Finite Group Databasebeta

← C2×#21458617 #30559719

Subgroups — C2×#21458617 #30559719

Order 15616.

1778 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840488C211yesmaximal, characteristic
2C2×#16840487C211yesmaximal, characteristic
3#21458617C2116yesmaximal, direct product factor
4C2×#16840130C211yesmaximal, characteristic
5C2×#16840129C211yesmaximal, characteristic
6C2×#16840111C211yesmaximal, characteristic
7C2×#16840071C211yesmaximal, characteristic
8C2×#16840070C211yesmaximal, characteristic
9C2×#16840045C211yesmaximal, characteristic
10C2×#16839970C211yesmaximal, characteristic
11C2×#16839970C211yesmaximal, characteristic
12C2×#16839891C211yesmaximal, characteristic
13C2×#16839869C211yesmaximal, characteristic
14C2×#16839867C211yesmaximal, characteristic
15C2×Q16×Dic61C211yesmaximal, characteristic
16C2×C61×#462C211yesmaximal, characteristic
17C22×#11758202C2211yescharacteristic
18#16840488—24no—
19#16840488C2218yes—
20#16840487—24no—
21#16840487C2218yes—
22C2×#14519776—21no—
23C2×#14519776C2211yescharacteristic
24C2×#14519775C2211yescharacteristic
25C2×#14519775C2211yescharacteristic
26C2×#14519769—21no—
27C2×#14519769C2211yescharacteristic
28C2×#14519768—21no—
29C2×#14519768—21no—
30C2×#14519768C2211yescharacteristic
31C2×#14519767—21no—
32C2×#14519767—21no—
33C2×Q8×Dic61C2211yescharacteristic
34C2×Q8×Dic61C2211yescharacteristic
35C2×Q8×Dic61C2211yescharacteristic
36C2×Q8×Dic61C2211yescharacteristic
37C2×C4×Dic122—21no—
38C2×C4×Dic122C2211yescharacteristic
39C2×C4×Dic122C2211yescharacteristic
40C2×#14519742—21no—
41#16840130C2218yes—
42#16840129C2218yes—
43#16840111C2218yes—
44#16840071C2218yes—
45#16840070C2218yes—
46#16840045C2218yes—
47#16839970C2218yes—
48#16839970C2218yes—
49#16839891C2218yes—
50#16839869C2218yes—
51#16839867C2218yes—
52Q16×Dic61C2218yes—
53C2×#14519711C2211yescharacteristic
54C2×#14519709C2211yescharacteristic
55C2×#14519708C2211yescharacteristic
56C2×#14519706C2211yescharacteristic
57C2×#14519706C2211yescharacteristic
58C2×#14519702C2211yescharacteristic
59C2×#14519702C2211yescharacteristic
60C2×#14519701C2211yescharacteristic
61C2×#14519701C2211yescharacteristic
62C2×#14519701C2211yescharacteristic
63C2×#14519701C2211yescharacteristic
64C2×#14519701C2211yescharacteristic
65C2×#14519700C2211yescharacteristic
66C2×#14519688C2211yescharacteristic
67C2×#14519687C2211yescharacteristic
68C2×C8×Dic61C2211yescharacteristic
69C22×Q16×C61C2211yescharacteristic
70C22×Q16×C61C2211yescharacteristic
71C2×C4⋊Q8×C61C2211yescharacteristic
72C2×C4×Q8×C61C2211yescharacteristic
