Finite Group Databasebeta

← #33089172

Subgroups — #33089172

Order 18688.

938 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×Dic73C211yesmaximal, characteristic
2#22612568C211yesmaximal, characteristic, complemented
3#22612292C211yesmaximal, characteristic, complemented
4#22610561C211yesmaximal, characteristic, complemented
5#22610555C211yesmaximal, characteristic, complemented
6#22610503C211yesmaximal, characteristic, complemented
7#22610501C211yesmaximal, characteristic, complemented
8#22610281C211yesmaximal, characteristic
9#22610192C211yesmaximal, characteristic, complemented
10#22610190C211yesmaximal, characteristic, complemented
11#22609819C211yesmaximal, characteristic, complemented
12#22609758C211yesmaximal, characteristic, complemented
13#22609758C211yesmaximal, characteristic, complemented
14#22609681C211yesmaximal, characteristic, complemented
15C73×#1879C211yesmaximal, characteristic
16C2×D4×Dic73C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19D8×Dic73C418yescomplemented
20C2×#14556525C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988490C2211yescharacteristic
30#17988485C2211yescharacteristic
31#17988483C2211yescharacteristic
32#17988481—21no—
33#17988475—21no—
34#17988448—21no—
35#17988445—21no—
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40C22⋊C4×Dic73—21no—
41C22⋊C4×Dic73—21no—
42C4×C22⋊Dic73—21no—
43#17988433C2211yescharacteristic
44C2×#14556498—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988342C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C22×D8×C73C2211yescharacteristic
54C2×C4×D4×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57#390×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C23×Dic73—21no—
61C23×Dic73—21no—
62C23×Dic73—21no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65D4×Dic73—22no—
66D4×Dic73—22no—
67D4×Dic73—22no—
68D4×Dic73—22no—
69D4×Dic73C2×C414yes—
70D4×Dic73C2×C414yescomplemented
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88#14556525C2×C414yescomplemented
89#14556522—22no—
90#14556522—22no—
91#14556512—22no—
92#14556512—22no—
93#14556512C2×C414yes—
94#14556512C2×C414yescomplemented
95#14556501—22no—
96C8×Dic73C2×C414yescomplemented
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498D411yescharacteristic
104#14556498D411yescharacteristic
105#14556498D411yescharacteristic
106#14556498D411yescharacteristic
107C22×D4×C73C2311yescharacteristic
108C22×D4×C73C2311yescharacteristic
109C23×C4×C73—21no—
110C2×D8×C73—22no—
111C2×D8×C73—22no—
112C2×D8×C73C2×C414yes—
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C4×D4×C73—22no—
116C4×D4×C73—22no—
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—21no—
122C2×C42×C73C2311yescharacteristic
123#102×C73—22no—
124#102×C73—22no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—24no—
129C22×Dic73—24no—
130C22×Dic73—24no—
131C22×Dic73—21no—
132C22×Dic73—81no—
133C22×Dic73—81no—
134C22×Dic73—81no—
135C22×Dic73—81no—
136C22×Dic73—81no—
137C22×Dic73—81no—
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831C22×C412yes—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73C22×C412yes—
154C4⋊Dic73C22×C412yes—
155C4⋊Dic73#4112yes—
156#11924007—81no—
157#11924007—81no—
158C22⋊Dic73—24no—
159C22⋊Dic73—24no—
160C22⋊Dic73—24no—
161C22⋊Dic73—81no—
162C22⋊Dic73—81no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—81no—
166C4×Dic73—81no—
167C4×Dic73—81no—
168C4×Dic73—81no—
169C4×Dic73C22×C412yes—
170C4×Dic73#4112yes—
171C24×C73—21no—
172C24×C73—21no—
173C2×D4×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73C22×C412yes—
182C2×D4×C73C22×C412yes—
183C2×D4×C73#4112yes—
184C2×D4×C73#4112yes—
185C22×C4×C73—21no—
186C22×C4×C73—24no—
187C22×C4×C73—21no—
188C22×C4×C73—81no—
189C22×C4×C73—81no—
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2411yescharacteristic
193D8×C73—28no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73C22×C412yes—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73#4112yes—
204C22⋊C4×C73—24no—
205C22⋊C4×C73—81no—
206C22⋊C4×C73—81no—
207C42×C73—21no—
208C42×C73—21no—
209C23×C73—22no—
