Finite Group Databasebeta

← #33022631

Subgroups — #33022631

Order 18688.

1814 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988523C212yesmaximal
2C2×#17988484C212yesmaximal, complemented
3C2×#17988456C211yesmaximal, characteristic, complemented
4C2×#17988445C212yesmaximal
5C2×C22⋊C4×Dic73C211yesmaximal, characteristic
6#22608197C212yesmaximal, complemented
7#22608196C211yesmaximal, characteristic, complemented
8#22608194C211yesmaximal, characteristic, complemented
9#22608176C211yesmaximal, characteristic
10#22608159C211yesmaximal, characteristic, complemented
11C73×#1439C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2212yes—
14C22×C4⋊Dic73C2211yescharacteristic
15C22×#11924007C2212yes—
16C22×#11924007C2212yes—
17C22×#11924007C2212yes—
18C22×#11924007C2212yes—
19C22×C22⋊Dic73C2212yes—
20C22×C22⋊Dic73C2211yescharacteristic
21C22×C4×Dic73C2211yescharacteristic
22C22×C4×Dic73C2211yescharacteristic
23#17988523—28no—
24#17988484—28no—
25#17988456—24no—
26#17988445—28no—
27C22⋊C4×Dic73—24no—
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2212yes—
30C2×#14556498C2211yescharacteristic
31C2×#14556498C2211yescharacteristic
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C22⋊C4×C73C2212yes—
37C22×C22⋊C4×C73C2211yescharacteristic
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2212yes—
40C23×Dic73—22no—
41C23×Dic73—24no—
42C23×Dic73—24no—
43C23×Dic73—22no—
44C23×Dic73Q814yes—
45C23×Dic73C2312yes—
46C23×Dic73C2312yes—
47C23×Dic73C2311yescharacteristic
48C23×Dic73C2311yescharacteristic
49C23×Dic73C2311yescharacteristic
50C23×Dic73C2311yescharacteristic
51C2×C4⋊Dic73—24no—
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—24no—
54C2×C4⋊Dic73—42no—
55C2×C4⋊Dic73—42no—
56C2×#11924007—24no—
57C2×#11924007—24no—
58C2×#11924007—24no—
59C2×#11924007—24no—
60C2×#11924007—24no—
61C2×#11924007—24no—
62C2×#11924007—24no—
63C2×#11924007—24no—
64C2×#11924007—24no—
65C2×#11924007—24no—
66C2×#11924007—42no—
67C2×#11924007D418yes—
68C2×C22⋊Dic73—22no—
69C2×C22⋊Dic73—22no—
70C2×C22⋊Dic73—24no—
71C2×C22⋊Dic73—24no—
72C2×C22⋊Dic73—42no—
73C2×C22⋊Dic73D414yes—
74C2×C4×Dic73—24no—
75C2×C4×Dic73—22no—
76C2×C4×Dic73—24no—
77C2×C4×Dic73D414yes—
78#14556498—22no—
79#14556498—24no—
80#14556498—24no—
81#14556498—24no—
82#14556498—22no—
83#14556498—24no—
84#14556498—22no—
85C25×C73C2311yescharacteristic
86C23×C4×C73C2312yes—
87C23×C4×C73C2312yes—
88C23×C4×C73C2311yescharacteristic
89C23×C4×C73C2311yescharacteristic
90C2×C4⋊C4×C73—22no—
91C2×C4⋊C4×C73—42no—
92C2×C22⋊C4×C73—22no—
93C2×C22⋊C4×C73—22no—
94C2×C22⋊C4×C73—24no—
95C2×C22⋊C4×C73—24no—
96C2×C22⋊C4×C73—42no—
97C2×C22⋊C4×C73Q814yes—
98#95×C73—24no—
99#95×C73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—21no—
103C22×Dic73—22no—
104C22×Dic73—21no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—21no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—21no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—28no—
117C22×Dic73—216no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—44no—
126C22×Dic73—48no—
127C22×Dic73—44no—
128C22×Dic73C2×D414yes—
129C22×Dic73C2×D412yes—
130C22×Dic73C2×D412yes—
131C22×Dic73C2×D412yes—
132C22×Dic73C2×D412yes—
133C22×Dic73C2×D414yes—
134C22×Dic73C2×Q812yes—
135C22×Dic73C2×Q812yes—
136C22×Dic73#4112yes—