73C61×#462C2218yes—
74C2×C4⋊C8×C61C2211yescharacteristic
75C2×#103×C61C2211yescharacteristic
76C2×#103×C61C2211yescharacteristic
77C22×Dic122—21no—
78C22×Dic122D411yescharacteristic
79C22×Dic122C2311yescharacteristic
80C2×#11758202—24no—
81C2×#11758202C2314yes—
82C22×#92418C2311yescharacteristic
83C22×#92418C2311yescharacteristic
84#14519776—28no—
85#14519776—22no—
86#14519776C2314yes—
87#14519775—22no—
88#14519775—22no—
89#14519775C2314yes—
90#14519775C2314yes—
91#14519769—22no—
92#14519769—28no—
93#14519769C2314yes—
94#14519768—28no—
95#14519768—28no—
96#14519768—22no—
97#14519768C2314yes—
98#14519767—28no—
99#14519767—28no—
100Q8×Dic61—22no—
101Q8×Dic61—22no—
102Q8×Dic61—22no—
103Q8×Dic61—22no—
104Q8×Dic61C2314yes—
105Q8×Dic61C2314yes—
106Q8×Dic61C2314yes—
107Q8×Dic61C2314yes—
108C4×Dic122—22no—
109C4×Dic122—28no—
110C4×Dic122D414yes—
111C4×Dic122C2314yes—
112C4×Dic122C2314yes—
113#14519742—28no—
114C2×C4⋊Dic61—21no—
115C2×C4⋊Dic61—21no—
116C2×C4⋊Dic61—21no—
117C2×C4⋊Dic61—21no—
118C2×C4⋊Dic61—21no—
119C2×C4⋊Dic61D411yescharacteristic
120C2×C4⋊Dic61C2311yescharacteristic
121C2×C4⋊Dic61C2311yescharacteristic
122C2×C4⋊Dic61C2311yescharacteristic
123C2×#11758196—21no—
124C2×#11758196—21no—
125C2×#11758196—21no—
126C2×#11758196—21no—
127C2×#11758196—21no—
128C2×#11758196—21no—
129C2×#11758196—21no—
130C2×#11758196—21no—
131C2×#11758196—21no—
132C2×#11758196D411yescharacteristic
133C2×#11758196D411yescharacteristic
134C2×C4×Dic61—21no—
135C2×C4×Dic61—21no—
136C2×C4×Dic61—21no—
137C2×C4×Dic61—21no—
138C2×C4×Dic61—21no—
139C2×C4×Dic61C2311yescharacteristic
140C2×C4×Dic61C2311yescharacteristic
141#14519711C2314yes—
142#14519709C2314yes—
143#14519708C2314yes—
144#14519706C2314yes—
145#14519706C2314yes—
146#14519702C2314yes—
147#14519702C2314yes—
148#14519701C2314yes—
149#14519701C2314yes—
150#14519701C2314yes—
151#14519701C2314yes—
152#14519701C2314yes—
153#14519700C2314yes—
154#14519688C2314yes—
155#14519687C2314yes—
156C8×Dic61C2314yes—
157C22×Q8×C61C2311yescharacteristic
158C22×Q8×C61C2311yescharacteristic
159C22×Q8×C61C2311yescharacteristic
160C2×Q16×C61—24no—
161C2×Q16×C61—24no—
162C2×Q16×C61C2314yes—
163C2×Q16×C61C2314yes—
164C22×C8×C61C2311yescharacteristic
165C22×C8×C61C2311yescharacteristic
166C4⋊Q8×C61—22no—
167C4⋊Q8×C61C2314yes—
168C4×Q8×C61—22no—
169C4×Q8×C61C2314yes—
170C2×C4⋊C4×C61—21no—
171C2×C4⋊C4×C61—21no—
172C2×C4⋊C4×C61—21no—
173C2×C4⋊C4×C61C2311yescharacteristic
174C2×C42×C61—21no—
175C2×C42×C61C2311yescharacteristic
176C4⋊C8×C61C2314yes—
177#103×C61C2314yes—
178#103×C61C2314yes—
179C2×Q8×C61—21no—