210C23×C73—22no—
211C23×C73—22no—
212C23×C73—22no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73C22×D411yescharacteristic
224D4×C73—22no—
225D4×C73—22no—
226D4×C73—22no—
227D4×C73—22no—
228D4×C73—22no—
229D4×C73—41no—
230D4×C73—41no—
231D4×C73#11714yes—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—22no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—84no—
245C2×C4×C73C23×C411yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—22no—
256C2×Dic73—22no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—22no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—84no—
274C2×Dic73—84no—
275C2×Dic73—84no—
276C2×Dic73C4×D412yes—
277C2×Dic73C4×D412yes—
278C8×C73—22no—
279C8×C73—41no—
280#10596831—22no—
281#10596831—41no—
282C22×C73—24no—
283C22×C73—42no—
284C22×C73—42no—
285C22×C73—42no—
286C22×C73—42no—
287C22×C73—42no—
288C22×C73—42no—
289C22×C73—42no—
290C22×C73—42no—
291C22×C73—42no—
292C22×C73—81no—
293C22×C73—81no—
294C22×C73—81no—
295C22×C73—81no—
296C22×C73C2×C4×D411yescharacteristic
297C22×C73#52511yescharacteristic
298C22×C73#52511yescharacteristic
299C22×C73#57611yescharacteristic
300C22×C73#57611yescharacteristic
301C22×C73#57711yescharacteristic
302C22×C73#57711yescharacteristic
303C4×C73—41no—
304C4×C73—42no—
305C4×C73—41no—
306C4×C73—81no—
307C4×C73C2×#11711yescharacteristic
308C4×C73C4×#4111yescharacteristic
309C4×C73#52911yescharacteristic
310C4×C73#52911yescharacteristic
311Dic73—22no—
312Dic73—22no—
313Dic73—41no—
314Dic73—42no—
315Dic73—41no—
316Dic73—42no—
317Dic73—42no—
318Dic73—81no—
319#21134—731nomaximal, Sylow
320C2×C73—44no—
321C2×C73—82no—
322C2×C73#229011yescharacteristic
323C2×C73#294811yescharacteristic
324C2×C73#295211yescharacteristic
325C2×C73#311811yescharacteristic
326C2×C73#311811yescharacteristic
327C2×C73#311911yescharacteristic
328C2×C73#311911yescharacteristic
329C2×C4×D8—731no—
330#2319—731no—
331#2275—731no—
332#1926—731no—
333#1926—731no—
334#1918—731no—
335#1918—731no—
336#1879—731no—
337#1879D7311yescharacteristic
338#1864—731no—
339#1864—731no—
340C4×#106—731no—
341#1762—731no—
342#1762—731no—
343#1746—731no—
344C73#2113411yescharacteristic, complemented, Sylow
345C22×D8C2×D7311yescharacteristic
346C2×C4×D4—731no—
347C2×C4×D4—731no—
348C2×C4×D4—731no—
349C2×C4×D4C2×D7311yescharacteristic
350C4×D8—738no—
351C2×#107—731no—
352C2×#106—731no—
353C2×#106—731no—
354C2×#102—731no—
355C2×#102—731no—
356C2×#102—731no—
357C2×#102—731no—
358C2×#102C2×D7311yescharacteristic
359C2×#102C2×D7311yescharacteristic
360C2×#97—731no—
361C2×C4×C8—731no—
362#390—731no—
363#390C2×D7311yescharacteristic
364#389—731no—
365#389—731no—
366#388—1461no—
367#388—1461no—
368#381—1461no—
369#380—1461no—
370C4×C4⋊C4—731no—
371C4×C4⋊C4—731no—
372C4×C4⋊C4—731no—
373C4×C4⋊C4—731no—
374C4×C22⋊C4—1461no—
375C4×C22⋊C4—1461no—
376C4×C22⋊C4—1461no—
377#376—731no—
378C2×#95—1461no—
379#340—731no—
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340C2×D7311yescharacteristic
385#336—731no—
386#336—731no—
387#336—731no—
388#336C2×D7311yescharacteristic
389C22×D4C22×D7311yescharacteristic
390C22×D4C22×D7311yescharacteristic
391C23×C4—21no—
392C23×C4—1461no—
393C23×C4—1461no—
394C23×C4—1461no—
395C2×D8—22no—
396C2×D8—22no—
397C2×D8C4×D7314yes—
398C22×C8—731no—
399C22×C8—731no—
400C22×C8C22×D7311yescharacteristic
401C22×C8C22×D7311yescharacteristic
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—734no—
405C4×D4—734no—
406C4×D4—1462no—
407C4×D4—1462no—
408C4×D4—1462no—
409C4×D4—1462no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4—1461no—
416C2×C4⋊C4C22×D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1461no—
422C2×C22⋊C4—1461no—
423C2×C22⋊C4—1461no—
424C2×C22⋊C4—1461no—
425C2×C22⋊C4—1461no—
426C2×C42—731no—
427C2×C42—731no—
428C2×C42—1461no—
429C2×C42—1461no—
430C2×C42—1461no—
431C2×C42—1461no—
432C2×C42C22×D7311yescharacteristic
433#107—734no—
434#106—1462no—
435#106—1462no—
436#102—22no—
437#102—22no—
438#102—734no—
439#102—734no—
440#102—1462no—
441#102—1462no—
442#97—1462no—
443C4×C8—734no—
444#95—731no—
445#95—731no—
