Finite Group Databasebeta

← C2×#22615117 #33257255

Subgroups — C2×#22615117 #33257255

Order 18688.

2486 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556581C211yesmaximal, characteristic
2C22×#14556556C211yesmaximal, characteristic
3#22615117C2116yesmaximal, direct product factor
4C2×#17988520C211yesmaximal, characteristic
5C2×#17988520C211yesmaximal, characteristic
6C2×#17988517C211yesmaximal, characteristic
7C2×#17988491C211yesmaximal, characteristic
8C2×#17988491C211yesmaximal, characteristic
9C2×#17988474C211yesmaximal, characteristic
10C2×#17988473C211yesmaximal, characteristic
11C2×#17988470C211yesmaximal, characteristic
12C2×#17988469C211yesmaximal, characteristic
13C2×#17988456C211yesmaximal, characteristic
14C2×#17988451C211yesmaximal, characteristic
15C2×C4⋊C4×Dic73C211yesmaximal, characteristic
16C2×#398×C73C211yesmaximal, characteristic
17C23×Dic146C2211yescharacteristic
18C2×#14556581—28no—
19C2×#14556581C2218yes—
20C2×#14556556—28no—
21C2×#14556556C2218yes—
22C22×C4⋊Dic73C2211yescharacteristic
23C22×C4⋊Dic73C2211yescharacteristic
24C22×C4⋊Dic73C2211yescharacteristic
25C22×#11924007C2211yescharacteristic
26C22×#11924007C2211yescharacteristic
27C22×#11924007C2211yescharacteristic
28C22×#11924007C2211yescharacteristic
29C22×#11924007C2211yescharacteristic
30C22×#11924007C2211yescharacteristic
31C22×#11924007C2211yescharacteristic
32C22×#11924007C2211yescharacteristic
33C22×C4×Dic73C2211yescharacteristic
34C22×C4×Dic73C2211yescharacteristic
35C22×C4×Dic73C2211yescharacteristic
36C22×C4×Dic73C2211yescharacteristic
37C22×C4×Dic73C2211yescharacteristic
38#17988520C2218yes—
39#17988520C2218yes—
40#17988517C2218yes—
41#17988491C2218yes—
42#17988491C2218yes—
43#17988474C2218yes—
44#17988473C2218yes—
45#17988470C2218yes—
46#17988469C2218yes—
47#17988456C2218yes—
48#17988451C2218yes—
49C4⋊C4×Dic73C2218yes—
50C2×#14556498C2211yescharacteristic
51C2×#14556498C2211yescharacteristic
52C2×#14556498C2211yescharacteristic
53C2×#14556498C2211yescharacteristic
54C2×#14556498C2211yescharacteristic
55C2×#14556498C2211yescharacteristic
56C2×#14556498C2211yescharacteristic
57C2×#14556498C2211yescharacteristic
58C2×#14556498C2211yescharacteristic
59C2×#14556498C2211yescharacteristic
60C2×#14556498C2211yescharacteristic
61C22×C4⋊C4×C73C2211yescharacteristic
62C22×C4⋊C4×C73C2211yescharacteristic
63C22×C4⋊C4×C73C2211yescharacteristic
64C22×C4⋊C4×C73C2211yescharacteristic
65#398×C73C2218yes—
66C2×#95×C73C2211yescharacteristic
