Finite Group Databasebeta

← #33143140

Subgroups — #33143140

Order 18688.

946 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
1C2×#17988892C211yesmaximal, characteristic, complemented
2#22613649C211yesmaximal, characteristic
3Q8×C22⋊Dic73C211yesmaximal, characteristic
4#22611481C211yesmaximal, characteristic, complemented
5#22611322C211yesmaximal, characteristic, complemented
6#22611310C211yesmaximal, characteristic, complemented
7#22611080C211yesmaximal, characteristic, complemented
8#22610201C211yesmaximal, characteristic, complemented
9#22610183C211yesmaximal, characteristic, complemented
10#22610119C211yesmaximal, characteristic
11#22610109C211yesmaximal, characteristic
12#22609956C211yesmaximal, characteristic
13#22609944C211yesmaximal, characteristic, complemented
14#22609866C211yesmaximal, characteristic
15C22×#103×C73C211yesmaximal, characteristic
16C2×#14556582C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×Q8×Dic73C2211yescharacteristic
19#17988892—24no—
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556520C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556515C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31#17988526—21no—
32#17988520—21no—
33#17988494—21no—
34#17988494—21no—
35#17988494C2211yescharacteristic
36#17988494C2211yescharacteristic
37#17988486C2211yescharacteristic
38#17988483C2211yescharacteristic
39#17988460—21no—
40#17988452C2211yescharacteristic
41#17988447—21no—
42#17988447—21no—
43#17988447C2211yescharacteristic
44#17988447C2211yescharacteristic
45C4×C22⋊Dic73—21no—
46C4×C22⋊Dic73C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C23×Q8×C73C2211yescharacteristic
54C23×C8×C73C2211yescharacteristic
55C22×C4⋊C4×C73C2211yescharacteristic
56C2×#103×C73—24no—
57C2×#103×C73C2211yescharacteristic
58C2×#103×C73C2211yescharacteristic
59C2×#103×C73C2211yescharacteristic
60C2×#103×C73C2211yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556582—22no—
64#14556582—41no—
65Q8×Dic73—22no—
66Q8×Dic73—22no—
67Q8×Dic73—41no—
68Q8×Dic73—41no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73D411yescharacteristic
82C2×C22⋊Dic73D411yescharacteristic
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73C2311yescharacteristic
86C2×C4×Dic73C2311yescharacteristic
87#14556524—22no—
88#14556524—41no—
89#14556523—22no—
90#14556523—41no—
91#14556520—22no—
92#14556520—41no—
93#14556515—22no—
94#14556515—22no—
95#14556515—41no—
96#14556515—41no—
97C22⋊#10596831—22no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498D411yescharacteristic
109#14556498D411yescharacteristic
110C22×Q8×C73—21no—
111C22×Q8×C73—22no—
112C22×Q8×C73—22no—
113C22×Q8×C73—22no—
114C22×Q8×C73—41no—
115C22×Q8×C73—41no—
116C22×Q8×C73Q814yes—
117C22×Q8×C73C2311yescharacteristic
118C22×Q8×C73C2311yescharacteristic
119C23×C4×C73—21no—
120C23×C4×C73—21no—
121C23×C4×C73C2311yescharacteristic
122C22×C8×C73—22no—
123C22×C8×C73—22no—
124C22×C8×C73—41no—
125C22×C8×C73C2311yescharacteristic
126C22×C8×C73C2311yescharacteristic
127C2×C4⋊C4×C73—22no—
128C2×C4⋊C4×C73—21no—
129C2×C4⋊C4×C73—41no—
130C2×C4⋊C4×C73C2311yescharacteristic
131C2×C4⋊C4×C73C2311yescharacteristic
132#103×C73—22no—
133#103×C73—22no—
134#103×C73—22no—
135#103×C73D414yes—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143C22×Dic73C2×D411yescharacteristic
