Finite Group Databasebeta

← #33089049

Subgroups — #33089049

Order 18688.

922 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612396C211yesmaximal, characteristic
2C4×Q8×Dic73C211yesmaximal, characteristic, complemented
3#22610560C211yesmaximal, characteristic
4#22610553C211yesmaximal, characteristic
5#22610525C211yesmaximal, characteristic, complemented
6#22610504C211yesmaximal, characteristic, complemented
7#22610288C211yesmaximal, characteristic
8#22610288C211yesmaximal, characteristic
9#22610201C211yesmaximal, characteristic, complemented
10#22610187C211yesmaximal, characteristic, complemented
11#22609828C211yesmaximal, characteristic, complemented
12#22609761C211yesmaximal, characteristic, complemented
13#22609756C211yesmaximal, characteristic
14#22609679C211yesmaximal, characteristic, complemented
15C2×#440×C73C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×#14556550—21no—
19C2×#14556525C2211yescharacteristic
20C2×#14556523C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556502C2211yescharacteristic
26#17988490C2211yescharacteristic
27#17988490C2211yescharacteristic
28#17988486C2211yescharacteristic
29#17988486C2211yescharacteristic
30#17988480—21no—
31#17988480—21no—
32C4⋊C4×Dic73—21no—
33C4⋊C4×Dic73—21no—
34C4⋊C4×Dic73C2211yescharacteristic
35C4⋊C4×Dic73C2211yescharacteristic
36C4×C4⋊Dic73—21no—
37C4×C4⋊Dic73C2211yescharacteristic
38C4×C4⋊Dic73C2211yescharacteristic
39C4×C22⋊Dic73—21no—
40C42×Dic73—21no—
41C42×Dic73C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C22×SD16×C73C2211yescharacteristic
53C2×C4×Q8×C73C2211yescharacteristic
54C2×C4×D4×C73C2211yescharacteristic
55#440×C73C418yescomplemented
56C2×#107×C73C2211yescharacteristic
57C2×#103×C73C2211yescharacteristic
58C2×#102×C73C2211yescharacteristic
59C2×#97×C73C2211yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62Q8×Dic73—22no—
63Q8×Dic73—22no—
64Q8×Dic73—22no—
65Q8×Dic73—22no—
66#14556550—44no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73D411yescharacteristic
86C2×C4×Dic73D411yescharacteristic
87C2×C4×Dic73D411yescharacteristic
88C2×C4×Dic73D411yescharacteristic
89C2×C4×Dic73C2311yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91#14556525—22no—
92#14556523—22no—
93#14556515—22no—
94#14556515—22no—
95#14556512—22no—
96#14556512—22no—
97#14556502—22no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104C22×Q8×C73C2311yescharacteristic
105C22×D4×C73C2311yescharacteristic
106C23×C4×C73—21no—
107C2×SD16×C73—22no—
108C2×SD16×C73—22no—
109C2×SD16×C73C2×C414yes—
110C22×C8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C4×Q8×C73—22no—
113C4×Q8×C73C2×C414yes—
114C4×D4×C73—22no—
115C4×D4×C73C2×C414yes—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C42×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#107×C73C2×C414yes—
123#103×C73C2×C414yes—
124#102×C73C2×C414yes—
125#97×C73C2×C414yes—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73C2×D411yescharacteristic
132C22×Dic73C2×D411yescharacteristic
133C22×Dic73C2×D411yescharacteristic
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C4⋊Dic73—24no—
141C4⋊Dic73—24no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—42no—
149C4⋊Dic73—42no—
150C4⋊Dic73#4112yes—
151C4⋊Dic73#4112yes—
152C22⋊Dic73—42no—
153C22⋊Dic73—42no—
154C22⋊Dic73—42no—
155C22⋊Dic73—42no—
156C4×Dic73—24no—
157C4×Dic73—24no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—21no—
161C4×Dic73—21no—
162C4×Dic73—22no—
163C4×Dic73—22no—
164C4×Dic73—41no—
165C4×Dic73—41no—
166C4×Dic73—42no—
167C4×Dic73—42no—
168C4×Dic73—42no—
169C4×Dic73—42no—
170C24×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73—21no—
174C2×Q8×C73C22×C412yes—
175C2×Q8×C73#4112yes—
176C2×Q8×C73#4112yes—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73C22×C412yes—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—24no—
187C22×C4×C73—42no—
188C22×C4×C73—42no—
