Finite Group Databasebeta

← #33090631

Subgroups — #33090631

Order 18688.

1338 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989212C211yesmaximal, characteristic
2#22612544C211yesmaximal, characteristic
3#22612283C211yesmaximal, characteristic
4#22610917C211yesmaximal, characteristic, complemented
5#22610916C211yesmaximal, characteristic, complemented
6#22610915C211yesmaximal, characteristic, complemented
7#22610914C214yesmaximal, complemented
8#342×D73C214yesmaximal
9C73×#1996C211yesmaximal, characteristic, complemented
10#17989212—22no—
11#17989212—22no—
12#17989212—22no—
13#17989212—22no—
14C2×#14556571C412yescomplemented
15C2×#14556570C2211yescharacteristic
16C2×#14556569C412yescomplemented
17C2×#14556569C412yescomplemented
18C2×#14556568C2212yes—
19C2×#14556567C412yescomplemented
20C22×#11924005C2211yescharacteristic
21C2×C22⋊C4×D73C2212yes—
22C2×C22⋊C4×D73C2212yes—
23C2×C22⋊C4×D73C2212yes—
24#17988461—22no—
25#17988458—22no—
26#17988447C2211yescharacteristic
27#17988445C2212yes—
28C22⋊C4×Dic73C2212yes—
29C4×#11924005—22no—
30C4×C22⋊Dic73C2211yescharacteristic
31#17988434—22no—
32#17988354C2214yes—
33#17988353C2214yes—
34#17988353C2214yes—
35C2×C22⋊Q8×C73C2211yescharacteristic
36#380×C73C2211yescharacteristic
37C4×C22⋊C4×C73C2211yescharacteristic
38#342×C73C2214yes—
39C24×D73C2311yescharacteristic
40C22×Dic146C2×C412yescomplemented
41C22×Dic146C2×C411yescharacteristic, complemented
42C23×Dic73C2311yescharacteristic
43C22×C4×D73—22no—
44C22×C4×D73—22no—
45C22×C4×D73C2×C412yes—
46C22×C4×D73C2×C412yes—
47#14556571—24no—
48#14556571—24no—
49#14556570—21no—
50#14556570—21no—
51#14556570—21no—
52#14556570—21no—
53#14556569—24no—
54#14556569—24no—
55#14556569—24no—
56#14556569—24no—
57#14556568—24no—
58#14556568—24no—
59#14556567—24no—
60#14556567—24no—
61C2×C4⋊Dic73—21no—
62C2×C4⋊Dic73C2×C412yescomplemented
63C2×C4⋊Dic73C2×C411yescharacteristic, complemented
64C2×#11924007—22no—
65C2×#11924007C2×C412yescomplemented
66C2×#11924007C2×C412yescomplemented
67C2×#11924007C2×C412yescomplemented
68C2×C22⋊Dic73C2312yes—
69C2×C22⋊Dic73C2312yes—
70C2×C22⋊Dic73C2312yes—
71C2×#11924005—22no—
72C2×#11924005—22no—
73C2×#11924005C2×C412yes—
74C2×#11924005C2×C412yes—
75C2×#11924005C2×C412yes—
76C2×#11924005C2×C412yes—
77C2×#11924005D412yes—
78C2×#11924005D412yes—
79C2×#11924005D412yes—
80C2×#11924005D412yes—
81C22⋊C4×D73—24no—
82C22⋊C4×D73—24no—
83C22⋊C4×D73—24no—
84C22⋊C4×D73—28no—
85C22⋊C4×D73Q818yescomplemented
86C2×C4×Dic73—22no—
87C2×C4×Dic73—21no—
88#14556498—22no—
89#14556498—22no—
90#14556498—21no—
91#14556498—21no—
92C22×Q8×C73C2×C411yescharacteristic, complemented
93C23×C4×C73C2311yescharacteristic
94C22⋊Q8×C73—21no—
95C22⋊Q8×C73—21no—
96C22⋊Q8×C73—21no—
97C22⋊Q8×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73C2×C411yescharacteristic, complemented
100C2×C4⋊C4×C73C2×C412yescomplemented
101C2×C22⋊C4×C73C2312yes—
102C2×C22⋊C4×C73C2312yes—
103C2×C22⋊C4×C73C2312yes—
104C2×C42×C73—21no—
105#95×C73—21no—
106#95×C73—21no—
107C23×D73—81no—
108C23×D73—81no—
109C23×D73C22×C412yes—
110C23×D73C2×D412yes—
111C23×D73C2×D412yes—
112C23×D73C2×Q814yes—
113C23×D73#4114yes—
114C2×Dic146—21no—
115C2×Dic146—21no—
116C2×Dic146—24no—
117C2×Dic146—82no—
118C2×Dic146—81no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—81no—