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—42no—
159C4⋊Dic73—42no—
160C4⋊Dic73—42no—
161C4⋊Dic73—81no—
162#11924007—42no—
163#11924007—81no—
164C22⋊Dic73—22no—
165C22⋊Dic73—22no—
166C22⋊Dic73—42no—
167C22⋊Dic73—42no—
168C4×Dic73—21no—
169C4×Dic73—21no—
170C4×Dic73—21no—
171C4×Dic73—21no—
172C4×Dic73—42no—
173C4×Dic73—42no—
174C24×C73C2×D411yescharacteristic
175C2×Q8×C73—24no—
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73—28no—
181C2×Q8×C73—21no—
182C2×Q8×C73—21no—
183C2×Q8×C73—21no—
184C2×Q8×C73—21no—
185C2×Q8×C73—42no—
186C2×Q8×C73—42no—
187C2×Q8×C73—44no—
188C2×Q8×C73—44no—
189C2×Q8×C73—44no—
190C2×Q8×C73C2×D412yes—
191C2×Q8×C73C2×Q812yes—
192C2×Q8×C73C2×Q812yes—
193C2×Q8×C73#4112yes—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—21no—
198C22×C4×C73—21no—
199C22×C4×C73—24no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—42no—
204C22×C4×C73—42no—
205C22×C4×C73—42no—
206C22×C4×C73—42no—
207C22×C4×C73—81no—
208C22×C4×C73C2×D411yescharacteristic
209C22×C4×C73C2×Q812yes—
210C22×C4×C73#4112yes—
211C22×C4×C73C2411yescharacteristic
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73—21no—
215C2×C8×C73—21no—
216C2×C8×C73—21no—
217C2×C8×C73—44no—
218C2×C8×C73C2×D412yes—
219C4⋊C4×C73—21no—
220C4⋊C4×C73—21no—
221C4⋊C4×C73—21no—
222C4⋊C4×C73—42no—
223C4⋊C4×C73C2×D412yes—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73C22⋊Q812yes—
229C23×C73#13512yes—
230C23×C73#13712yes—
231C23×C73C22×D411yescharacteristic
232Q8×C73—22no—
233Q8×C73—22no—
234Q8×C73—22no—
235Q8×C73—22no—
236Q8×C73—22no—
237Q8×C73—48no—
238Q8×C73—41no—
239Q8×C73—41no—
240Q8×C73C22⋊Q814yes—
241C2×C4×C73—22no—
242C2×C4×C73—21no—
243C2×C4×C73—22no—
244C2×C4×C73—21no—
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—48no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—84no—
261C2×C4×C73C22⋊Q812yes—
262C2×C4×C73C22×D411yescharacteristic
263C2×C4×C73C22×Q811yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C73H(5,𝔽2)11yescharacteristic
267C2×C4×C73H(5,𝔽2)11yescharacteristic
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—21no—
273C2×Dic73—21no—
274C2×Dic73—21no—
275C2×Dic73—21no—
276C2×Dic73—21no—
277C2×Dic73—21no—
278C2×Dic73—21no—
279C2×Dic73—21no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288C2×Dic73—41no—
289C2×Dic73—41no—
290C2×Dic73—41no—
291C2×Dic73—41no—
292C8×C73—22no—
293C8×C73—41no—
294#10596831—41no—
295#10596831—41no—
296C22×C73—22no—
297C22×C73—22no—
298C22×C73—22no—
299C22×C73—22no—
300C22×C73—22no—
301C22×C73—22no—
302C22×C73#47914yescomplemented
303C22×C73C2×C22⋊Q811yescharacteristic
304C22×C73D4211yescharacteristic
305C22×C73#54611yescharacteristic
306C22×C73C2×#13511yescharacteristic
307C22×C73C2×#13711yescharacteristic
308C22×C73#57711yescharacteristic
309C22×C73#57811yescharacteristic
310C4×C73—22no—
311C4×C73—24no—
312C4×C73—42no—
313C4×C73—42no—
314C4×C73—81no—
315C4×C73C2×C22⋊Q811yescharacteristic
316C4×C73#54311yescharacteristic
317C4×C73#54611yescharacteristic
318C4×C73#55111yescharacteristic
319Dic73—41no—
320Dic73—41no—
321Dic73—41no—
322Dic73—41no—
323Dic73—81no—
324Dic73—81no—
325Dic73—81no—
326Dic73—81no—
327#25957—731nomaximal, Sylow
328C2×C73—24no—
329C2×C73#243711yescharacteristic
330C2×C73C2×#47911yescharacteristic
331C2×C73#308211yescharacteristic