180C2×Q8×C61—21no—
181C2×Q8×C61—21no—
182C2×Q8×C61—22no—
183C2×Q8×C61—22no—
184C2×Q8×C61—22no—
185C2×Q8×C61—22no—
186C2×Q8×C61—22no—
187C2×Q8×C61#4114yes—
188C2×Q8×C61C2412yes—
189C2×Q8×C61C2412yes—
190C2×Q8×C61C2412yes—
191C22×C4×C61—21no—
192C22×C4×C61—21no—
193C22×C4×C61—21no—
194C22×C4×C61—21no—
195C22×C4×C61C2×D411yescharacteristic
196C22×C4×C61C2×D411yescharacteristic
197C22×C4×C61C2411yescharacteristic
198C22×Dic61—21no—
199C22×Dic61—21no—
200C22×Dic61—21no—
201C22×Dic61—21no—
202C22×Dic61C2×D411yescharacteristic
203C22×Dic61C2×D411yescharacteristic
204C22×Dic61C2×D411yescharacteristic
205C22×Dic61C2×D411yescharacteristic
206C2×Dic122—24no—
207C2×Dic122—22no—
208C2×Dic122—22no—
209C2×Dic122—21no—
210C2×Dic122—24no—
211C2×Dic122—44no—
212C2×Dic122C2×D414yes—
213C2×Dic122C2412yes—
214Q16×C61—28no—
215Q16×C61—28no—
216C2×C8×C61—22no—
217C2×C8×C61—22no—
218C2×C8×C61C2412yes—
219C2×C8×C61C2412yes—
220C4⋊C4×C61—24no—
221C4⋊C4×C61—24no—
222C4⋊C4×C61—24no—
223C4⋊C4×C61—21no—
224C4⋊C4×C61C2412yes—
225C42×C61—24no—
226C42×C61C2×D412yes—
227C42×C61C2412yes—
228#11758202—28no—
229C4⋊Dic61—21no—
230C4⋊Dic61—21no—
231C4⋊Dic61—21no—
232C4⋊Dic61—24no—
233C4⋊Dic61—24no—
234C4⋊Dic61—24no—
235C4⋊Dic61—24no—
236C4⋊Dic61—24no—
237C4⋊Dic61C2×D414yes—
238C4⋊Dic61C2412yes—
239C4⋊Dic61C2412yes—
240C4⋊Dic61C2412yes—
241#11758196—24no—
242#11758196—24no—
243#11758196—24no—
244#11758196—24no—
245#11758196—24no—
246#11758196—24no—
247#11758196—24no—
248#11758196—24no—
249#11758196—24no—
250#11758196C2×D414yes—
251#11758196C2×D414yes—
252C4×Dic61—24no—
253C4×Dic61—24no—
254C4×Dic61—24no—
255C4×Dic61—24no—
256C4×Dic61—24no—
257C4×Dic61C2×D412yes—
258C4×Dic61C2×D412yes—
259C4×Dic61C2412yes—
260C4×Dic61C2412yes—
261C2×#92418—22no—
262C2×#92418—22no—
263C2×#92418C2412yes—
264C2×#92418C2412yes—
265C23×C61C22×D411yescharacteristic
266Q8×C61—24no—
267Q8×C61—22no—
268Q8×C61—22no—
269Q8×C61—22no—
270Q8×C61—22no—
271Q8×C61—22no—
272Q8×C61—42no—
273Q8×C61—42no—
274Q8×C61C2×#4114yes—
275C2×C4×C61—24no—
276C2×C4×C61—22no—
277C2×C4×C61—22no—
278C2×C4×C61—22no—
279C2×C4×C61—22no—
280C2×C4×C61—22no—
281C2×C4×C61—22no—
282C2×C4×C61—42no—
283C2×C4×C61—42no—
284C2×C4×C61—42no—
285C2×C4×C61C22×D412yes—
286C2×C4×C61C22×D412yes—
287C2×C4×C61C22×D411yescharacteristic
288C2×C4×C61C2×#4112yes—
289C2×C4×C612−1+412yes—
290C2×C4×C61C2511yescharacteristic
291C2×Dic61—22no—
292C2×Dic61—22no—
293C2×Dic61—22no—
294C2×Dic61—22no—
295C2×Dic61—22no—
296C2×Dic61—22no—
297C2×Dic61—22no—
298C2×Dic61—42no—
299C2×Dic61—42no—
300C2×Dic61—42no—
301C2×Dic61—42no—