446#95—731no—
447#95—731no—
448#95—1461no—
449#95—1461no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453#95—1461no—
454C24—21no—
455C24—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4C2×C4×D7312yes—
465C2×D4C2×C4×D7312yes—
466C2×D4#1192402712yes—
467C2×D4#1192402712yes—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—5841no—
474C22×C4—5841no—
475C22×C4—5841no—
476C22×C4—5841no—
477C22×C4—5841no—
478C22×C4—5841no—
479C22×C4—731no—
480C22×C4—731no—
481C22×C4—731no—
482C22×C4—731no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1464no—
488C22×C4—1464no—
489C22×C4—1464no—
490C22×C4D4×D7311yescharacteristic
491C22×C4D4×D7311yescharacteristic
492C22×C4C23×D7311yescharacteristic
493D8—28no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—732no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8C2×C4×D7312yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—5841no—
510C4⋊C4—5841no—
511C4⋊C4—732no—
512C4⋊C4—732no—
513C4⋊C4—732no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4#1192402712yes—
520C22⋊C4—24no—
521C22⋊C4—81no—
522C22⋊C4—81no—
523C22⋊C4—5841no—
524C22⋊C4—5841no—
525C22⋊C4—1464no—
526C22⋊C4—1464no—
527C22⋊C4—1464no—
528C42—21no—
529C42—21no—
530C42—5841no—
531C42—5841no—
532C42—5841no—
533C42—5841no—
534C42—732no—
535C42—732no—
536C42—1461no—
537C42—1461no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23C2×D4×D7311yescharacteristic
553D4—22no—
554D4—22no—
555D4—22no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560D4#1455654714yes—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—84no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—5844no—
584C2×C4—5844no—
585C2×C4—5844no—
586C2×C4—732no—
587C2×C4—732no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4—1461no—
591C2×C4—1461no—
592C2×C4—1462no—
593C2×C4—1462no—
594C2×C4—1461no—
595C2×C4—1461no—
596C2×C4—1462no—
597C2×C4—1461no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4C22×C4×D7311yescharacteristic
602C2×C4#41×D7311yescharacteristic
603C2×C4#41×D7311yescharacteristic
604C2×C4C2×#1192402711yescharacteristic
605C2×C4C2×#1192402711yescharacteristic
606C2×C4C2×#1192402711yescharacteristic
607C8—22no—
608C8—41no—
609C8—2921no—
610C8—1462no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22C4×D4×D7311yescharacteristic
626C22#1798909311yescharacteristic
627C22#1798909711yescharacteristic
628C22#1798938811yescharacteristic
629C22#1798938911yescharacteristic
630C22#1798939711yescharacteristic
631C22#1798939711yescharacteristic
632C4—41no—
633C4—42no—
634C4—41no—
635C4—81no—
636C4—2922no—
637C4—2922no—
638C4—2922no—
639C4—2921no—
640C4—2921no—
641C4—5841no—
642C4—1462no—
643C4—1462no—
644C4C2×#1455654711yescharacteristic
645C4C4×#1192402711yescharacteristic
646C4#1798911411yescharacteristic
647C4#1798911811yescharacteristic
648C2—44no—
649C2—82no—
650C2#2261237111yescharacteristic
651C2#2261877811yescharacteristic
652C2#2261881311yescharacteristic
653C2#2262010811yescharacteristic
654C2#2262011211yescharacteristic
655C2#2262012311yescharacteristic
656C2#2262012711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1879C2
socleC23×C73C22×D4
prime radicalC22×D8×C73C22
cosocleC22×C4×C73C24
hypercenter#1879D73
hypocenterC73#21134
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

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

NH ≅ G/Ntype
2. #22612568C2semidirect
3. #22612292C2semidirect
4. #22610561C2semidirect
5. #22610555C2semidirect
6. #22610503C2semidirect
7. #22610501C2semidirect
9. #22610192C2semidirect
10. #22610190C2semidirect
11. #22609819C2semidirect
12. #22609758C2semidirect
13. #22609758C2semidirect
14. #22609681C2semidirect
19. D8×Dic73C4semidirect
70. D4×Dic73C2×C4semidirect
88. #14556525C2×C4semidirect
94. #14556512C2×C4semidirect
96. C8×Dic73C2×C4semidirect
344. C73#21134semidirect

Maximal subgroups

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

subgroupindex
C2×D8×Dic732
#226125682
#226122922
#226105612
#226105552
#226105032
#226105012
#226102812
#226101922
#226101902
#226098192
#226097582
#226096812
C73×#18792
#2113473