Finite Group Databasebeta

← C2×C61×#3461 #29566115

Subgroups — C2×C61×#3461 #29566115

Order 15616.

7130 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3461C2132yesmaximal, direct product factor
2C2×C61×#544C211yesmaximal, characteristic, complemented
3C2×C61×#542C214yesmaximal, complemented
4C2×C61×#541C214yesmaximal, complemented
5C2×C61×#539C214yesmaximal, complemented
6C2×C61×#537C211yesmaximal, characteristic
7C2×C61×#536C212yesmaximal, complemented
8C2×C61×#533C214yesmaximal, complemented
9C2×C61×#529C214yesmaximal, complemented
10C2×C61×#529C214yesmaximal, complemented
11C2×C4×#41×C61C212yesmaximal, complemented
12C22×C4×Q8×C61C211yesmaximal, characteristic, complemented
13C22×#41×C61C2212yes—
14C23×Q8×C61C2211yescharacteristic
15C61×#544C22116yescomplemented
16C61×#542C22164yescomplemented
17C61×#541C22164yescomplemented
18C61×#539C22164yescomplemented
19C61×#537C22116yes—
20C61×#536C22132yescomplemented
21C61×#533C22164yescomplemented
22C2×C4⋊Q8×C61C2214yes—
23C2×C4⋊Q8×C61C2214yes—
24C61×#529C22164yescomplemented
25C61×#529C22164yescomplemented
26C2×#126×C61C2218yes—
27C2×#126×C61C2218yes—
28C2×#125×C61C2214yes—
29C2×#125×C61C2214yes—
30C2×#125×C61C2218yes—
31C2×#124×C61C2214yes—
32C2×#124×C61C2214yes—
33C2×#123×C61C2218yes—
34C2×#123×C61C2218yes—
35C2×C22⋊Q8×C61C2214yes—
36C2×C22⋊Q8×C61C2214yes—
37C2×C22⋊Q8×C61C2214yes—
38C2×C22⋊Q8×C61C2214yes—
39C2×C22⋊Q8×C61C2218yes—
40C2×#121×C61C2214yes—
41C2×#121×C61C2214yes—
42C4×#41×C61C22132yescomplemented
43C2×C4×Q8×C61C2214yes—
44C2×C4×Q8×C61C2214yes—
45C2×C4×Q8×C61C2212yes—
46C2×C4×Q8×C61C2212yes—
47C2×C4×Q8×C61C2214yes—
48C2×C4×Q8×C61C22116yescomplemented
49C2×C4×D4×C61C2212yes—
50C2×C4×D4×C61C2214yes—
51C2×C4×D4×C61C2214yes—
52C2×C4×D4×C61C2212yes—
53C2×C4×D4×C61C2214yes—
54C2×#117×C61C2212yes—
55C2×#117×C61C2214yes—
56C2×#117×C61C2218yes—
57C22×C4⋊C4×C61C2211yescharacteristic
58C22×C4⋊C4×C61C2212yes—
59C22×C42×C61C2211yescharacteristic
60C22×C42×C61C2212yes—
61C22×C42×C61C2214yes—
62C2×#41×C61—216no—
63C2×#41×C61C23116yes—
64C22×Q8×C61—28no—
65C22×Q8×C61C2312yes—
66C22×Q8×C61C2314yes—
67C22×Q8×C61C2318yes—
68C22×Q8×C61C2314yes—
69C22×D4×C61C2314yes—
70C22×D4×C61C2312yes—
71C22×D4×C61C2314yes—
72C23×C4×C61C2312yes—
73C23×C4×C61C2314yes—
74C23×C4×C61C2311yescharacteristic
75C23×C4×C61C2314yes—
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2312yes—
78C23×C4×C61C2314yes—
79C23×C4×C61C2311yescharacteristic
80C4⋊Q8×C61C23132yes—
81C4⋊Q8×C61C23132yes—