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2411yescharacteristic
192SD16×C73—28no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73C22×C412yes—
198C2×C8×C73C22×C412yes—
199C4⋊C4×C73—24no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73C22×C412yes—
203C4⋊C4×C73C22×C412yes—
204C22⋊C4×C73—24no—
205C42×C73—24no—
206C42×C73C22×C412yes—
207C42×C73#4112yes—
208C23×C73—22no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73C22×D411yescharacteristic
216Q8×C73—22no—
217Q8×C73—22no—
218Q8×C73—22no—
219Q8×C73#11714yescomplemented
220D4×C73—22no—
221D4×C73—22no—
222D4×C73—41no—
223D4×C73—41no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—48no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73C4×D412yes—
239C2×C4×C73C4×D412yes—
240C2×C4×C73C23×C411yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—22no—
253C2×Dic73—22no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73AGL(1,ℤ/8)12yes—
269C2×Dic73#13712yes—
270C8×C73—22no—
271C8×C73—22no—
272#10596831—41no—
273#10596831—41no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—42no—
277C22×C73—81no—
278C22×C73—81no—
279C22×C73—81no—
280C22×C73—81no—
281C22×C73C2×C4×D411yescharacteristic
282C22×C73#52511yescharacteristic
283C22×C73#52511yescharacteristic
284C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C73C2×#13711yescharacteristic
286C22×C73#57511yescharacteristic
287C22×C73#57511yescharacteristic
288C4×C73—22no—
289C4×C73—22no—
290C4×C73—24no—
291C4×C73—42no—
292C4×C73—42no—
293C4×C73C2×#11711yescharacteristic
294C4×C73C4×#4111yescharacteristic
295C4×C73#52911yescharacteristic
296C4×C73#52911yescharacteristic
297Dic73—22no—
298Dic73—22no—
299Dic73—41no—
300Dic73—42no—
301Dic73—41no—
302Dic73—42no—
303Dic73—81no—
304Dic73—81no—
305#21124—731nomaximal, Sylow
306C2×C73—82no—
307C2×C73#229011yescharacteristic
308C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
309C2×C73C4×#13711yescharacteristic
310C2×C73#311111yescharacteristic
311C2×C73#311211yescharacteristic
312C2×C73#311511yescharacteristic
313C2×C73#311711yescharacteristic
314C2×#440D7311yescharacteristic
315#2293—731no—
316C42×Q8—731no—
317#1926—731no—
318#1925—731no—
319#1921—731no—
320#1918—731no—
321#1880—731no—
322#1880—731no—
323#1865—731no—
324#1863—731no—
325#1776—731no—
326#1763—731no—
327#1762—731no—
328#1745—731no—
329C73#2112411yescharacteristic, complemented, Sylow
330C22×SD16C2×D7311yescharacteristic
331C2×C4×Q8—731no—
332C2×C4×Q8—731no—
333C2×C4×Q8C2×D7311yescharacteristic
334C2×C4×D4C2×D7311yescharacteristic
335C2×#117—1461no—
336#440Dic7318yescomplemented
337C2×#107—731no—
338C2×#107C2×D7311yescharacteristic
339C2×#106—731no—
340C2×#103—731no—
341C2×#103—731no—
342C2×#103C2×D7311yescharacteristic
343C2×#102—731no—
344C2×#102—731no—
345C2×#102C2×D7311yescharacteristic
346C2×#97—731no—
347C2×#97C2×D7311yescharacteristic
348#390—731no—
349#390—731no—
350#389—731no—
351#389—731no—
352#388—1461no—
353#388—1461no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4—731no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4—1461no—
359C4×C4⋊C4—1461no—
360C4×C4⋊C4—1461no—
361C4×C22⋊C4—1461no—
362C43—731no—
363C43—1461no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#336—731no—
371#336—731no—
372#336—731no—
373#336—731no—
374C22×Q8C22×D7311yescharacteristic
375C22×D4C22×D7311yescharacteristic
376C23×C4—21no—
377C2×SD16—22no—
378C2×SD16—22no—
379C2×SD16C2×Dic7314yes—
380C22×C8—731no—
381C22×C8—731no—
382C22×C8C22×D7311yescharacteristic
383C22×C8C22×D7311yescharacteristic
384C4×Q8—22no—
385C4×Q8—1462no—
386C4×Q8—1462no—
387C4×Q8—1462no—
388C4×Q8—1462no—
389C4×Q8C2×Dic7314yes—
390C4×D4—22no—
391C4×D4C2×Dic7314yes—
392#117—2924no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4C22×D7311yescharacteristic
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1461no—
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1461no—
