Finite Group Databasebeta

← #33148896

Subgroups — #33148896

Order 18688.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988896C211yesmaximal, characteristic, complemented
2C2×#17988629C211yesmaximal, characteristic
3#22613656C211yesmaximal, characteristic
4#22612962C211yesmaximal, characteristic
5#22611481C211yesmaximal, characteristic, complemented
6#22611431C211yesmaximal, characteristic, complemented
7#22611310C211yesmaximal, characteristic, complemented
8#22611268C211yesmaximal, characteristic, complemented
9#22610702C211yesmaximal, characteristic, complemented
10#22610522C211yesmaximal, characteristic, complemented
11#22610239C211yesmaximal, characteristic
12#22610187C211yesmaximal, characteristic, complemented
13#22610128C211yesmaximal, characteristic
14#22610114C211yesmaximal, characteristic
15C73×#1894C211yesmaximal, characteristic
16C2×#14556582C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×#14556550C2211yescharacteristic
19C22×C4⋊Dic73C2211yescharacteristic
20#17988896—24no—
21#17988629—24no—
22C2×#14556525C2211yescharacteristic
23C2×#14556525C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556523C2211yescharacteristic
26C2×#14556523C2211yescharacteristic
27C2×#14556522C2211yescharacteristic
28C2×#14556520C2211yescharacteristic
29C2×#14556520C2211yescharacteristic
30C2×#14556515C2211yescharacteristic
31C2×#14556515C2211yescharacteristic
32C2×C22⋊#10596831C2211yescharacteristic
33#17988529—21no—
34#17988519—21no—
35#17988494C2211yescharacteristic
36#17988486C2211yescharacteristic
37#17988486C2211yescharacteristic
38#17988483C2211yescharacteristic
39#17988480—21no—
40#17988480—21no—
41#17988473—21no—
42#17988462—21no—
43#17988452C2211yescharacteristic
44#17988445—21no—
45C4⋊C4×Dic73—21no—
46C4⋊C4×Dic73C2211yescharacteristic
47C4×C22⋊Dic73C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988341C2211yescharacteristic
53#17988341C2211yescharacteristic
54C22×M16×C73C2211yescharacteristic
55C2×C22⋊Q8×C73C2211yescharacteristic
56C2×#103×C73C2211yescharacteristic
57C2×#103×C73C2211yescharacteristic
58#380×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61C23×Dic73—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64#14556582—22no—
65#14556582—41no—
66Q8×Dic73—22no—
67Q8×Dic73—41no—
68#14556550—22no—
69#14556550—41no—
70C2×C4⋊Dic73—22no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—41no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×C22⋊Dic73—21no—
84C2×C22⋊Dic73D411yescharacteristic
85C2×C22⋊Dic73D411yescharacteristic
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73C2311yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91#14556525—22no—
92#14556525D414yes—
93#14556524—41no—
94#14556523—22no—
95#14556523—22no—
96#14556522—22no—
97#14556520—22no—
98#14556520—22no—
99#14556515—22no—
100#14556515—41no—
101C22⋊#10596831—22no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498D411yescharacteristic
111#14556498D411yescharacteristic
112C22×Q8×C73C2311yescharacteristic
113C23×C4×C73C2311yescharacteristic
114C2×M16×C73—22no—
115C2×M16×C73—22no—
116C2×M16×C73—22no—
117C22×C8×C73C2311yescharacteristic
118C22×C8×C73C2311yescharacteristic
119C22⋊Q8×C73—22no—
120C22⋊Q8×C73Q814yes—
121C2×C4⋊C4×C73—21no—
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C22⋊C4×C73—21no—
126C2×C22⋊C4×C73—21no—
127#103×C73—22no—
128#103×C73—22no—
129#95×C73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—42no—
135C22×Dic73—42no—
136C22×Dic73—81no—
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C22×Dic73C2×D411yescharacteristic
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—42no—
154C4⋊Dic73—42no—
155C4⋊Dic73—42no—
156C4⋊Dic73—42no—
157C4⋊Dic73—81no—
158C4⋊Dic73C2×D412yes—
159C4⋊Dic73C2×D412yes—
160#11924007—42no—
161#11924007—81no—
162C22⋊Dic73—41no—
163C22⋊Dic73—41no—
164C22⋊Dic73—42no—
165C4×Dic73—21no—
