Finite Group Databasebeta

← #33114146

Subgroups — #33114146

Order 18688.

1240 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989144C211yesmaximal, characteristic, complemented
2C2×C4×#11924016C212yesmaximal, complemented
3#22611264C211yesmaximal, characteristic, complemented
4#22611177C211yesmaximal, characteristic, complemented
5#22611127C212yesmaximal, complemented
6#22611067C211yesmaximal, characteristic, complemented
7#22611013C211yesmaximal, characteristic, complemented
8#22610934C212yesmaximal
9C4×#14556514C212yesmaximal, complemented
10C4×C22⋊#10596831C211yesmaximal, characteristic, complemented
11C4×#121×C73C211yesmaximal, characteristic
12C22×#11924016C2212yes—
13#17989144—28no—
14C2×#14556591C2211yescharacteristic
15C2×#14556591C2211yescharacteristic
16C2×#14556583—21no—
17C2×#14556570—21no—
18C2×#14556570C2212yes—
19C2×C4×Dic146—21no—
20C2×C4×Dic146C2212yes—
21C2×#14556550—21no—
22C4×#11924016—28no—
23C2×#14556522C2212yes—
24C2×#14556514C2212yes—
25C2×#14556514C2212yes—
26C2×#14556514C2212yes—
27C2×#14556512C2212yes—
28C2×C22⋊#10596831C2212yes—
29C2×C4×#10596831C2212yes—
30#17988349C2212yes—
31#17988349C2212yes—
32#17988342C2211yescharacteristic
33#17988342C2211yescharacteristic
34C2×#121×C73C2211yescharacteristic
35C2×C4×D4×C73—21no—
36C2×C4×D4×C73C2211yescharacteristic
37C22×C42×C73C2211yescharacteristic
38#390×C73C2211yescharacteristic
39#389×C73C2211yescharacteristic
40#388×C73—21no—
41#386×C73—21no—
42#380×C73C2211yescharacteristic
43C4×C4⋊C4×C73C2211yescharacteristic
44C4×C22⋊C4×C73—21no—
45C22×Dic146—21no—
46C22×Dic146C2312yes—
47C2×#11924016—24no—
48C2×#11924016—24no—
49C2×#11924016—24no—
50C22×#10596831C2312yes—
51C22×#10596831C2312yes—
52#14556591—24no—
53#14556591—24no—
54#14556583—28no—
55#14556570—28no—
56#14556570—24no—
57#14556570—24no—
58C4×Dic146—28no—
59C4×Dic146—24no—
60C4×Dic146D418yescomplemented
61#14556550—28no—
62C2×C4⋊Dic73—21no—
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73C2312yes—
66C2×#11924007—22no—
67C2×#11924007—22no—
68C2×C22⋊Dic73—22no—
69C2×C4×Dic73—22no—
70#14556522—24no—
71#14556514—24no—
72#14556514—24no—
73#14556514—24no—
74#14556512—24no—
75C22⋊#10596831—24no—
76C4×#10596831—24no—
77C22×D4×C73—21no—
78C22×D4×C73C2311yescharacteristic
79C23×C4×C73—21no—
80C23×C4×C73D411yescharacteristic
81C23×C4×C73D411yescharacteristic
82C23×C4×C73C2311yescharacteristic
83#121×C73—22no—
84#121×C73—22no—
85C4×D4×C73—24no—
86C4×D4×C73—44no—
87C2×C4⋊C4×C73—21no—
88C2×C4⋊C4×C73C2311yescharacteristic
89C2×C4⋊C4×C73C2311yescharacteristic
90C2×C4⋊C4×C73C2311yescharacteristic
91C2×C22⋊C4×C73—21no—
92C2×C22⋊C4×C73—21no—
93C2×C22⋊C4×C73—21no—
94C2×C22⋊C4×C73—21no—
95C2×C22⋊C4×C73—21no—
96C2×C42×C73—24no—
97C2×C42×C73—21no—
98C2×C42×C73D411yescharacteristic
99C2×C42×C73D411yescharacteristic
100C2×C42×C73C2311yescharacteristic
101C2×C42×C73C2311yescharacteristic
102#95×C73—21no—
103#95×C73—21no—
104#95×C73—21no—
105C2×Dic146—22no—
106C2×Dic146—22no—
107C2×Dic146—22no—
108C2×Dic146—22no—
109C2×Dic146—24no—
110C2×Dic146—22no—