82#126×C61C23164yes—
83#126×C61C23164yes—
84#125×C61C23164yes—
85#125×C61C23132yes—
86#125×C61C23132yes—
87#124×C61C23132yes—
88#124×C61C23132yes—
89#123×C61C23164yes—
90#123×C61C23164yes—
91C22⋊Q8×C61C23132yes—
92C22⋊Q8×C61C23132yes—
93C22⋊Q8×C61C23164yes—
94C22⋊Q8×C61C23132yes—
95C22⋊Q8×C61C23132yes—
96#121×C61C23132yes—
97#121×C61C23132yes—
98C4×Q8×C61C23116yes—
99C4×Q8×C61C23132yes—
100C4×Q8×C61C23132yes—
101C4×Q8×C61C23116yes—
102C4×Q8×C61C23132yes—
103C4×D4×C61C23132yes—
104C4×D4×C61C23116yes—
105C4×D4×C61C23116yes—
106C4×D4×C61C23132yes—
107C4×D4×C61C23132yes—
108#117×C61C23164yes—
109#117×C61C23116yes—
110#117×C61C23132yes—
111C2×C4⋊C4×C61C2312yes—
112C2×C4⋊C4×C61C2314yes—
113C2×C4⋊C4×C61C2318yes—
114C2×C4⋊C4×C61C2318yes—
115C2×C4⋊C4×C61C2318yes—
116C2×C4⋊C4×C61C2318yes—
117C2×C4⋊C4×C61C23116yes—
118C2×C4⋊C4×C61C2318yes—
119C2×C4⋊C4×C61C2314yes—
120C2×C4⋊C4×C61C2312yes—
121C2×C4⋊C4×C61C2314yes—
122C2×C4⋊C4×C61C2314yes—
123C2×C4⋊C4×C61C2314yes—
124C2×C22⋊C4×C61C2314yes—
125C2×C22⋊C4×C61C2314yes—
126C2×C22⋊C4×C61C2314yes—
127C2×C22⋊C4×C61C2318yes—
128C2×C22⋊C4×C61C2318yes—
129C2×C22⋊C4×C61C2318yes—
130C2×C42×C61C2314yes—
131C2×C42×C61C23132yes—
132C2×C42×C61C2312yes—
133C2×C42×C61C2314yes—
134C2×C42×C61C2314yes—
135C2×C42×C61C2318yes—
136C2×C42×C61C23116yes—
137C2×C42×C61C2312yes—
138C2×C42×C61C2314yes—
139C2×C42×C61C2314yes—
140C24×C61C2414yes—
141C24×C61C2411yescharacteristic
142#41×C61—264no—
143C2×Q8×C61—216no—
144C2×Q8×C61—232no—
145C2×Q8×C61—28no—
146C2×Q8×C61—48no—
147C2×Q8×C61C24116yes—
148C2×Q8×C61C2418yes—
149C2×Q8×C61C24116yes—
150C2×D4×C61—216no—
151C2×D4×C61—28no—
152C2×D4×C61—48no—
153C2×D4×C61C24116yes—
154C2×D4×C61C2418yes—
155C2×D4×C61C24116yes—
156C22×C4×C61—28no—
157C22×C4×C61—216no—
158C22×C4×C61—24no—
159C22×C4×C61—216no—
160C22×C4×C61—42no—
161C22×C4×C61—48no—
162C22×C4×C61—44no—
163C22×C4×C61#4118yes—
164C22×C4×C61C2411yescharacteristic
165C22×C4×C61C2414yes—
166C22×C4×C61C2412yes—
167C22×C4×C61C2414yes—
168C22×C4×C61C2414yes—
169C22×C4×C61C2414yes—
170C22×C4×C61C2414yes—
171C22×C4×C61C2414yes—
172C22×C4×C61C2418yes—
173C22×C4×C61C24116yes—
174C22×C4×C61C24116yes—
175C22×C4×C61C2414yes—
176C22×C4×C61C2418yes—
177C22×C4×C61C24116yes—
178C22×C4×C61C2412yes—
179C22×C4×C61C2414yes—
180C22×C4×C61C2411yescharacteristic
181C4⋊C4×C61C24116yes—
182C4⋊C4×C61C24132yes—
183C4⋊C4×C61C24116yes—
184C4⋊C4×C61C2418yes—
185C4⋊C4×C61C2418yes—
186C4⋊C4×C61C24116yes—
187C4⋊C4×C61C24132yes—
188C4⋊C4×C61C24132yes—