67C2×#95×C73C2211yescharacteristic
68C2×#95×C73C2211yescharacteristic
69C22×Dic146—24no—
70C22×Dic146—24no—
71C22×Dic146Q818yes—
72C22×Dic146C2314yes—
73C23×Dic73C2311yescharacteristic
74C23×Dic73C2311yescharacteristic
75C23×Dic73C2311yescharacteristic
76C23×Dic73C2311yescharacteristic
77C23×Dic73C2311yescharacteristic
78C23×Dic73C2311yescharacteristic
79C23×Dic73C2311yescharacteristic
80C23×Dic73C2311yescharacteristic
81#14556581—232no—
82#14556556—232no—
83C2×C4⋊Dic73—24no—
84C2×C4⋊Dic73—24no—
85C2×C4⋊Dic73—24no—
86C2×C4⋊Dic73C2314yes—
87C2×C4⋊Dic73C2314yes—
88C2×C4⋊Dic73C2314yes—
89C2×#11924007—24no—
90C2×#11924007—24no—
91C2×#11924007—24no—
92C2×#11924007—24no—
93C2×#11924007—24no—
94C2×#11924007—24no—
95C2×#11924007—24no—
96C2×#11924007—24no—
97C2×#11924007C2314yes—
98C2×#11924007C2314yes—
99C2×#11924007C2314yes—
100C2×#11924007C2314yes—
101C2×#11924007C2314yes—
102C2×#11924007C2314yes—
103C2×#11924007C2314yes—
104C2×#11924007C2314yes—
105C2×C4×Dic73—24no—
106C2×C4×Dic73—24no—
107C2×C4×Dic73—24no—
108C2×C4×Dic73—24no—
109C2×C4×Dic73D418yes—
110C2×C4×Dic73C2314yes—
111C2×C4×Dic73C2314yes—
112C2×C4×Dic73C2314yes—
113C2×C4×Dic73C2314yes—
114C2×C4×Dic73C2314yes—
115#14556498C2314yes—
116#14556498C2314yes—
117#14556498C2314yes—
118#14556498C2314yes—
119#14556498C2314yes—
120#14556498C2314yes—
121#14556498C2314yes—
122#14556498C2314yes—
123#14556498C2314yes—
124#14556498C2314yes—
125#14556498C2314yes—
126C23×C4×C73C2311yescharacteristic
127C23×C4×C73C2311yescharacteristic
128C23×C4×C73C2311yescharacteristic
129C23×C4×C73C2311yescharacteristic
130C23×C4×C73C2311yescharacteristic
131C23×C4×C73C2311yescharacteristic
132C23×C4×C73C2311yescharacteristic
133C2×C4⋊C4×C73—24no—
134C2×C4⋊C4×C73—24no—
135C2×C4⋊C4×C73—24no—
136C2×C4⋊C4×C73—24no—
137C2×C4⋊C4×C73C2314yes—
138C2×C4⋊C4×C73C2314yes—
139C2×C4⋊C4×C73C2314yes—
140C2×C4⋊C4×C73C2314yes—
141#95×C73C2314yes—
142#95×C73C2314yes—
143#95×C73C2314yes—
144C2×Dic146—232no—
145C2×Dic146—28no—
146C2×Dic146—28no—
147C2×Dic146C2×Q8116yes—
148C22×Dic73—22no—
149C22×Dic73—22no—
150C22×Dic73—22no—
151C22×Dic73—22no—
152C22×Dic73—22no—
153C22×Dic73—22no—
154C22×Dic73—22no—
155C22×Dic73—22no—
156C22×Dic73—22no—
157C22×Dic73—22no—
158C22×Dic73—22no—
159C22×Dic73—22no—
160C22×Dic73—22no—
161C22×Dic73—22no—
162C22×Dic73—22no—
163C22×Dic73—22no—
164C22×Dic73—22no—
165C22×Dic73—22no—
166C22×Dic73C2×D414yes—
167C22×Dic73C2×D414yes—
168C22×Dic73C2×Q814yes—
169C22×Dic73C2×Q814yes—
170C22×Dic73#4114yes—
171C22×Dic73#4114yes—