332C2×C73#327011yescharacteristic
333C2×C73#328211yescharacteristic
334C2×C73#330011yescharacteristic
335C2×C73#332011yescharacteristic
336C2×#479—731no—
337C22×#103D7311yescharacteristic
338#2442—731no—
339Q8×C22⋊C4—731no—
340#2097—731no—
341#2068—731no—
342#2066—731no—
343#2026—731no—
344#1865—731no—
345#1863—731no—
346#1852—731no—
347#1850—731no—
348#1807—731no—
349#1805—731no—
350#1787—731no—
351C73#2595711yescharacteristic, complemented, Sylow
352C23×Q8C2×D7311yescharacteristic
353C23×C8C2×D7311yescharacteristic
354C2×#125—731no—
355C2×C4×Q8—731no—
356C2×C4×Q8—731no—
357C22×C4⋊C4C2×D7311yescharacteristic
358#479—1464no—
359C2×#107—731no—
360C2×#107—731no—
361C2×#106—731no—
362C2×#106—731no—
363C2×C4⋊C8—731no—
364C2×C4⋊C8—731no—
365C2×#103—24no—
366C2×#103—731no—
367C2×#103—731no—
368C2×#103—731no—
369C2×#103—731no—
370C2×#103C2×D7311yescharacteristic
371C2×#103C2×D7311yescharacteristic
372C2×#103C2×D7311yescharacteristic
373C2×#103C2×D7311yescharacteristic
374C2×C22⋊C8—731no—
375#400—1461no—
376#398—1461no—
377#391—731no—
378#391—731no—
379#391—1461no—
380#391—1461no—
381#389—731no—
382#389—731no—
383#385—1461no—
384#382—731no—
385#380—731no—
386#380—731no—
387#380—1461no—
388#380—1461no—
389C4×C22⋊C4—731no—
390C4×C22⋊C4—1461no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#336—731no—
396#336—731no—
397C22×Q8—21no—
398C22×Q8—22no—
399C22×Q8—22no—
400C22×Q8—22no—
401C22×Q8—41no—
402C22×Q8—41no—
403C22×Q8Dic14614yes—
404C22×Q8C22×D7311yescharacteristic
405C22×Q8C22×D7311yescharacteristic
406C23×C4—21no—
407C23×C4—21no—
408C23×C4C22×D7311yescharacteristic
409C22×C8—22no—
410C22×C8—22no—
411C22×C8—41no—
412C22×C8—731no—
413C22×C8—731no—
414C22×C8C22×D7311yescharacteristic
415C22×C8C22×D7311yescharacteristic
416#125—2921no—
417#125—1462no—
418C4×Q8—2921no—
419C4×Q8—2921no—
420C4×Q8—1462no—
421C4×Q8—1462no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—41no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—731no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4—1461no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4—1461no—
434C2×C4⋊C4—1461no—
435C2×C4⋊C4C22×D7311yescharacteristic
436C2×C4⋊C4C22×D7311yescharacteristic
437C2×C22⋊C4—731no—
438C2×C22⋊C4—731no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4—1461no—
441C2×C42—731no—
442C2×C42—731no—
443C2×C42—1461no—
444C2×C42—1461no—
445#107—2921no—
446#107—1462no—
447#106—2921no—
448#106—1462no—
449C4⋊C8—2921no—
450C4⋊C8—1462no—
451#103—22no—
452#103—22no—
453#103—22no—
454#103—2921no—
455#103—2921no—
456#103—1462no—
457#103—1462no—
458#103C22⋊D7314yes—
459C22⋊C8—1462no—
460#95—731no—
461#95—731no—
462#95—1461no—
463#95—1461no—
464#95—1461no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95—1461no—
471#95—1461no—
472C24D4×D7311yescharacteristic
473C2×Q8—24no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—28no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—44no—
486C2×Q8—44no—
487C2×Q8—44no—
488C2×Q8C2×C22⋊D7312yes—
489C2×Q8C2×Dic14612yes—
490C2×Q8C2×Dic14612yes—
491C2×Q8#1192402812yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—24no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—81no—
506C22×C4—731no—
507C22×C4—731no—
508C22×C4—731no—
509C22×C4—731no—
510C22×C4—1461no—
511C22×C4—1461no—