122C22×Dic73C22×C412yes—
123C22×Dic73C22×C412yes—
124C22×Dic73C22×C412yes—
125C22×Dic73C2×D412yes—
126C22×Dic73C2×D412yes—
127C2×C4×D73—24no—
128C2×C4×D73—24no—
129C2×C4×D73—28no—
130C2×C4×D73—44no—
131C2×C4×D73—82no—
132C2×C4×D73—82no—
133C2×C4×D73—82no—
134C2×C4×D73—82no—
135C2×C4×D73—82no—
136C2×C4×D73—82no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—24no—
140C4⋊Dic73—81no—
141#11924007—22no—
142#11924007—22no—
143#11924007—24no—
144#11924007—24no—
145#11924007—82no—
146C22⋊Dic73—22no—
147C22⋊Dic73—22no—
148C22⋊Dic73—22no—
149C22⋊Dic73—24no—
150C22⋊Dic73C2×Q814yes—
151#11924005—24no—
152#11924005—24no—
153#11924005—28no—
154#11924005—44no—
155#11924005C22⋊C414yes—
156#11924005C22⋊C414yes—
157#11924005C22⋊C414yes—
158#11924005C22⋊C414yes—
159C4×Dic73—82no—
160C4×Dic73—81no—
161C24×C73C2411yescharacteristic
162C2×Q8×C73—21no—
163C2×Q8×C73—21no—
164C2×Q8×C73—81no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—81no—
168C22×C4×C73C22×C412yes—
169C22×C4×C73C22×C412yes—
170C22×C4×C73C22×C411yescharacteristic
171C22×C4×C73C22×C411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C4⋊C4×C73—21no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—22no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—81no—
181C22⋊C4×C73—22no—
182C22⋊C4×C73—22no—
183C22⋊C4×C73—22no—
184C22⋊C4×C73—24no—
185C22⋊C4×C73C2×Q814yes—
186C42×C73—81no—
187C23×C73—41no—
188C23×C73—41no—
189C23×C73C23×C411yescharacteristic
190C23×C73C22×D411yescharacteristic
191C23×C73C22×D411yescharacteristic
192C23×C73C22×Q812yes—
193C23×C73C2×#4112yes—
194C22×D73—42no—
195C22×D73—42no—
196C22×D73—42no—
197C22×D73—42no—
198C22×D73—42no—
199C22×D73—42no—
200C22×D73—84no—
201C22×D73—84no—
202C22×D73C2×C22⋊C414yes—
203C22×D73C4×Q818yes—
204C22×D73C22⋊Q814yes—
205C22×D73C22⋊Q814yes—
206C22×D73#12414yes—
207C22×D73C4⋊Q814yes—
208Q8×C73—81no—
209Q8×C73—81no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—24no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—42no—
220C2×C4×C73—42no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—42no—
224C2×C4×C73—42no—
225C2×C4×C73—42no—
226C2×C4×C73C2×C22⋊C411yescharacteristic
227C2×C4×C73C2×C22⋊C411yescharacteristic
228C2×C4×C73C2×C22⋊C411yescharacteristic
229C2×C4×C73C2×C22⋊C411yescharacteristic
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—24no—
233C2×Dic73—42no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—42no—
237C2×Dic73—42no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73C2×C22⋊C414yes—
244C4×D73—84no—
245C4×D73—84no—
246C4×D73—88no—
247C4×D73—88no—
248Dic146—84no—
249Dic146—81no—
250Dic146—81no—
251C22×C73—21no—
252C22×C73—21no—
253C22×C73—21no—
254C22×C73—21no—
255C22×C73—21no—
256C22×C73—21no—
257C22×C73—42no—
258C22×C73—42no—
259C22×C73C22×C22⋊C411yescharacteristic
260C22×C73C2×C4×Q812yes—
261C22×C73C2×C22⋊Q812yes—
262C22×C73C2×#12411yescharacteristic
263C22×C73C2×C4⋊Q811yescharacteristic
264C2×D73—24no—
265C2×D73—24no—
266C2×D73—44no—
267C2×D73—44no—
268C2×D73—48no—
269C2×D73—44no—
270C2×D73—44no—
271C2×D73#39118yes—
272C4×C73—21no—
273C4×C73—21no—
274C4×C73—21no—
275C4×C73—21no—
276C4×C73—44no—
277C4×C73—42no—
278C4×C73—42no—
279C4×C73—82no—
280Dic73—24no—
281Dic73—44no—
282Dic73—42no—
283Dic73—42no—
284Dic73—82no—
285#21333—731nomaximal, Sylow
286C2×C73—22no—
287C2×C73—21no—