172C22×Dic73C2412yes—
173C22×Dic73C2412yes—
174C22×Dic73C2412yes—
175C22×Dic73C2412yes—
176C22×Dic73C2412yes—
177C22×Dic73C2412yes—
178C22×Dic73C2412yes—
179C22×Dic73C2412yes—
180C4⋊Dic73—28no—
181C4⋊Dic73—28no—
182C4⋊Dic73—28no—
183#11924007—28no—
184#11924007—28no—
185#11924007—28no—
186#11924007—28no—
187#11924007—28no—
188#11924007—28no—
189#11924007—28no—
190#11924007—28no—
191C4×Dic73—28no—
192C4×Dic73—28no—
193C4×Dic73—28no—
194C4×Dic73—28no—
195C4×Dic73C2×D4116yes—
196C24×C73C2411yescharacteristic
197C22×C4×C73—22no—
198C22×C4×C73—22no—
199C22×C4×C73—22no—
200C22×C4×C73—22no—
201C22×C4×C73—22no—
202C22×C4×C73—22no—
203C22×C4×C73—22no—
204C22×C4×C73—22no—
205C22×C4×C73—22no—
206C22×C4×C73—22no—
207C22×C4×C73—22no—
208C22×C4×C73—22no—
209C22×C4×C73—22no—
210C22×C4×C73—22no—
211C22×C4×C73—22no—
212C22×C4×C73—22no—
213C22×C4×C73—22no—
214C22×C4×C73—22no—
215C22×C4×C73C2×D414yes—
216C22×C4×C73C2×Q814yes—
217C22×C4×C73#4114yes—
218C22×C4×C73C2412yes—
219C22×C4×C73C2412yes—
220C22×C4×C73C2412yes—
221C22×C4×C73C2412yes—
222C22×C4×C73C2412yes—
223C22×C4×C73C2412yes—
224C22×C4×C73C2412yes—
225C4⋊C4×C73—28no—
226C4⋊C4×C73—28no—
227C4⋊C4×C73—28no—
228C4⋊C4×C73—28no—
229C23×C73C22×D412yes—
230C23×C73C22×Q812yes—
231C23×C73C2×#4112yes—
232C23×C73C2×#4112yes—
233C23×C73C2×#4112yes—
234C23×C732−1+412yes—
235C23×C732−1+412yes—
236C23×C73C2511yescharacteristic
237C2×C4×C73—28no—
238C2×C4×C73—28no—
239C2×C4×C73—28no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—22no—
249C2×C4×C73—22no—
250C2×C4×C73—22no—
251C2×C4×C73—22no—
252C2×C4×C73—22no—
253C2×C4×C73—22no—
254C2×C4×C73—22no—
255C2×C4×C73—22no—
256C2×C4×C73—22no—
257C2×C4×C73—22no—
258C2×C4×C73—44no—
259C2×C4×C73—44no—
260C2×C4×C73—44no—
261C2×C4×C73—44no—
262C2×C4×C73C22×D414yes—
263C2×C4×C73C22×Q814yes—
264C2×C4×C73C2×#4114yes—
265C2×Dic73—28no—
266C2×Dic73—28no—
267C2×Dic73—28no—
268C2×Dic73—22no—
269C2×Dic73—22no—
270C2×Dic73—22no—
271C2×Dic73—22no—
272C2×Dic73—22no—
273C2×Dic73—22no—
274C2×Dic73—22no—
275C2×Dic73—22no—
276C2×Dic73—22no—
277C2×Dic73—22no—
278C2×Dic73—22no—
279C2×Dic73—22no—
280C2×Dic73—22no—
281C2×Dic73—22no—
282C2×Dic73—22no—
283C2×Dic73—22no—
284C2×Dic73—22no—
285C2×Dic73—22no—
286C2×Dic73—44no—
287C2×Dic73—44no—
288C2×Dic73—44no—
289C2×Dic73—44no—
290C2×Dic73C22⋊Q8116yes—
291C2×Dic73C22×D414yes—
292C2×Dic73C22×D414yes—
293C2×Dic73C22×Q814yes—
294C2×Dic73C22×Q814yes—
295C2×Dic73C2×#4114yes—
296C2×Dic73C2×#4114yes—
297Dic146—232no—
298C22×C73C2×C22⋊Q814yes—
299C22×C73#53214yes—