512C22×C4—1461no—
513C22×C4—1461no—
514C22×C4D4×D7311yescharacteristic
515C22×C4C2×Dic14612yes—
516C22×C4#1192402712yes—
517C22×C4C23×D7311yescharacteristic
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—44no—
524C2×C8—1461no—
525C2×C8—1461no—
526C2×C8—1461no—
527C2×C8—1461no—
528C2×C8—1461no—
529C2×C8—1461no—
530C2×C8C2×C22⋊D7312yes—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—42no—
535C4⋊C4—2922no—
536C4⋊C4—2922no—
537C4⋊C4—2922no—
538C4⋊C4—2922no—
539C4⋊C4—5841no—
540C4⋊C4—5841no—
541C4⋊C4—1461no—
542C4⋊C4—1461no—
543C4⋊C4—1461no—
544C4⋊C4—1461no—
545C4⋊C4—1461no—
546C4⋊C4—1461no—
547C4⋊C4—1461no—
548C4⋊C4—1461no—
549C4⋊C4C2×C22⋊D7312yes—
550C22⋊C4—2922no—
551C22⋊C4—2922no—
552C22⋊C4—1462no—
553C22⋊C4—1462no—
554C42—2922no—
555C42—2922no—
556C42—1461no—
557C42—1461no—
558C42—1461no—
559C42—1461no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#1455656812yes—
565C23#1455661212yes—
566C23#1455662312yes—
567C23C2×D4×D7311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—22no—
573Q8—48no—
574Q8—41no—
575Q8—41no—
576Q8#1455657014yes—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—84no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4—1461no—
614C2×C4—1461no—
615C2×C4—1461no—
616C2×C4—1461no—
617C2×C4—1461no—
618C2×C4—1461no—
619C2×C4—1461no—
620C2×C4—1461no—
621C2×C4#1455656812yes—
622C2×C4C22×C22⋊D7311yescharacteristic
623C2×C4C22×Dic14611yescharacteristic
624C2×C4C2×#1192402711yescharacteristic
625C2×C4C2×#1192402811yescharacteristic
626C2×C4#1455663611yescharacteristic
627C2×C4#1455663611yescharacteristic
628C8—22no—
629C8—41no—
630C8—2921no—
631C8—2921no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22#1798889214yescomplemented
639C22C2×#1455656811yescharacteristic
640C22D4×C22⋊D7311yescharacteristic
641C22#1798922711yescharacteristic
642C22C2×#1455661211yescharacteristic
643C22C2×#1455662311yescharacteristic
644C22#1798939311yescharacteristic
645C22#1798940011yescharacteristic
646C4—22no—
647C4—24no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—2921no—
652C4—2921no—
653C4—2921no—
654C4—2921no—
655C4—5841no—
656C4—5841no—
657C4—5841no—
658C4—5841no—
659C4C2×#1455657011yescharacteristic
660C4#1798921311yescharacteristic
661C4#1798923111yescharacteristic
662C4#1798926311yescharacteristic
663C2—24no—
664C2#2261359211yescharacteristic
665C2C2×#1798889211yescharacteristic
666C2#2261976911yescharacteristic
667C2#2262150111yescharacteristic
668C2#2262163811yescharacteristic
669C2#2262181411yescharacteristic
670C2#2262209811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC22×#103×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenterC22×#103D73
hypocenterC73#25957
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
1. C2×#17988892C2semidirect
4. #22611481C2semidirect
5. #22611322C2semidirect
6. #22611310C2semidirect
7. #22611080C2semidirect
8. #22610201C2semidirect
9. #22610183C2semidirect
13. #22609944C2semidirect
302. C22×C73#479semidirect
351. C73#25957semidirect
638. C22#17988892semidirect

Maximal subgroups

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

subgroupindex
C2×#179888922
#226136492
Q8×C22⋊Dic732
#226114812
#226113222
#226113102
#226110802
#226102012
#226101832
#226101192
#226101092
#226099562
#226099442
#226098662
C22×#103×C732
#2595773