302C2×Dic61#13714yes—
303C2×Dic61C22×D412yes—
304C2×Dic61C22×D412yes—
305C2×Dic61C22×D412yes—
306C2×Dic61C22×D412yes—
307Dic122—28no—
308Dic122—22no—
309Dic122—22no—
310Dic122—42no—
311Dic122—48no—
312C8×C61—22no—
313C8×C61—22no—
314#92418—22no—
315#92418—22no—
316C2×#3374—611nomaximal, Sylow
317C22×C61#54812yes—
318C22×C61C2×#13712yes—
319C22×C61#57812yes—
320C22×C61C23×D411yescharacteristic
321C4×C61—24no—
322C4×C61—22no—
323C4×C61—22no—
324C4×C61—42no—
325C4×C61—42no—
326C4×C61—42no—
327C4×C61#54812yes—
328C4×C61C22×#4111yescharacteristic
329C4×C61C2×2−1+411yescharacteristic
330Dic61—22no—
331Dic61—22no—
332Dic61—22no—
333Dic61—42no—
334Dic61—42no—
335Dic61—42no—
336Dic61—42no—
337Dic61C2×#13714yes—
338C2×#557—611no—
339C2×#557—611no—
340#3374—6116no—
341C2×#491—611no—
342C2×#491—611no—
343C2×#487—611no—
344C2×#479—611no—
345C2×#479—611no—
346C2×#475—611no—
347C2×#462—611no—
348C2×#462—611no—
349C2×#462D6111yescharacteristic
350C2×#446—611no—
351C2×#441—611no—
352C2×#441—611no—
353C2×C4×Q16—611no—
354C2×C61#337414yescomplemented
355C2×C61C2×#54811yescharacteristic
356C2×C61C22×#13711yescharacteristic
357C2×C61C2×#57811yescharacteristic
358C22×Q16—611no—
359C22×Q16C2×D6111yescharacteristic
360C22×Q16C2×D6111yescharacteristic
361#557—618no—
362#557—618no—
363#557—1224no—
364#557—1224no—
365C2×C4⋊Q8—611no—
366C2×C4⋊Q8—611no—
367C2×C4⋊Q8—611no—
368C2×C4⋊Q8—1221no—
369C2×C4⋊Q8C2×D6111yescharacteristic
370C2×#125—611no—
371C2×#125—611no—
372C2×#125—1221no—
373C2×#125—1221no—
374C2×#125—1221no—
375C2×#125—1221no—
376C2×#125—1221no—
377C2×C4×Q8—611no—
378C2×C4×Q8—611no—
379C2×C4×Q8—611no—
380C2×C4×Q8—611no—
381C2×C4×Q8—611no—
382C2×C4×Q8—611no—
383C2×C4×Q8—1221no—
384C2×C4×Q8—1221no—
385C2×C4×Q8C2×D6111yescharacteristic
386#491—618no—
387#491—618no—
388#487—618no—
389#479—618no—
390#479—618no—
391#475—618no—
392#462—618no—
393#462—618no—
394#462C2×D6118yes—
395#446—618no—
396#441—618no—
397#441—618no—
398C4×Q16—618no—
399C2×#107—611no—
400C2×#106—611no—
401C2×#106—611no—
402C2×C4⋊C8—611no—
403C2×C4⋊C8—611no—
404C2×C4⋊C8C2×D6111yescharacteristic
405C2×#103—611no—
406C2×#103—611no—
407C2×#103—611no—
408C2×#103—611no—
409C2×#103—611no—
410C2×#103—611no—
411C2×#103—611no—
412C2×#103—611no—
413C2×#103C2×D6111yescharacteristic
414C2×#103C2×D6111yescharacteristic
415C2×#97—611no—
416C2×#97—611no—
417C2×C4×C8—611no—
418C61C2×#337411yescharacteristic, complemented, Sylow
419C22×Q8—611no—
420C22×Q8—611no—
421C22×Q8—1221no—
422C22×Q8C22×D6111yescharacteristic
423C22×Q8C22×D6111yescharacteristic
424C22×Q8C22×D6111yescharacteristic