288C2×C73—22no—
289C2×C73—21no—
290C2×C73C2×#39111yescharacteristic
291C2×C73#288311yescharacteristic
292C2×C73#288311yescharacteristic
293D73—28no—
294C2×#543—731no—
295#2315—731no—
296#2273—731no—
297#1996—731no—
298#1996—731no—
299#1996—734no—
300#1996—731no—
301#1996D7311yescharacteristic, complemented
302C2×#342—734no—
303C73#2133311yescharacteristic, complemented, Sylow
304#543—1462no—
305#543—1462no—
306#543—1462no—
307#543—1462no—
308C2×C22⋊Q8—732no—
309C2×C22⋊Q8—731no—
310C2×C22⋊Q8—732no—
311C2×C22⋊Q8—732no—
312C2×C22⋊Q8—732no—
313C2×C22⋊Q8—732no—
314C2×C22⋊Q8C2×D7311yescharacteristic
315C22×C22⋊C4—731no—
316C22×C22⋊C4—732no—
317C22×C22⋊C4—732no—
318C22×C22⋊C4—732no—
319#385—1462no—
320#385—1462no—
321#380—731no—
322#380—732no—
323#380C2×D7311yescharacteristic
324C4×C22⋊C4—731no—
325C4×C22⋊C4—732no—
326C4×C22⋊C4—1462no—
327C4×C22⋊C4—1462no—
328C4×C22⋊C4C2×D7311yescharacteristic
329#342—734no—
330#342—734no—
331#342—734no—
332#342C2×D7314yes—
333C25—731no—
334C22×Q8—732no—
335C22×Q8—731no—
336C22×Q8C4×D7311yescharacteristic, complemented
337C23×C4—731no—
338C23×C4—732no—
339C23×C4—732no—
340C23×C4—1462no—
341C23×C4—1462no—
342C23×C4C22×D7311yescharacteristic
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346C22⋊Q8—21no—
347C22⋊Q8—1464no—
348C22⋊Q8—1464no—
349C22⋊Q8—1464no—
350C22⋊Q8—1464no—
351C22⋊Q8—1464no—
352C22⋊Q8—1464no—
353C22⋊Q8—1464no—
354C22⋊Q8—1464no—
355C22⋊Q8—1464no—
356C22⋊Q8—1464no—
357C22⋊Q8—1461no—
358C22⋊Q8—1461no—
359C22⋊Q8—1461no—
360C22⋊Q8—1461no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—732no—
363C2×C4⋊C4—732no—
364C2×C4⋊C4—732no—
365C2×C4⋊C4—732no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—1462no—
368C2×C4⋊C4—1461no—
369C2×C4⋊C4C4×D7312yescomplemented
370C2×C4⋊C4C4×D7311yescharacteristic, complemented
371C2×C22⋊C4—732no—
372C2×C22⋊C4—732no—
373C2×C22⋊C4—732no—
374C2×C22⋊C4—732no—
375C2×C22⋊C4—732no—
376C2×C22⋊C4—732no—
377C2×C22⋊C4—732no—
378C2×C22⋊C4—738no—
379C2×C22⋊C4—732no—
380C2×C22⋊C4—732no—
381C2×C22⋊C4—732no—
382C2×C22⋊C4—732no—
383C2×C22⋊C4—1468no—
384C2×C22⋊C4—1464no—
385C2×C22⋊C4—1464no—
386C2×C22⋊C4—1464no—
387C2×C22⋊C4—1462no—
388C2×C22⋊C4—1462no—
389C2×C22⋊C4C22×D7312yes—
390C2×C22⋊C4C22×D7312yes—
391C2×C22⋊C4C22×D7312yes—
392C2×C42—21no—
393C2×C42—1462no—
394C2×C42—1461no—
395#95—21no—
396#95—21no—
397#95—1462no—
398#95—1462no—
399#95—1461no—
400#95—1461no—
401C24—5841no—
402C24—5841no—
403C24—732no—
404C24—732no—
405C24—732no—
406C24—734no—
407C24—734no—
408C24C23×D7311yescharacteristic
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—81no—
412C2×Q8—5841no—
413C2×Q8—5842no—
414C2×Q8—1464no—
415C2×Q8—1461no—
416C2×Q8—1461no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—81no—
420C22×C4—2924no—
421C22×C4—5842no—
422C22×C4—5842no—
423C22×C4—5842no—
424C22×C4—5842no—
425C22×C4—5842no—
426C22×C4—5842no—
427C22×C4—5841no—
428C22×C4—732no—
429C22×C4—732no—
430C22×C4—732no—
431C22×C4—732no—
432C22×C4—732no—
433C22×C4—1468no—
434C22×C4—1464no—
435C22×C4—1464no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4C2×C4×D7312yes—
439C22×C4C2×C4×D7312yes—
440C22×C4C2×C4×D7311yescharacteristic
441C22×C4C2×C4×D7311yescharacteristic
442C22×C4D4×D7311yescharacteristic
443C22×C4D4×D7311yescharacteristic
444C22×C4D4×D7311yescharacteristic
445C22×C4D4×D7311yescharacteristic