189C4⋊C4×C61C24116yes—
190C4⋊C4×C61C24116yes—
191C4⋊C4×C61C24132yes—
192C22⋊C4×C61C24116yes—
193C22⋊C4×C61C24116yes—
194C22⋊C4×C61C24116yes—
195C22⋊C4×C61C24132yes—
196C22⋊C4×C61C24132yes—
197C22⋊C4×C61C24132yes—
198C42×C61C2418yes—
199C42×C61C2418yes—
200C42×C61C24116yes—
201C42×C61C24116yes—
202C42×C61C24116yes—
203C42×C61C24116yes—
204C42×C61C24116yes—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—28no—
208C23×C61—28no—
209C23×C61—28no—
210C23×C61C2×#4114yes—
211C23×C61C2518yes—
212C23×C61C2512yes—
213C23×C61C2511yescharacteristic
214Q8×C61—232no—
215Q8×C61—216no—
216Q8×C61—416no—
217D4×C61—232no—
218D4×C61—216no—
219D4×C61—416no—
220C2×C4×C61—28no—
221C2×C4×C61—216no—
222C2×C4×C61—28no—
223C2×C4×C61—22no—
224C2×C4×C61—28no—
225C2×C4×C61—24no—
226C2×C4×C61—28no—
227C2×C4×C61—28no—
228C2×C4×C61—232no—
229C2×C4×C61—28no—
230C2×C4×C61—232no—
231C2×C4×C61—44no—
232C2×C4×C61—416no—
233C2×C4×C61—48no—
234C2×C4×C61C2×#4118yes—
235C2×C4×C61C2×#41116yes—
236C2×C4×C612−1+414yes—
237C2×C4×C61C2512yes—
238C2×C4×C61C2514yes—
239C2×C4×C61C2518yes—
240C2×C4×C61C2518yes—
241C2×C4×C61C2518yes—
242C2×C4×C61C2518yes—
243C2×C4×C61C2518yes—
244C2×C4×C61C2514yes—
245C2×C4×C61C2512yes—
246C2×#3461C6111yesmaximal, characteristic, direct product factor, Sylow
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—28no—
250C22×C61—28no—
251C22×C61—28no—
252C22×C61—28no—
253C22×C61—416no—
254C22×C61C22×#4114yes—
255C22×C61C22×#4112yes—
256C22×C61C2×2−1+412yes—
257C22×C61#58512yes—
258C22×C61C2611yescharacteristic
259C4×C61—28no—
260C4×C61—28no—
261C4×C61—22no—
262C4×C61—28no—
263C4×C61—24no—
264C4×C61—28no—
265C4×C61—28no—
266C4×C61C22×#4118yes—
267C4×C61C2×2−1+414yes—
268#3461C2×C61132yesdirect product factor
269C2×#544C2×C6111yescharacteristic, complemented
270C2×#542C2×C6114yescomplemented
271C2×#541C2×C6114yescomplemented
272C2×#539C2×C6114yescomplemented
273C2×#537C2×C6111yescharacteristic
274C2×#536C2×C6112yescomplemented
275C2×#533C2×C6114yescomplemented
276C2×#529C2×C6114yescomplemented
277C2×#529C2×C6114yescomplemented
278C2×C4×#41C2×C6112yescomplemented
279C22×C4×Q8C2×C6111yescharacteristic, complemented
280C2×C61—24no—
281C2×C61—48no—
282C2×C61#346114yesdirect product factor
283C2×C61C23×#4111yescharacteristic
284C2×C61C22×2−1+411yescharacteristic
285C2×C61C2×#58511yescharacteristic
286C22×#41C22×C6112yes—
287C23×Q8C22×C6111yescharacteristic
288#544C22×C61116yescomplemented
289#542C22×C61164yescomplemented
290#541C22×C61164yescomplemented