300C22×C73#53614yes—
301C22×C73#55214yes—
302C22×C73#55214yes—
303C22×C73#55714yes—
304C22×C73#55714yes—
305C22×C73C23×D411yescharacteristic
306C22×C73C23×Q811yescharacteristic
307C22×C73C22×#4111yescharacteristic
308C22×C73C22×#4111yescharacteristic
309C22×C73C22×#4111yescharacteristic
310C22×C73C2×2−1+411yescharacteristic
311C22×C73C2×2−1+411yescharacteristic
312C4×C73—24no—
313C4×C73—24no—
314C4×C73—24no—
315C4×C73—42no—
316C4×C73—42no—
317C4×C73—42no—
318C4×C73—42no—
319Dic73—24no—
320Dic73—24no—
321Dic73—24no—
322Dic73—42no—
323Dic73—42no—
324Dic73—42no—
325Dic73—42no—
326Dic73C2×C22⋊Q818yes—
327C2×#2565—731nomaximal, Sylow
328C2×C73#256518yescomplemented
329C2×C73C22×C22⋊Q811yescharacteristic
330C2×C73C2×#53211yescharacteristic
331C2×C73C2×#53611yescharacteristic
332C2×C73C2×#55211yescharacteristic
333C2×C73C2×#55211yescharacteristic
334C2×C73C2×#55711yescharacteristic
335C2×C73C2×#55711yescharacteristic
336C22×#125—731no—
337C22×C4×Q8—731no—
338#2565—7316no—
339C2×#398—731no—
340C2×#398—731no—
341C2×#398—731no—
342C2×#398D7311yescharacteristic
343C2×#391—731no—
344C2×#391—731no—
345C2×#387—731no—
346C2×#387—731no—
347C2×#387—731no—
348C2×#387—731no—
349C2×#384—731no—
350C2×#382—731no—
351C2×C4×C4⋊C4—731no—
352C73C2×#256511yescharacteristic, complemented, Sylow
353C23×Q8—731no—
354C2×#125—738no—
355C2×#125—1468no—
356C2×C4×Q8—738no—
357C2×C4×Q8—1468no—
358C22×C4⋊C4—731no—
359C22×C4⋊C4—731no—
360C22×C4⋊C4—731no—
361C22×C4⋊C4—731no—
362C22×C4⋊C4—731no—
363C22×C4⋊C4—731no—
364C22×C4⋊C4—731no—
365C22×C4⋊C4—731no—
366C22×C4⋊C4—731no—
367C22×C4⋊C4—731no—
368C22×C4⋊C4—731no—
369C22×C4⋊C4C2×D7311yescharacteristic
370C22×C4⋊C4C2×D7311yescharacteristic
371C22×C4⋊C4C2×D7311yescharacteristic
372C22×C4⋊C4C2×D7311yescharacteristic
373C22×C42—731no—
374C22×C42—731no—
375C22×C42—731no—
376C22×C42—731no—
377C22×C42—731no—
378#398—738no—
379#398—738no—
380#398—738no—
381#398C2×D7318yes—
382#391—738no—
383#391—738no—
384#387—738no—
385#387—738no—
386#387—738no—
387#387—738no—
388#384—738no—
389#382—738no—
390C4×C4⋊C4—738no—
391C2×#95—731no—
392C2×#95—731no—
393C2×#95—731no—
394C2×#95—731no—
395C2×#95—731no—
396C2×#95—731no—
397C2×#95—731no—
398C2×#95—731no—
399C2×#95—731no—
400C2×#95—731no—
401C2×#95—731no—
402C2×#95C2×D7311yescharacteristic
403C2×#95C2×D7311yescharacteristic
404C2×#95C2×D7311yescharacteristic
405C22×Q8—738no—
406C22×Q8—734no—
407C22×Q8—1464no—
408C22×Q8—1464no—
409C23×C4—731no—
410C23×C4—731no—
411C23×C4—731no—
412C23×C4—731no—