137C22×Dic73#4114yes—
138C22×Dic73#4112yes—
139C4⋊Dic73—48no—
140C4⋊Dic73—44no—
141#11924007—216no—
142#11924007—48no—
143#11924007—48no—
144#11924007—48no—
145C22⋊Dic73—28no—
146C22⋊Dic73—48no—
147C4×Dic73—28no—
148C4×Dic73—44no—
149C24×C73—21no—
150C24×C73—22no—
151C24×C73—21no—
152C24×C73—21no—
153C24×C73—21no—
154C24×C73—22no—
155C24×C73—22no—
156C24×C73—22no—
157C24×C73C2×D412yes—
158C24×C73C2×Q812yes—
159C24×C73C2×Q812yes—
160C24×C73C2411yescharacteristic
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—21no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—21no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73C2×D414yes—
181C22×C4×C73C2×D412yes—
182C22×C4×C73C2×Q812yes—
183C22×C4×C73C2×Q812yes—
184C22×C4×C73#4112yes—
185C4⋊C4×C73—44no—
186C22⋊C4×C73—28no—
187C22⋊C4×C73—48no—
188C23×C73—22no—
189C23×C73—22no—
190C23×C73—22no—
191C23×C73—24no—
192C23×C73—22no—
193C23×C73—22no—
194C23×C73—22no—
195C23×C73—24no—
196C23×C73—24no—
197C23×C73—24no—
198C23×C73—24no—
199C23×C73—24no—
200C23×C73—42no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—42no—
206C23×C73—42no—
207C23×C73—42no—
208C23×C73—42no—
209C23×C73—42no—
210C23×C73C4⋊Q814yes—
211C23×C73C22×D411yescharacteristic
212C23×C73C22×D411yescharacteristic
213C23×C73C22×D412yes—
214C23×C73C22×Q811yescharacteristic
215C23×C73C22×Q811yescharacteristic
216C23×C73C2×#4111yescharacteristic
217C23×C73C2×#4111yescharacteristic
218C23×C73C2×#4112yes—
219C23×C73C2×#4111yescharacteristic
220C23×C73H(5,𝔽2)12yes—
221C23×C732−1+412yes—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—24no—
225C2×C4×C73—24no—
226C2×C4×C73—22no—
227C2×C4×C73—24no—
228C2×C4×C73—24no—
229C2×C4×C73—24no—
230C2×C4×C73—24no—
231C2×C4×C73—41no—
232C2×C4×C73—42no—
233C2×C4×C73—41no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73C22⋊Q814yes—
246C2×Dic73—22no—
247C2×Dic73—22no—
248C2×Dic73—24no—
249C2×Dic73—24no—
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—24no—
253C2×Dic73—24no—
254C2×Dic73—24no—
255C2×Dic73—24no—
256C2×Dic73—24no—
257C2×Dic73—42no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—42no—
262C2×Dic73—41no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—416no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—42no—
274C2×Dic73—42no—
275C2×Dic73#12114yes—
276C2×Dic73C22⋊Q814yes—
277C2×Dic73C22⋊Q818yes—
278C2×Dic73#12314yes—
279C2×Dic73#12318yes—
280C2×Dic73#12714yes—
281C22×C73—24no—
282C22×C73—28no—
283C22×C73—42no—
284C22×C73—42no—
285C22×C73—44no—
286C22×C73—44no—
287C22×C73—44no—
288C22×C73—42no—
289C22×C73—42no—
290C22×C73—42no—
291C22×C73—42no—
292C22×C73C2×#12111yescharacteristic
293C22×C73C2×C22⋊Q811yescharacteristic
294C22×C73C2×C22⋊Q812yes—
295C22×C73C2×C22⋊Q811yescharacteristic
296C22×C73C2×#12312yes—
297C22×C73C2×#12311yescharacteristic
298C22×C73C2×#12711yescharacteristic
299C22×C73C2×C4⋊Q811yescharacteristic
300C22×C73#53211yescharacteristic
301C22×C73#53311yescharacteristic
302C22×C73#53311yescharacteristic
303C22×C73#53511yescharacteristic
304C22×C73#53611yescharacteristic
305C22×C73#54212yes—
306C22×C73#54412yes—
307C22×C73D4212yes—
308C22×C73#54712yes—
309C22×C73#54712yes—
310C22×C73D4×Q812yes—
311C22×C73#55012yes—
312C22×C73#55212yes—
313C22×C73#55312yes—
314C22×C73#55412yes—
315C4×C73—24no—
316C4×C73—44no—
317C4×C73—42no—