291#539C22×C61164yescomplemented
292#537C22×C61116yes—
293#536C22×C61132yescomplemented
294#533C22×C61164yescomplemented
295C2×C4⋊Q8C22×C6114yes—
296C2×C4⋊Q8C22×C6114yes—
297#529C22×C61164yescomplemented
298#529C22×C61164yescomplemented
299C2×#126C22×C6118yes—
300C2×#126C22×C6118yes—
301C2×#125C22×C6114yes—
302C2×#125C22×C6114yes—
303C2×#125C22×C6118yes—
304C2×#124C22×C6114yes—
305C2×#124C22×C6114yes—
306C2×#123C22×C6118yes—
307C2×#123C22×C6118yes—
308C2×C22⋊Q8C22×C6114yes—
309C2×C22⋊Q8C22×C6114yes—
310C2×C22⋊Q8C22×C6114yes—
311C2×C22⋊Q8C22×C6114yes—
312C2×C22⋊Q8C22×C6118yes—
313C2×#121C22×C6114yes—
314C2×#121C22×C6114yes—
315C4×#41C22×C61132yescomplemented
316C2×C4×Q8C22×C6114yes—
317C2×C4×Q8C22×C6114yes—
318C2×C4×Q8C22×C6112yes—
319C2×C4×Q8C22×C6112yes—
320C2×C4×Q8C22×C6114yes—
321C2×C4×Q8C22×C61116yescomplemented
322C2×C4×D4C22×C6114yes—
323C2×C4×D4C22×C6112yes—
324C2×C4×D4C22×C6114yes—
325C2×C4×D4C22×C6112yes—
326C2×C4×D4C22×C6114yes—
327C2×#117C22×C6112yes—
328C2×#117C22×C6114yes—
329C2×#117C22×C6118yes—
330C22×C4⋊C4C22×C6111yescharacteristic
331C22×C4⋊C4C22×C6112yes—
332C22×C42C22×C6111yescharacteristic
333C22×C42C22×C6112yes—
334C22×C42C22×C6114yes—
335C61C2×#346111yescharacteristic, direct product factor, Sylow
336C2×#41—216no—
337C2×#41C23×C61116yes—
338C22×Q8—28no—
339C22×Q8C23×C6114yes—
340C22×Q8C23×C6112yes—
341C22×Q8C23×C6118yes—
342C22×Q8C23×C6114yes—
343C22×D4C23×C6114yes—
344C22×D4C23×C6112yes—
345C22×D4C23×C6114yes—
346C23×C4C23×C6112yes—
347C23×C4C23×C6114yes—
348C23×C4C23×C6114yes—
349C23×C4C23×C6111yescharacteristic
350C23×C4C23×C6111yescharacteristic
351C23×C4C23×C6112yes—
352C23×C4C23×C6114yes—
353C23×C4C23×C6111yescharacteristic
354C4⋊Q8C23×C61132yes—
355C4⋊Q8C23×C61132yes—
356#126C23×C61164yes—
357#126C23×C61164yes—
358#125C23×C61132yes—
359#125C23×C61132yes—
360#125C23×C61164yes—
361#124C23×C61132yes—
362#124C23×C61132yes—
363#123C23×C61164yes—
364#123C23×C61164yes—
365C22⋊Q8C23×C61132yes—
366C22⋊Q8C23×C61132yes—
367C22⋊Q8C23×C61132yes—
368C22⋊Q8C23×C61132yes—
369C22⋊Q8C23×C61164yes—
370#121C23×C61132yes—
371#121C23×C61132yes—
372C4×Q8C23×C61132yes—
373C4×Q8C23×C61132yes—
374C4×Q8C23×C61116yes—
375C4×Q8C23×C61116yes—
376C4×Q8C23×C61132yes—
377C4×D4C23×C61116yes—
378C4×D4C23×C61132yes—
379C4×D4C23×C61132yes—
380C4×D4C23×C61116yes—
381C4×D4C23×C61132yes—
382#117C23×C61116yes—
383#117C23×C61132yes—
384#117C23×C61164yes—
385C2×C4⋊C4C23×C6114yes—
386C2×C4⋊C4C23×C6118yes—
387C2×C4⋊C4C23×C6118yes—