111C2×Dic146—42no—
112C2×Dic146—42no—
113C2×Dic146C2×D414yes—
114C22×Dic73—22no—
115C22×Dic73—22no—
116#11924016—216no—
117C2×#10596831—22no—
118C2×#10596831—22no—
119C2×#10596831—22no—
120C2×#10596831—22no—
121C2×#10596831—22no—
122C2×#10596831#4114yes—
123C4⋊Dic73—24no—
124C4⋊Dic73—24no—
125C4⋊Dic73—24no—
126C4⋊Dic73—22no—
127C4⋊Dic73C2×D414yes—
128#11924007—28no—
129#11924007—28no—
130C22⋊Dic73—28no—
131C4×Dic73—28no—
132C24×C73—21no—
133C24×C73C2×D411yescharacteristic
134C2×D4×C73—24no—
135C2×D4×C73—22no—
136C2×D4×C73—22no—
137C2×D4×C73—21no—
138C2×D4×C73—42no—
139C2×D4×C73—42no—
140C2×D4×C73#4112yes—
141C22×C4×C73—22no—
142C22×C4×C73—22no—
143C22×C4×C73—21no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—22no—
150C22×C4×C73—21no—
151C22×C4×C73—21no—
152C22×C4×C73—41no—
153C22×C4×C73—42no—
154C22×C4×C73—42no—
155C22×C4×C73—42no—
156C22×C4×C73SD1614yescomplemented
157C22×C4×C73C2×D411yescharacteristic
158C22×C4×C73C2×D411yescharacteristic
159C22×C4×C73C2×D411yescharacteristic
160C22×C4×C73C2×D411yescharacteristic
161C22×C4×C73C2×D411yescharacteristic
162C22×C4×C73C2411yescharacteristic
163C4⋊C4×C73—21no—
164C4⋊C4×C73—22no—
165C4⋊C4×C73—22no—
166C4⋊C4×C73—42no—
167C4⋊C4×C73#4112yes—
168C22⋊C4×C73—24no—
169C22⋊C4×C73—42no—
170C22⋊C4×C73—42no—
171C22⋊C4×C73—42no—
172C22⋊C4×C73—42no—
173C42×C73—22no—
174C42×C73—22no—
175C42×C73—22no—
176C42×C73—42no—
177C42×C73C2×D414yes—
178C23×C73—21no—
179C23×C73—21no—
180C23×C73—21no—
181C23×C73—21no—
182C23×C73—21no—
183C23×C73—21no—
184C23×C73—22no—
185C23×C73—41no—
186C23×C73—41no—
187C23×C73—41no—
188C23×C73—41no—
189C23×C73—41no—
190C23×C73—41no—
191C23×C73C2×SD1612yes—
192C23×C73C22×D411yescharacteristic
193D4×C73—24no—
194D4×C73—48no—
195D4×C73—42no—
196C2×C4×C73—21no—
197C2×C4×C73—21no—
198C2×C4×C73—21no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—28no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—22no—
206C2×C4×C73—21no—
207C2×C4×C73—44no—
208C2×C4×C73—41no—
209C2×C4×C73—41no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—84no—
219C2×C4×C73#12012yes—
220C2×C4×C73#12012yes—
221C2×C4×C73C2×SD1612yes—
222C2×C4×C73C2×SD1612yes—
223C2×C4×C73#13512yes—
224C2×C4×C73#13712yes—
225C2×C4×C73C22×D412yes—
226C2×C4×C73C2×#4112yes—
227C2×C4×C73C2×#4111yescharacteristic
228C2×C4×C73C2×#4111yescharacteristic
229C2×Dic73—24no—
230C2×Dic73—24no—
231C2×Dic73—42no—
232C2×Dic73—42no—
233C2×Dic73—42no—
234C2×Dic73—42no—
235Dic146—24no—
236Dic146—24no—
237Dic146—48no—
238Dic146—42no—
239Dic146—42no—
240#10596831—24no—
241#10596831—42no—
242C22×C73—22no—
243C22×C73—22no—
244C22×C73—22no—
245C22×C73—41no—
246C22×C73—41no—
247C22×C73—41no—
248C22×C73—41no—
249C22×C73—42no—
250C22×C73—42no—
251C22×C73—42no—
252C22×C73—81no—
253C22×C73—81no—
254C22×C73—81no—
255C22×C73—81no—
256C22×C73C2×#12011yescharacteristic
257C22×C73#52511yescharacteristic
258C22×C73#53211yescharacteristic
259C22×C73C22×SD1611yescharacteristic
260C22×C73C2×#13511yescharacteristic
261C22×C73C2×#13711yescharacteristic
262C22×C73#57511yescharacteristic
