidirect
263. C4×D61C2×Q16semidirect
305. C2×#462D61semidirect
309. C2×C61#3209semidirect
346. #462C2×D61semidirect
358. C61C2×#3209semidirect
599. C2#21456766direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×C4⋊Q8×D612
#214567662
C2×#168401402
C2×#168401112
C2×#168399732
C2×#168399722
C2×#168399692
C2×#168399192
C2×#168399162
C2×#168398142
C2×C61×#4622
C2×#320961