388C2×C4⋊C4C23×C61116yes—
389C2×C4⋊C4C23×C6114yes—
390C2×C4⋊C4C23×C6112yes—
391C2×C4⋊C4C23×C6112yes—
392C2×C4⋊C4C23×C6114yes—
393C2×C4⋊C4C23×C6118yes—
394C2×C4⋊C4C23×C6118yes—
395C2×C4⋊C4C23×C6114yes—
396C2×C4⋊C4C23×C6114yes—
397C2×C4⋊C4C23×C6118yes—
398C2×C22⋊C4C23×C6114yes—
399C2×C22⋊C4C23×C6114yes—
400C2×C22⋊C4C23×C6114yes—
401C2×C22⋊C4C23×C6118yes—
402C2×C22⋊C4C23×C6118yes—
403C2×C22⋊C4C23×C6118yes—
404C2×C42C23×C6112yes—
405C2×C42C23×C6114yes—
406C2×C42C23×C6114yes—
407C2×C42C23×C6114yes—
408C2×C42C23×C6112yes—
409C2×C42C23×C6114yes—
410C2×C42C23×C6114yes—
411C2×C42C23×C6118yes—
412C2×C42C23×C61116yes—
413C2×C42C23×C61132yes—
414C24C24×C6114yes—
415C24C24×C6111yescharacteristic
416#41—264no—
417C2×Q8—216no—
418C2×Q8—232no—
419C2×Q8—28no—
420C2×Q8—48no—
421C2×Q8C24×C61116yes—
422C2×Q8C24×C6118yes—
423C2×Q8C24×C61116yes—
424C2×D4—216no—
425C2×D4—28no—
426C2×D4—48no—
427C2×D4C24×C61116yes—
428C2×D4C24×C6118yes—
429C2×D4C24×C61116yes—
430C22×C4—28no—
431C22×C4—216no—
432C22×C4—24no—
433C22×C4—216no—
434C22×C4—42no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4#41×C6118yes—
438C22×C4C24×C6111yescharacteristic
439C22×C4C24×C6114yes—
440C22×C4C24×C6112yes—
441C22×C4C24×C6114yes—
442C22×C4C24×C6114yes—
443C22×C4C24×C6114yes—
444C22×C4C24×C6114yes—
445C22×C4C24×C6114yes—
446C22×C4C24×C6118yes—
447C22×C4C24×C61116yes—
448C22×C4C24×C61116yes—
449C22×C4C24×C6114yes—
450C22×C4C24×C6118yes—
451C22×C4C24×C61116yes—
452C22×C4C24×C6112yes—
453C22×C4C24×C6114yes—
454C22×C4C24×C6111yescharacteristic
455C4⋊C4C24×C61116yes—
456C4⋊C4C24×C61132yes—
457C4⋊C4C24×C61116yes—
458C4⋊C4C24×C6118yes—
459C4⋊C4C24×C6118yes—
460C4⋊C4C24×C61116yes—
461C4⋊C4C24×C61132yes—
462C4⋊C4C24×C61132yes—
463C4⋊C4C24×C61116yes—
464C4⋊C4C24×C61116yes—
465C4⋊C4C24×C61132yes—
466C22⋊C4C24×C61116yes—
467C22⋊C4C24×C61116yes—
468C22⋊C4C24×C61116yes—
469C22⋊C4C24×C61132yes—
470C22⋊C4C24×C61132yes—
471C22⋊C4C24×C61132yes—
472C42C24×C6118yes—
473C42C24×C6118yes—
474C42C24×C61116yes—
475C42C24×C61116yes—
476C42C24×C61116yes—
477C42C24×C61116yes—
478C42C24×C61116yes—
479C23—22no—
480C23—22no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23C2×#41×C6114yes—
485C23C25×C6118yes—
486C23C25×C6112yes—
487C23C25×C6111yescharacteristic
488Q8—232no—
489Q8—216no—
490Q8—416no—
491D4—232no—
492D4—216no—
493D4—416no—
494C2×C4—28no—
495C2×C4—216no—
496C2×C4—28no—
497C2×C4—22no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—232no—
503C2×C4—28no—
504C2×C4—232no—
505C2×C4—44no—
506C2×C4—416no—