425C2×Q16—24no—
426C2×Q16—24no—
427C2×Q16—614no—
428C2×Q16—1224no—
429C2×Q16C22×D6114yes—
430C2×Q16C22×D6114yes—
431C22×C8—611no—
432C22×C8—611no—
433C22×C8C22×D6111yescharacteristic
434C22×C8C22×D6111yescharacteristic
435C4⋊Q8—22no—
436C4⋊Q8—614no—
437C4⋊Q8—614no—
438C4⋊Q8—614no—
439C4⋊Q8—1222no—
440C4⋊Q8—1222no—
441C4⋊Q8—1222no—
442C4⋊Q8—1228no—
443C4⋊Q8C22×D6114yes—
444#125—614no—
445#125—614no—
446#125—1228no—
447#125—1222no—
448#125—1228no—
449#125—1228no—
450#125—1228no—
451#125—1228no—
452#125—1222no—
453C4×Q8—22no—
454C4×Q8—614no—
455C4×Q8—614no—
456C4×Q8—614no—
457C4×Q8—614no—
458C4×Q8—614no—
459C4×Q8—614no—
460C4×Q8—614no—
461C4×Q8—1228no—
462C4×Q8—1222no—
463C4×Q8—1222no—
464C4×Q8—1222no—
465C4×Q8—1222no—
466C4×Q8—1228no—
467C4×Q8—1222no—
468C4×Q8C22×D6114yes—
469C2×C4⋊C4—21no—
470C2×C4⋊C4—21no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—611no—
473C2×C4⋊C4—611no—
474C2×C4⋊C4—611no—
475C2×C4⋊C4—611no—
476C2×C4⋊C4—611no—
477C2×C4⋊C4—611no—
478C2×C4⋊C4—1221no—
479C2×C4⋊C4—1221no—
480C2×C4⋊C4—1221no—
481C2×C4⋊C4—1221no—
482C2×C4⋊C4—1221no—
483C2×C4⋊C4—1221no—
484C2×C4⋊C4—1221no—
485C2×C4⋊C4—1221no—
486C2×C4⋊C4—1221no—
487C2×C4⋊C4—1221no—
488C2×C4⋊C4—1221no—
489C2×C4⋊C4—1221no—
490C2×C4⋊C4—1221no—
491C2×C4⋊C4—1221no—
492C2×C4⋊C4C22×D6111yescharacteristic
493C2×C42—21no—
494C2×C42—611no—
495C2×C42—611no—
496C2×C42—1221no—
497C2×C42—1221no—
498C2×C42—1221no—
499C2×C42—1221no—
500C2×C42—1221no—
501C2×C42C22×D6111yescharacteristic
502#107—614no—
503#106—614no—
504#106—614no—
505C4⋊C8—614no—
506C4⋊C8—614no—
507C4⋊C8C22×D6114yes—
508#103—614no—
509#103—614no—
510#103—614no—
511#103—614no—
512#103—614no—
513#103—614no—
514#103—614no—
515#103—614no—
516#103C22×D6114yes—
517#103C22×D6114yes—
518#97—614no—
519#97—614no—
520C4×C8—614no—
521C2×Q8—21no—
522C2×Q8—21no—
523C2×Q8—21no—
524C2×Q8—22no—
525C2×Q8—22no—
526C2×Q8—22no—
527C2×Q8—22no—
528C2×Q8—22no—
529C2×Q8—614no—
530C2×Q8—612no—
531C2×Q8—1224no—
532C2×Q8—1224no—
533C2×Q8—1221no—
534C2×Q8—1222no—
535C2×Q8—1222no—
536C2×Q8—2444no—
537C2×Q8#1175822414yes—
538C2×Q8C23×D6112yes—
539C2×Q8C23×D6112yes—
540C2×Q8C23×D6112yes—
541C22×C4—21no—
542C22×C4—21no—
543C22×C4—21no—
544C22×C4—21no—
545C22×C4—611no—
546C22×C4—611no—
547C22×C4—611no—
548C22×C4—611no—
549C22×C4—1221no—
550C22×C4—1221no—
551C22×C4—1221no—
552C22×C4—1221no—
553C22×C4D4×D6111yescharacteristic
554C22×C4D4×D6111yescharacteristic
555C22×C4C23×D6111yescharacteristic
556Q16—28no—
557Q16—28no—
558Q16—1228no—
559C2×C8—22no—
560C2×C8—22no—
561C2×C8—612no—
562C2×C8—612no—