263C4×C73—22no—
264C4×C73—22no—
265C4×C73—22no—
266C4×C73—42no—
267C4×C73—41no—
268C4×C73—81no—
269C4×C73—81no—
270C4×C73#44814yes—
271C4×C73#52512yes—
272C4×C73#53212yes—
273Dic73—44no—
274Dic73—44no—
275#23568—731nomaximal, Sylow
276C2×C73—42no—
277C2×C73—82no—
278C2×C73#238811yescharacteristic
279C2×C73C2×#44811yescharacteristic
280C2×C73#300211yescharacteristic
281C2×C73#310111yescharacteristic
282C2×C73#310711yescharacteristic
283C2×C73#316711yescharacteristic
284C2×C73#317211yescharacteristic
285C2×#533—731no—
286C2×C4×SD16—732no—
287C4×#121D7311yescharacteristic
288#2058—731no—
289#2043—731no—
290#2035—732no—
291#2023—731no—
292#2013—731no—
293#2000—732no—
294C4×#103—732no—
295C4×C22⋊C8—731no—
296C73#2356811yescharacteristic, complemented, Sylow
297C22×SD16—732no—
298#533—1468no—
299C2×C4⋊Q8—731no—
300C2×C4⋊Q8—731no—
301C2×#125—1461no—
302C2×C22⋊Q8—732no—
303C2×C22⋊Q8—1461no—
304C2×#121C2×D7311yescharacteristic
305C2×C4×Q8—732no—
306C2×C4×Q8—1461no—
307C2×C4×D4—21no—
308C2×C4×D4C2×D7311yescharacteristic
309C2×#117—1461no—
310C22×C42C2×D7311yescharacteristic
311C4×SD16—1468no—
312C2×#106—732no—
313C2×#103—732no—
314C2×#103—732no—
315C2×#103—732no—
316C2×#102—732no—
317C2×C22⋊C8—732no—
318C2×C4×C8—732no—
319#390C2×D7311yescharacteristic
320#389C2×D7311yescharacteristic
321#388—21no—
322#386—21no—
323#380C2×D7311yescharacteristic
324C4×C4⋊C4C2×D7311yescharacteristic
325C4×C22⋊C4—21no—
326#340—732no—
327#340—732no—
328#336—731no—
329#336—731no—
330C22×Q8—732no—
331C22×Q8—1461no—
332C22×D4—21no—
333C22×D4C22×D7311yescharacteristic
334C23×C4—21no—
335C23×C4C22⋊D7311yescharacteristic
336C23×C4C22⋊D7311yescharacteristic
337C23×C4C22×D7311yescharacteristic
338C2×SD16—1464no—
339C2×SD16—1464no—
340C2×SD16—1464no—
341C22×C8—732no—
342C22×C8—732no—
343C4⋊Q8—1464no—
344C4⋊Q8—1464no—
345#125—1468no—
346C22⋊Q8—1464no—
347C22⋊Q8—1464no—
348C22⋊Q8—1468no—
349#121—22no—
350#121—22no—
351C4×Q8—738no—
352C4×Q8—1468no—
353C4×Q8—1464no—
354C4×D4—24no—
355C4×D4—44no—
356#117—1468no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—732no—
359C2×C4⋊C4—1461no—
360C2×C4⋊C4—1461no—
361C2×C4⋊C4—1462no—
362C2×C4⋊C4—1462no—
363C2×C4⋊C4—1461no—
364C2×C4⋊C4C22×D7311yescharacteristic
365C2×C4⋊C4C22×D7311yescharacteristic
366C2×C4⋊C4C22×D7311yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1462no—
373C2×C42—24no—
374C2×C42—21no—
375C2×C42—1462no—
376C2×C42C22⋊D7311yescharacteristic
377C2×C42C22⋊D7311yescharacteristic
378C2×C42C22×D7311yescharacteristic
379C2×C42C22×D7311yescharacteristic
380#106—1464no—
381#103—1464no—
382#103—1464no—
383#103—1464no—
384#102—1464no—
385C22⋊C8—1464no—
386C4×C8—1464no—
387#95—21no—
388#95—21no—
389#95—21no—
390C24—21no—
391C24C2×C22⋊D7311yescharacteristic
392C2×Q8—2922no—
393C2×Q8—2922no—
394C2×Q8—734no—
395C2×Q8—1464no—
396C2×Q8—1462no—
397C2×Q8—1462no—
398C2×Q8—1462no—
399C2×Q8—1462no—
400C2×Q8—1462no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4#1192402812yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—1462no—
424C22×C4—1462no—