507C2×C4—48no—
508C2×C4C2×#41×C6118yes—
509C2×C4C2×#41×C61116yes—
510C2×C42−1+4×C6114yes—
511C2×C4C25×C6112yes—
512C2×C4C25×C6114yes—
513C2×C4C25×C6118yes—
514C2×C4C25×C6118yes—
515C2×C4C25×C6118yes—
516C2×C4C25×C6118yes—
517C2×C4C25×C6118yes—
518C2×C4C25×C6114yes—
519C2×C4C25×C6112yes—
520C22—22no—
521C22—22no—
522C22—28no—
523C22—28no—
524C22—28no—
525C22—28no—
526C22—416no—
527C22C22×#41×C6114yes—
528C22C22×#41×C6112yes—
529C22C2×2−1+4×C6112yes—
530C22C61×#58512yes—
531C22C26×C6111yescharacteristic
532C4—28no—
533C4—28no—
534C4—22no—
535C4—28no—
536C4—24no—
537C4—28no—
538C4—28no—
539C4C22×#41×C6118yes—
540C4C2×2−1+4×C6114yes—
541C2—24no—
542C2—48no—
543C2C61×#346114yesdirect product factor
544C2C23×#41×C6111yescharacteristic
545C2C22×2−1+4×C6111yescharacteristic
546C2C2×C61×#58511yescharacteristic
subgroupG/H
centerC22×C4×C61C24
commutator, Frattini, cosocleC22C26×C61
socleC23×C61C25
prime radicalC2×C61×#537C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3461C2direct (N × H)
2. C2×C61×#544C2semidirect
3. C2×C61×#542C2semidirect
4. C2×C61×#541C2semidirect
5. C2×C61×#539C2semidirect
7. C2×C61×#536C2semidirect
8. C2×C61×#533C2semidirect
9. C2×C61×#529C2semidirect
10. C2×C61×#529C2semidirect
11. C2×C4×#41×C61C2semidirect
12. C22×C4×Q8×C61C2semidirect
15. C61×#544C22semidirect
16. C61×#542C22semidirect
17. C61×#541C22semidirect
18. C61×#539C22semidirect
20. C61×#536C22semidirect
21. C61×#533C22semidirect
24. C61×#529C22semidirect
25. C61×#529C22semidirect
42. C4×#41×C61C22semidirect
48. C2×C4×Q8×C61C22semidirect
246. C2×#3461C61direct (N × H)
268. #3461C2×C61direct (N × H)
269. C2×#544C2×C61semidirect
270. C2×#542C2×C61semidirect
271. C2×#541C2×C61semidirect
272. C2×#539C2×C61semidirect
274. C2×#536C2×C61semidirect
275. C2×#533C2×C61semidirect
276. C2×#529C2×C61semidirect
277. C2×#529C2×C61semidirect
278. C2×C4×#41C2×C61semidirect
279. C22×C4×Q8C2×C61semidirect
282. C2×C61#3461direct (N × H)
288. #544C22×C61semidirect
289. #542C22×C61semidirect
290. #541C22×C61semidirect
291. #539C22×C61semidirect
293. #536C22×C61semidirect
294. #533C22×C61semidirect
297. #529C22×C61semidirect
298. #529C22×C61semidirect
315. C4×#41C22×C61semidirect
321. C2×C4×Q8C22×C61semidirect
335. C61C2×#3461direct (N × H)
543. C2C61×#3461direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#34612
C2×C61×#5442
C2×C61×#5422
C2×C61×#5412
C2×C61×#5392
C2×C61×#5372
C2×C61×#5362
C2×C61×#5332
C2×C61×#5292
C2×C4×#41×C612
C22×C4×Q8×C612
C2×#346161