166C4×Dic73—21no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C4×Dic73—42no—
170C4×Dic73—42no—
171C4×Dic73—42no—
172C24×C73C2×D411yescharacteristic
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73C2×Q812yes—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×Q812yes—
189C22×C4×C73#4112yes—
190C22×C4×C73C2411yescharacteristic
191M16×C73—28no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73C2×D412yes—
197C2×C8×C73#4112yes—
198C4⋊C4×C73—24no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73C2×Q812yes—
205C22⋊C4×C73—24no—
206C22⋊C4×C73—42no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73C22⋊Q812yes—
214C23×C73C22×D411yescharacteristic
215Q8×C73—22no—
216Q8×C73—22no—
217Q8×C73—41no—
218Q8×C73—41no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73C22⋊Q812yes—
238C2×C4×C73C22×D411yescharacteristic
239C2×C4×C73C22×Q811yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C73H(5,𝔽2)11yescharacteristic
243C2×C4×C732−1+411yescharacteristic
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—22no—
247C2×Dic73—22no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
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×Dic73—84no—
269C8×C73—22no—
270C8×C73—22no—
271#10596831—41no—
272#10596831—41no—
273C22×C73—22no—
274C22×C73—22no—
275C22×C73—22no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73C2×C22⋊Q811yescharacteristic
281C22×C73D4211yescharacteristic
282C22×C73#54711yescharacteristic
283C22×C73#57511yescharacteristic
284C22×C73#57511yescharacteristic
285C22×C73#57711yescharacteristic
286C22×C73#57811yescharacteristic
287C4×C73—22no—
288C4×C73—42no—
289C4×C73—42no—
290C4×C73—81no—
291C4×C73—81no—
292C4×C73#52511yescharacteristic
293C4×C73#53911yescharacteristic
294C4×C73D4×Q811yescharacteristic
295C4×C73#55411yescharacteristic
296Dic73—41no—
297Dic73—41no—
298Dic73—41no—
299Dic73—42no—
300Dic73—42no—
301Dic73—41no—
302Dic73—81no—
303Dic73—81no—
304#26365—731nomaximal, Sylow
305C2×C73—42no—
306C2×C73#244911yescharacteristic
307C2×C73#308011yescharacteristic
308C2×C73#308211yescharacteristic
309C2×C73#326711yescharacteristic
310C2×C73#327711yescharacteristic
311C2×C73#335011yescharacteristic
312C2×C73#335211yescharacteristic
313C2×#479—731no—
314C2×#428—731no—
315#2442—731no—
316#2362—731no—
317#2097—731no—
318#2088—731no—
319#2066—731no—
320#2059—731no—
321#1949—731no—
322#1921—731no—
323#1894D7311yescharacteristic
324#1872—731no—
325#1863—731no—
326#1854—731no—
327#1851—731no—
328C73#2636511yescharacteristic, complemented, Sylow
329C22×M16C2×D7311yescharacteristic
330C2×#125—731no—
331C2×C22⋊Q8C2×D7311yescharacteristic
332C2×C4×Q8—731no—
333C2×#117—731no—
334C22×C4⋊C4—731no—
335#479—1464no—
336#428—1464no—
337C2×#107—731no—
338C2×#107—731no—
339C2×#107—731no—
340C2×#106—731no—
341C2×#106—731no—
342C2×#106—731no—
343C2×C4⋊C8—731no—
344C2×C4⋊C8—731no—
345C2×#103—731no—
346C2×#103—731no—
347C2×#103C2×D7311yescharacteristic
348C2×#103C2×D7311yescharacteristic
349C2×C22⋊C8—731no—
350#400—1461no—
351#398—1461no—
352#391—731no—
353#389—731no—
354#389—731no—
355#389—731no—
356#388—1461no—
357#388—1461no—
358#387—1461no—
359#385—1461no—
360#382—731no—
361#380—1461no—
362#380C2×D7311yescharacteristic
363C4×C4⋊C4—731no—
364C4×C4⋊C4—1461no—
365C4×C22⋊C4—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340C2×D7311yescharacteristic
371#340C2×D7311yescharacteristic
372#336—731no—
373#336—731no—
374C22×Q8C22×D7311yescharacteristic
375C23×C4—1461no—
376C23×C4C22×D7311yescharacteristic
377C2×M16—22no—
378C2×M16—22no—
379C2×M16—22no—
380C22×C8—731no—
381C22×C8—731no—
382C22×C8C22×D7311yescharacteristic
383C22×C8C22×D7311yescharacteristic
384#125—2921no—
385#125—1462no—
386C22⋊Q8—22no—
387C22⋊Q8Dic14614yes—
388C4×Q8—2921no—
389C4×Q8—1462no—
390#117—2921no—
391#117—1462no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—2921no—
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⋊C4—1461no—