318C4×C73—82no—
319Dic73—24no—
320Dic73—42no—
321Dic73—44no—
322Dic73—42no—
323Dic73—82no—
324Dic73#39918yescomplemented
325#15135—731nomaximal, Sylow
326C2×C73—44no—
327C2×C73C2×#39911yescharacteristic
328C2×C73#242212yes—
329C2×C73#242312yes—
330C2×C73#244012yes—
331C2×C73#244112yes—
332C2×C73#248611yescharacteristic
333C2×C73#248711yescharacteristic
334C2×C73#254111yescharacteristic
335C2×C73#254211yescharacteristic
336C2×C73#262811yescharacteristic
337C2×C73#285711yescharacteristic
338C2×#399—732no—
339C2×#389—732no—
340C2×#384—731no—
341C2×#380—732no—
342C2×C4×C22⋊C4—731no—
343#1445—732no—
344#1444—731no—
345#1444—731no—
346#1439—731no—
347#1439D7311yescharacteristic
348C4⋊#95—731no—
349C73#1513511yescharacteristic, complemented, Sylow
350C24×C4—731no—
351C22×C4⋊C4—731no—
352C22×C4⋊C4—732no—
353C22×C4⋊C4—732no—
354C22×C4⋊C4—732no—
355C22×C4⋊C4—732no—
356C22×C4⋊C4—732no—
357C22×C4⋊C4C2×D7311yescharacteristic
358C22×C22⋊C4—732no—
359C22×C22⋊C4—731no—
360C22×C22⋊C4C2×D7312yes—
361C22×C22⋊C4C2×D7311yescharacteristic
362C22×C42—731no—
363C22×C42—731no—
364#399—1468no—
365#389—1468no—
366#384—1464no—
367#380—1468no—
368C4×C22⋊C4—1464no—
369C2×#95—732no—
370C2×#95—731no—
371C2×#95—732no—
372C2×#95—731no—
373C2×#95—732no—
374C2×#95—732no—
375C2×#95—731no—
376C2×#95C2×D7311yescharacteristic
377C2×#95C2×D7312yes—
378C25C22×D7311yescharacteristic
379C23×C4—731no—
380C23×C4—731no—
381C23×C4—734no—
382C23×C4—731no—
383C23×C4—732no—
384C23×C4—731no—
385C23×C4—732no—
386C23×C4—1462no—
387C23×C4—1464no—
388C23×C4—1464no—
389C23×C4—1462no—
390C23×C4C22×D7312yes—
391C23×C4C22×D7312yes—
392C23×C4C22×D7311yescharacteristic
393C23×C4C22×D7311yescharacteristic
394C2×C4⋊C4—22no—
395C2×C4⋊C4—42no—
396C2×C4⋊C4—2922no—
397C2×C4⋊C4—2922no—
398C2×C4⋊C4—2922no—
399C2×C4⋊C4—738no—
400C2×C4⋊C4—1464no—
401C2×C4⋊C4—1464no—
402C2×C4⋊C4—1464no—
403C2×C4⋊C4—1464no—
404C2×C4⋊C4—1464no—
405C2×C4⋊C4—1464no—
406C2×C4⋊C4—1464no—
407C2×C4⋊C4—1464no—
408C2×C4⋊C4—1464no—
409C2×C4⋊C4—1462no—
410C2×C4⋊C4—1464no—
411C2×C4⋊C4—1464no—
412C2×C4⋊C4—1464no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—24no—
416C2×C22⋊C4—24no—
417C2×C22⋊C4—42no—
418C2×C22⋊C4—2922no—
419C2×C22⋊C4—734no—
420C2×C22⋊C4—1464no—
421C2×C22⋊C4—1464no—
422C2×C22⋊C4—1462no—
423C2×C22⋊C4—1462no—
424C2×C22⋊C4Dic14614yes—
425C2×C42—734no—
426C2×C42—1462no—
427C2×C42—1464no—
428C2×C42—1464no—
429#95—24no—
430#95—22no—
431#95—1462no—
432#95—1464no—
433#95—1464no—
434#95—1464no—
435#95—1462no—
436#95—1464no—
437#95—1462no—
438C24—22no—
439C24—21no—
440C24—21no—
441C24—21no—
442C24—22no—
443C24—21no—
444C24—22no—
445C24—22no—
446C24D4×D7312yes—
447C24Q8×D7312yes—
448C24C2×Dic14612yes—
449C24C23×D7311yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—2924no—
470C22×C4—2924no—
471C22×C4—2928no—
472C22×C4—732no—
473C22×C4—734no—
474C22×C4—732no—
475C22×C4—734no—
476C22×C4—732no—
477C22×C4—732no—
478C22×C4—732no—
479C22×C4—732no—
480C22×C4—734no—
481C22×C4—732no—
482C22×C4—732no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4—1468no—
491C22×C4—14616no—
492C22×C4—1461no—
493C22×C4—1462no—
494C22×C4—1461no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1461no—
498C22×C4—1462no—
499C22×C4—1462no—
500C22×C4—1461no—
501C22×C4—1462no—
502C22×C4—1462no—
503C22×C4—1462no—