563C2×C8—1222no—
564C2×C8—1222no—
565C2×C8C23×D6112yes—
566C2×C8C23×D6112yes—
567C4⋊C4—24no—
568C4⋊C4—24no—
569C4⋊C4—24no—
570C4⋊C4—21no—
571C4⋊C4—614no—
572C4⋊C4—614no—
573C4⋊C4—614no—
574C4⋊C4—612no—
575C4⋊C4—612no—
576C4⋊C4—612no—
577C4⋊C4—1224no—
578C4⋊C4—1224no—
579C4⋊C4—1224no—
580C4⋊C4—1224no—
581C4⋊C4—1224no—
582C4⋊C4—1224no—
583C4⋊C4—1224no—
584C4⋊C4—1224no—
585C4⋊C4—1224no—
586C4⋊C4—1224no—
587C4⋊C4—1224no—
588C4⋊C4—1221no—
589C4⋊C4—1221no—
590C4⋊C4—1221no—
591C4⋊C4—1224no—
592C4⋊C4—1224no—
593C4⋊C4—1224no—
594C4⋊C4C23×D6112yes—
595C42—24no—
596C42—612no—
597C42—612no—
598C42—612no—
599C42—612no—
600C42—1224no—
601C42—1224no—
602C42—1224no—
603C42—1224no—
604C42—1224no—
605C42D4×D6112yes—
606C42C23×D6112yes—
607C23C2×D4×D6111yescharacteristic
608Q8—22no—
609Q8—24no—
610Q8—22no—
611Q8—22no—
612Q8—22no—
613Q8—22no—
614Q8—42no—
615Q8—42no—
616Q8—1228no—
617Q8—1222no—
618Q8—1222no—
619Q8—2448no—
620Q8—2442no—
621Q8C2×#1175822414yes—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—614no—
633C2×C4—612no—
634C2×C4—612no—
635C2×C4—612no—
636C2×C4—612no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—1222no—
641C2×C4—1222no—
642C2×C4—1222no—
643C2×C4—1222no—
644C2×C4—2442no—
645C2×C4—2442no—
646C2×C4—2442no—
647C2×C4—2442no—
648C2×C4C2×D4×D6112yes—
649C2×C4C2×D4×D6112yes—
650C2×C4C2×D4×D6111yescharacteristic
651C2×C4C2×#1175822412yes—
652C2×C4#1451982412yes—
653C2×C4C24×D6111yescharacteristic
654C8—22no—
655C8—22no—
656C8—1222no—
657C8—1222no—
658C22#1684041812yes—
659C22#137×D6112yes—
660C22#1684057512yes—
661C22C22×D4×D6111yescharacteristic
662C4—22no—
663C4—24no—
664C4—22no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—614no—
669C4—1222no—
670C4—1222no—
671C4—1222no—
672C4—2442no—
673C4—2442no—
674C4—2442no—
675C4—2442no—
676C4#1684041812yes—
677C4C22×#1175822411yescharacteristic
678C4C2×#1451982411yescharacteristic
679C2#2145861714yesdirect product factor
680C2C2×#1684041811yescharacteristic
681C2C2×#137×D6111yescharacteristic
682C2C2×#1684057511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#462C2
socle, prime radicalC23×C61C22×D4
hypercenterC2×#462D61
hypocenterC61C2×#3374
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. #21458617C2direct (N × H)
354. C2×C61#3374semidirect
418. C61C2×#3374semidirect
679. C2#21458617direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404882
C2×#168404872
#214586172
C2×#168401302
C2×#168401292
C2×#168401112
C2×#168400712
C2×#168400702
C2×#168400452
C2×#168399702
C2×#168398912
C2×#168398692
C2×#168398672
C2×Q16×Dic612
C2×C61×#4622
C2×#337461