425C22×C4#1192401614yescomplemented
426C22×C4C2×C22⋊D7311yescharacteristic
427C22×C4C2×C22⋊D7311yescharacteristic
428C22×C4C2×C22⋊D7311yescharacteristic
429C22×C4C2×C22⋊D7311yescharacteristic
430C22×C4C2×C22⋊D7311yescharacteristic
431C22×C4C23×D7311yescharacteristic
432SD16—14616no—
433C2×C8—734no—
434C2×C8—1462no—
435C2×C8—1462no—
436C2×C8—1462no—
437C2×C8—1462no—
438C2×C8—1462no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—734no—
444C4⋊C4—1468no—
445C4⋊C4—1464no—
446C4⋊C4—1464no—
447C4⋊C4—1464no—
448C4⋊C4—1468no—
449C4⋊C4—1462no—
450C4⋊C4#1192402812yes—
451C22⋊C4—24no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—1468no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—42no—
461C42—1468no—
462C42D4×D7314yes—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23C2×#1192401612yes—
477C23C22×C22⋊D7311yescharacteristic
478Q8—2928no—
479Q8—2922no—
480Q8—2922no—
481Q8—1464no—
482Q8—1464no—
483D4—24no—
484D4—48no—
485D4—42no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—28no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—84no—
509C2×C4—2922no—
510C2×C4—2922no—
511C2×C4—2922no—
512C2×C4—2922no—
513C2×C4—1464no—
514C2×C4—1464no—
515C2×C4#1455655912yes—
516C2×C4#1455655912yes—
517C2×C4C2×#1192401612yes—
518C2×C4C2×#1192401612yes—
519C2×C4#1455661112yes—
520C2×C4#1455662112yes—
521C2×C4C2×D4×D7312yes—
522C2×C4#41×D7312yes—
523C2×C4C2×#1192402811yescharacteristic
524C2×C4C2×#1192402811yescharacteristic
525C8—2922no—
526C8—1464no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22C2×#1455655911yescharacteristic
542C22#1798909811yescharacteristic
543C22#1798913311yescharacteristic
544C22C22×#1192401611yescharacteristic
545C22C2×#1455661111yescharacteristic
546C22C2×#1455662111yescharacteristic
547C22#1798938511yescharacteristic
548C4—22no—
549C4—22no—
550C4—22no—
551C4—42no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4—2924no—
556C4—2924no—
557C4#1798872514yes—
558C4#1798909612yes—
559C4#1798913412yes—
560C2—42no—
561C2—82no—
562C2#2261314611yescharacteristic
563C2C2×#1798872511yescharacteristic
564C2#2261914711yescharacteristic
565C2#2261993911yescharacteristic
566C2#2262000211yescharacteristic
567C2#2262048611yescharacteristic
568C2#2262053811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC4×#121×C73C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenterC4×#121D73
hypocenterC73#23568
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#17989144C2semidirect
2. C2×C4×#11924016C2semidirect
3. #22611264C2semidirect
4. #22611177C2semidirect
5. #22611127C2semidirect
6. #22611067C2semidirect
7. #22611013C2semidirect
9. C4×#14556514C2semidirect
10. C4×C22⋊#10596831C2semidirect
60. C4×Dic146D4semidirect
156. C22×C4×C73SD16semidirect
296. C73#23568semidirect
425. C22×C4#11924016semidirect

Maximal subgroups

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

subgroupindex
C2×#179891442
C2×C4×#119240162
#226112642
#226111772
#226111272
#226110672
#226110132
#226109342
C4×#145565142
C4×C22⋊#105968312
C4×#121×C732
#2356873