413C23×C4—731no—
414C23×C4—731no—
415C23×C4—731no—
416C23×C4—731no—
417C23×C4C22×D7311yescharacteristic
418C23×C4C22×D7311yescharacteristic
419C23×C4C22×D7311yescharacteristic
420C23×C4C22×D7311yescharacteristic
421C23×C4C22×D7311yescharacteristic
422C23×C4C22×D7311yescharacteristic
423C23×C4C22×D7311yescharacteristic
424#125—14632no—
425C4×Q8—14632no—
426C2×C4⋊C4—24no—
427C2×C4⋊C4—24no—
428C2×C4⋊C4—24no—
429C2×C4⋊C4—24no—
430C2×C4⋊C4—734no—
431C2×C4⋊C4—734no—
432C2×C4⋊C4—734no—
433C2×C4⋊C4—734no—
434C2×C4⋊C4—734no—
435C2×C4⋊C4—734no—
436C2×C4⋊C4—734no—
437C2×C4⋊C4—734no—
438C2×C4⋊C4—734no—
439C2×C4⋊C4—734no—
440C2×C4⋊C4—734no—
441C2×C4⋊C4—1464no—
442C2×C4⋊C4—1464no—
443C2×C4⋊C4—1464no—
444C2×C4⋊C4—1464no—
445C2×C4⋊C4—1464no—
446C2×C4⋊C4—1464no—
447C2×C4⋊C4—1464no—
448C2×C4⋊C4—1464no—
449C2×C4⋊C4—1464no—
450C2×C4⋊C4—1464no—
451C2×C4⋊C4—1464no—
452C2×C4⋊C4C22×D7314yes—
453C2×C4⋊C4C22×D7314yes—
454C2×C4⋊C4C22×D7314yes—
455C2×C4⋊C4C22×D7314yes—
456C2×C42—738no—
457C2×C42—734no—
458C2×C42—734no—
459C2×C42—734no—
460C2×C42—734no—
461C2×C42—734no—
462C2×C42—1464no—
463C2×C42—1464no—
464C2×C42—1464no—
465C2×C42—1464no—
466#95—734no—
467#95—734no—
468#95—734no—
469#95—734no—
470#95—734no—
471#95—734no—
472#95—734no—
473#95—734no—
474#95—734no—
475#95—734no—
476#95—734no—
477#95C22×D7314yes—
478#95C22×D7314yes—
479#95C22×D7314yes—
480C24C23×D7311yescharacteristic
481C2×Q8—7316no—
482C2×Q8—1468no—
483C2×Q8—1468no—
484C2×Q8—14632no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—734no—
504C22×C4—734no—
505C22×C4—734no—
506C22×C4—732no—
507C22×C4—734no—
508C22×C4—734no—
509C22×C4—734no—
510C22×C4—732no—
511C22×C4—732no—
512C22×C4—732no—
513C22×C4—732no—
514C22×C4—732no—
515C22×C4—732no—
516C22×C4—732no—
517C22×C4—1462no—
518C22×C4—1462no—
519C22×C4—1462no—
520C22×C4—1462no—
521C22×C4—1462no—
522C22×C4—1462no—
523C22×C4—1462no—
524C22×C4—1462no—
525C22×C4—1462no—
526C22×C4—1462no—
527C22×C4—1462no—
528C22×C4—1462no—
529C22×C4—1462no—
530C22×C4—1462no—
531C22×C4—1462no—
532C22×C4—1462no—
533C22×C4—1462no—
534C22×C4—1462no—
535C22×C4D4×D7314yes—
536C22×C4Q8×D7314yes—
537C22×C4#1192402714yes—
538C22×C4C23×D7312yes—
539C22×C4C23×D7312yes—
540C22×C4C23×D7312yes—
541C22×C4C23×D7312yes—
542C22×C4C23×D7312yes—
543C22×C4C23×D7312yes—
544C22×C4C23×D7312yes—
545C4⋊C4—28no—
546C4⋊C4—28no—
547C4⋊C4—28no—
548C4⋊C4—28no—
549C4⋊C4—1468no—
550C4⋊C4—1468no—
551C4⋊C4—1468no—
552C4⋊C4—1468no—
553C4⋊C4—1468no—
554C4⋊C4—1468no—
555C4⋊C4—1468no—
556C4⋊C4—1468no—