408C2×C22⋊C4—1461no—
409C2×C42—21no—
410C2×C42—731no—
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—731no—
414C2×C42—731no—
415C2×C42—731no—
416C2×C42—1461no—
417C2×C42—1461no—
418C2×C42—1461no—
419C2×C42—1461no—
420C2×C42—1461no—
421C2×C42—1461no—
422C2×C42C22×D7311yescharacteristic
423#107—1462no—
424#107C2×Dic7314yes—
425#106—1462no—
426#103—1462no—
427#103—1462no—
428#103C2×Dic7314yes—
429#102—1462no—
430#102—1462no—
431#102C2×Dic7314yes—
432#97—1462no—
433#97C2×Dic7314yes—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C22×Dic7312yes—
445C2×Q8#1192402812yes—
446C2×Q8#1192402812yes—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C22×Dic7312yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—731no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4D4×D7311yescharacteristic
468C22×C4D4×D7311yescharacteristic
469C22×C4C23×D7311yescharacteristic
470SD16—28no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8C22×Dic7312yes—
482C2×C8C22×Dic7312yes—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—2922no—
487C4⋊C4—2922no—
488C4⋊C4—732no—
489C4⋊C4—732no—
490C4⋊C4—1464no—
491C4⋊C4—1464no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4C22×Dic7312yes—
499C4⋊C4C22×Dic7312yes—
500C22⋊C4—24no—
501C22⋊C4—2922no—
502C22⋊C4—2922no—
503C22⋊C4—2922no—
504C22⋊C4—2922no—
505C42—24no—
506C42—2922no—
507C42—2922no—
508C42—2921no—
509C42—2921no—
510C42—2922no—
511C42—2922no—
512C42—1462no—
513C42—1462no—
514C42—1464no—
515C42—1464no—
516C42—1461no—
517C42—1461no—
518C42—1461no—
519C42—1461no—
520C42C22×Dic7312yes—
521C42#1192402712yes—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×D4×D7311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1455655014yescomplemented
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—732no—
563C2×C4—732no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1462no—
568C2×C4—1462no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4D4×Dic7312yes—
577C2×C4D4×Dic7312yes—
578C2×C4C23×Dic7311yescharacteristic
579C2×C4#41×D7311yescharacteristic
580C2×C4#41×D7311yescharacteristic
581C2×C4C2×#1192402711yescharacteristic
582C2×C4C2×#1192402811yescharacteristic
583C2×C4C2×#1192402811yescharacteristic
584C8—22no—
585C8—22no—
586C8—2921no—
587C8—2921no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C2×D4×Dic7311yescharacteristic
596C22#1798909611yescharacteristic
597C22#1798909611yescharacteristic
598C22AGL(1,ℤ/8)×D7311yescharacteristic
599C22#137×D7311yescharacteristic
600C22#1798938311yescharacteristic
601C22#1798938611yescharacteristic
602C4—22no—
603C4—22no—
604C4—24no—
605C4—42no—
606C4—42no—
607C4—2922no—
608C4—2921no—
609C4—2922no—
610C4—2921no—
611C4—5841no—
612C4—5841no—
613C4—1462no—
614C4—1462no—
615C4C2×#1455655011yescharacteristic
616C4#41×Dic7311yescharacteristic
617C4#1798911711yescharacteristic
618C4#1798911711yescharacteristic
619C2—82no—
620C2#2261237611yescharacteristic
621C2AGL(1,ℤ/8)×Dic7311yescharacteristic
622C2#137×Dic7311yescharacteristic
623C2#2262003811yescharacteristic
624C2#2262005111yescharacteristic
625C2#2262008011yescharacteristic
626C2#2262010011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#440×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenterC2×#440D73
hypocenterC73#21124
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×Dic73C2semidirect
5. #22610525C2semidirect
6. #22610504C2semidirect
9. #22610201C2semidirect
10. #22610187C2semidirect
11. #22609828C2semidirect
12. #22609761C2semidirect
14. #22609679C2semidirect
55. #440×C73C4semidirect
219. Q8×C73#117semidirect
329. C73#21124semidirect
336. #440Dic73semidirect
533. Q8#14556550semidirect

Maximal subgroups

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

subgroupindex
#226123962
C4×Q8×Dic732
#226105602
#226105532
#226105252
#226105042
#226102882
#226102012
#226101872
#226098282
#226097612
#226097562
#226096792
C2×#440×C732
#2112473