504C22×C4—1462no—
505C22×C4—1462no—
506C22×C4—1462no—
507C22×C4—1462no—
508C22×C4D4×D7314yes—
509C22×C4D4×D7312yes—
510C22×C4C2×Dic14612yes—
511C22×C4C2×Dic14612yes—
512C22×C4#1192402712yes—
513C4⋊C4—44no—
514C4⋊C4—2924no—
515C4⋊C4—2928no—
516C4⋊C4—2928no—
517C4⋊C4—2928no—
518C4⋊C4—2928no—
519C4⋊C4—14616no—
520C22⋊C4—28no—
521C22⋊C4—48no—
522C22⋊C4—2928no—
523C22⋊C4—1468no—
524C42—2924no—
525C42—1468no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23#1455659014yes—
549C23C2×D4×D7311yescharacteristic
550C23C2×D4×D7312yes—
551C23C2×D4×D7311yescharacteristic
552C23C2×Q8×D7311yescharacteristic
553C23C22×Dic14611yescharacteristic
554C23#41×D7311yescharacteristic
555C23#41×D7311yescharacteristic
556C23#41×D7312yes—
557C23C2×#1192402711yescharacteristic
558C23#1455663612yes—
559C23#1455663812yes—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2922no—
586C2×C4—2921no—
587C2×C4—2922no—
588C2×C4—2922no—
589C2×C4—2922no—
590C2×C4—2922no—
591C2×C4—2922no—
592C2×C4—2921no—
593C2×C4—29216no—
594C2×C4—2922no—
595C2×C4—2922no—
596C2×C4—2922no—
597C2×C4—2922no—
598C2×C4—2922no—
599C2×C4—2922no—
600C2×C4—2922no—
601C2×C4—734no—
602C2×C4—738no—
603C2×C4—734no—
604C2×C4—734no—
605C2×C4—738no—
606C2×C4—734no—
607C2×C4—1464no—
608C2×C4—1464no—
609C2×C4—1464no—
610C2×C4—1462no—
611C2×C4—1464no—
612C2×C4—1462no—
613C2×C4—1464no—
614C2×C4—1462no—
615C2×C4—1464no—
616C2×C4—1464no—
617C2×C4—1462no—
618C2×C4#1455656814yes—
619C22—28no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22#121×D7311yescharacteristic
631C22C22⋊Q8×D7312yes—
632C22C22⋊Q8×D7311yescharacteristic
633C22C2×#1455656811yescharacteristic
634C22#123×D7312yes—
635C22#123×D7311yescharacteristic
636C22#127×D7311yescharacteristic
637C22C2×#1455659011yescharacteristic
638C22#1798913211yescharacteristic
639C22#1798913911yescharacteristic
640C22#1798914011yescharacteristic
641C22#1798915211yescharacteristic
642C22#1798915711yescharacteristic
643C22#1798920112yes—
644C22#1798921612yes—
645C22#1798921912yes—
646C22#1798923512yes—
647C22#1798923712yes—
648C22D4×Dic14612yes—
649C22#1798925512yes—
650C22#1798927312yes—
651C22#1798928012yes—
652C22#1798929612yes—
653C4—24no—
654C4—42no—
655C4—44no—
656C4—82no—
657C4—2922no—
658C4—2922no—
659C4—2924no—
660C4—5842no—
661C4—738no—
662C4—1464no—
663C2—44no—
664C2#399×D7311yescharacteristic
665C2#2261343812yes—
666C2#2261345312yes—
667C2#2261363212yes—
668C2#2261363912yes—
669C2#2261423211yescharacteristic
670C2#2261424511yescharacteristic
671C2#2261487211yescharacteristic
672C2#2261488211yescharacteristic
673C2#2261575111yescharacteristic
674C2#2261828111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutator, socle, cosocleC24×C73C24
FittingC73×#1439C2
prime radicalC25×C73C23
hypercenter#1439D73
hypocenterC73#15135
LCSG ▷ C24×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#17988484C2semidirect
3. C2×#17988456C2semidirect
6. #22608197C2semidirect
7. #22608196C2semidirect
8. #22608194C2semidirect
10. #22608159C2semidirect
324. Dic73#399semidirect
349. C73#15135semidirect

Maximal subgroups

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

subgroupindex
C2×#179885232
C2×#179884842
C2×#179884562
C2×#179884452
C2×C22⋊C4×Dic732
#226081972
#226081962
#226081942
#226081762
#226081592
C73×#14392
#1513573