403C2×C4⋊C4—1462no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4C22×D7311yescharacteristic
407C2×C4⋊C4C22×D7311yescharacteristic
408C2×C4⋊C4C22×D7311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—731no—
412C2×C22⋊C4—731no—
413C2×C22⋊C4—1461no—
414C2×C42—731no—
415C2×C42—731no—
416C2×C42—1461no—
417C2×C42—1461no—
418C2×C42—1461no—
419#107—2921no—
420#107—734no—
421#107—1462no—
422#106—1462no—
423#106—1462no—
424#106—1462no—
425C4⋊C8—1462no—
426C4⋊C8—1462no—
427#103—22no—
428#103—22no—
429#103—2921no—
430#103—1462no—
431C22⋊C8—1462no—
432#95—21no—
433#95—731no—
434#95—731no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443C24D4×D7311yescharacteristic
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8C2×Dic14612yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—2922no—
459C22×C4—2922no—
460C22×C4—5841no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—731no—
464C22×C4—731no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4D4×D7311yescharacteristic
470C22×C4C2×Dic14612yes—
471C22×C4#1192402712yes—
472C22×C4C23×D7311yescharacteristic
473M16—28no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8D4×D7312yes—
485C2×C8#1192402712yes—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—2922no—
493C4⋊C4—2922no—
494C4⋊C4—2922no—
495C4⋊C4—2922no—
496C4⋊C4—2922no—
497C4⋊C4—5841no—
498C4⋊C4—5841no—
499C4⋊C4—732no—
500C4⋊C4—732no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4C2×Dic14612yes—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—2922no—
511C22⋊C4—2921no—
512C22⋊C4—2921no—
513C42—2922no—
514C42—2922no—
515C42—2922no—
516C42—1461no—
517C42—1461no—
518C42—1461no—
519C42—1461no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1455656812yes—
527C23C2×D4×D7311yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—2921no—
551C2×C4—2921no—
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—5844no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1462no—
571C2×C4—1461no—
572C2×C4—1462no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4#1455656812yes—
576C2×C4C2×D4×D7311yescharacteristic
577C2×C4C22×Dic14611yescharacteristic
578C2×C4C2×#1192402711yescharacteristic
579C2×C4C2×#1192402711yescharacteristic
580C2×C4#1455663711yescharacteristic
581C2×C4#1455663811yescharacteristic
582C8—22no—
583C8—22no—
584C8—2921no—
585C8—2921no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1455656811yescharacteristic
594C22#1798922111yescharacteristic
595C22#1798923711yescharacteristic
596C22#1798938311yescharacteristic
597C22#1798938611yescharacteristic
598C22#1798939611yescharacteristic
599C22#1798940311yescharacteristic
600C4—22no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—2921no—
606C4—2922no—
607C4—2922no—
608C4—2921no—
609C4—2921no—
610C4—2921no—
611C4—5841no—
612C4—5841no—
613C4#1798909311yescharacteristic
614C4#1798918111yescharacteristic
615C4D4×Dic14611yescharacteristic
616C4#1798929411yescharacteristic
617C2—42no—
618C2#2261374611yescharacteristic
619C2#2261975111yescharacteristic
620C2#2261977511yescharacteristic
621C2#2262146311yescharacteristic
622C2#2262158511yescharacteristic
623C2#2262247111yescharacteristic
624C2#2262250111yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1894C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1894D73
hypocenterC73#26365
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
1. C2×#17988896C2semidirect
5. #22611481C2semidirect
6. #22611431C2semidirect
7. #22611310C2semidirect
8. #22611268C2semidirect
9. #22610702C2semidirect
10. #22610522C2semidirect
12. #22610187C2semidirect
328. C73#26365semidirect

Maximal subgroups

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

subgroupindex
C2×#179888962
C2×#179886292
#226136562
#226129622
#226114812
#226114312
#226113102
#226112682
#226107022
#226105222
#226102392
#226101872
#226101282
#226101142
C73×#18942
#2636573