557C4⋊C4—1468no—
558C4⋊C4—1468no—
559C4⋊C4—1468no—
560C42—7316no—
561C42—1468no—
562C42—1468no—
563C42—1468no—
564C42—1468no—
565C23C2×D4×D7312yes—
566C23C2×Q8×D7312yes—
567C23#41×D7312yes—
568C23#41×D7312yes—
569C23C2×#1192402712yes—
570C23#1455663812yes—
571C23#1455663912yes—
572C23C24×D7311yescharacteristic
573Q8—14632no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—2924no—
600C2×C4—2924no—
601C2×C4—2924no—
602C2×C4—2924no—
603C2×C4—734no—
604C2×C4—734no—
605C2×C4—734no—
606C2×C4—7316no—
607C2×C4—734no—
608C2×C4—734no—
609C2×C4—734no—
610C2×C4—1462no—
611C2×C4—1462no—
612C2×C4—1462no—
613C2×C4—1462no—
614C2×C4—1462no—
615C2×C4—1462no—
616C2×C4—1462no—
617C2×C4—1462no—
618C2×C4—1462no—
619C2×C4—1462no—
620C2×C4—1462no—
621C2×C4—1462no—
622C2×C4—1462no—
623C2×C4—1462no—
624C2×C4—1462no—
625C2×C4—1468no—
626C2×C4—1462no—
627C2×C4—1462no—
628C2×C4—1462no—
629C2×C4—1468no—
630C2×C4—1468no—
631C2×C4C2×D4×D7314yes—
632C2×C4C2×Q8×D7314yes—
633C2×C4C2×#1192402714yes—
634C22C22⋊Q8×D7314yes—
635C22#1798913214yes—
636C22#1798916014yes—
637C22#1798926914yes—
638C22#1798927414yes—
639C22#1798931214yes—
640C22#1798931414yes—
641C22C22×D4×D7311yescharacteristic
642C22C22×Q8×D7311yescharacteristic
643C22C2×#41×D7311yescharacteristic
644C22C2×#41×D7311yescharacteristic
645C22C22×#1192402711yescharacteristic
646C22C2×#1455663811yescharacteristic
647C22C2×#1455663911yescharacteristic
648C4—24no—
649C4—24no—
650C4—24no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—2922no—
656C4—2922no—
657C4—2922no—
658C4—2922no—
659C4—738no—
660C4—1464no—
661C4—1464no—
662C4—1464no—
663C2#2261511718yesdirect product factor
664C2C2×C22⋊Q8×D7311yescharacteristic
665C2C2×#1798913211yescharacteristic
666C2C2×#1798916011yescharacteristic
667C2C2×#1798926911yescharacteristic
668C2C2×#1798927411yescharacteristic
669C2C2×#1798931211yescharacteristic
670C2C2×#1798931411yescharacteristic
subgroupG/H
centerC24C23×D73
commutator, cosocleC23×C73C25
FrattiniC23C24×D73
FittingC2×#398×C73C2
socle, prime radicalC24×C73C24
hypercenterC2×#398D73
hypocenterC73C2×#2565
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
3. #22615117C2direct (N × H)
328. C2×C73#2565semidirect
352. C73C2×#2565semidirect
663. C2#22615117direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145565812
C22×#145565562
#226151172
C2×#179885202
C2×#179885172
C2×#179884912
C2×#179884742
C2×#179884732
C2×#179884702
C2×#179884692
C2×#179884562
C2×#179884512
C2×C4⋊C4×Dic732
C2×#398×C732
C2×#256573