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—81no—
451C4⋊C4—5842no—
452C4⋊C4—5841no—
453C4⋊C4—1464no—
454C4⋊C4—1464no—
455C4⋊C4—1464no—
456C4⋊C4—1462no—
457C4⋊C4—1462no—
458C4⋊C4—1461no—
459C4⋊C4—1461no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—24no—
464C22⋊C4—2924no—
465C22⋊C4—734no—
466C22⋊C4—734no—
467C22⋊C4—734no—
468C22⋊C4—734no—
469C22⋊C4—734no—
470C22⋊C4—1468no—
471C22⋊C4—1464no—
472C22⋊C4—1464no—
473C22⋊C4—1464no—
474C22⋊C4—1462no—
475C22⋊C4—1462no—
476C22⋊C4—1462no—
477C22⋊C4Q8×D7314yes—
478C42—81no—
479C42—5841no—
480C42—5842no—
481C23—41no—
482C23—41no—
483C23—2922no—
484C23—2922no—
485C23—2922no—
486C23—2922no—
487C23—2922no—
488C23—2922no—
489C23—5844no—
490C23—5844no—
491C23—734no—
492C23—738no—
493C23—734no—
494C23—734no—
495C23—734no—
496C23—734no—
497C23C22×C4×D7311yescharacteristic
498C23C2×D4×D7311yescharacteristic
499C23C2×D4×D7311yescharacteristic
500C23C2×Q8×D7312yes—
501C23#41×D7312yes—
502Q8—81no—
503Q8—81no—
504Q8—5841no—
505Q8—5841no—
506Q8—5844no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—2922no—
524C2×C4—2922no—
525C2×C4—2922no—
526C2×C4—2922no—
527C2×C4—2922no—
528C2×C4—2922no—
529C2×C4—2922no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—2922no—
533C2×C4—5848no—
534C2×C4—5844no—
535C2×C4—5844no—
536C2×C4—5848no—
537C2×C4—734no—
538C2×C4—1462no—
539C2×C4—1462no—
540C2×C4—1464no—
541C2×C4C22⋊C4×D7311yescharacteristic
542C2×C4C22⋊C4×D7311yescharacteristic
543C2×C4C22⋊C4×D7311yescharacteristic
544C2×C4C22⋊C4×D7311yescharacteristic
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—42no—
552C22—42no—
553C22—2924no—
554C22—2924no—
555C22—2924no—
556C22—2924no—
557C22—2928no—
558C22—738no—
559C22—1464no—
560C22—1464no—
561C22C2×C22⋊C4×D7311yescharacteristic
562C22C4×Q8×D7312yes—
563C22C22⋊Q8×D7312yes—
564C22#124×D7311yescharacteristic
565C22C4⋊Q8×D7311yescharacteristic
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—82no—
574C4—2924no—
575C4—2922no—
576C4—2922no—
577C4—5842no—
578C4—1464no—
579C2—22no—
580C2—21no—
581C2—21no—
582C2—22no—
583C2—1468no—
584C2#391×D7311yescharacteristic
585C2#2261838911yescharacteristic
586C2#2261839011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D73
commutatorC23×C73C23×C4
FrattiniC24C23×D73
FittingC73×#1996C2
socleC22×C73C22×C22⋊C4
prime radicalC24×D73C23
cosocleC24×C73C24
hypercenter#1996D73
hypocenterC73#21333
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
4. #22610917C2semidirect
5. #22610916C2semidirect
6. #22610915C2semidirect
7. #22610914C2semidirect
9. C73×#1996C2semidirect
14. C2×#14556571C4semidirect
16. C2×#14556569C4semidirect
17. C2×#14556569C4semidirect
19. C2×#14556567C4semidirect
40. C22×Dic146C2×C4semidirect
41. C22×Dic146C2×C4semidirect
62. C2×C4⋊Dic73C2×C4semidirect
63. C2×C4⋊Dic73C2×C4semidirect
65. C2×#11924007C2×C4semidirect
66. C2×#11924007C2×C4semidirect
67. C2×#11924007C2×C4semidirect
85. C22⋊C4×D73Q8semidirect
92. C22×Q8×C73C2×C4semidirect
99. C2×C4⋊C4×C73C2×C4semidirect
100. C2×C4⋊C4×C73C2×C4semidirect
301. #1996D73semidirect
303. C73#21333semidirect
336. C22×Q8C4×D73semidirect
369. C2×C4⋊C4C4×D73semidirect
370. C2×C4⋊C4C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179892122
#226125442
#226122832
#226109172
#226109162
#226109152
#226109142
#342×D732
C73×#19962
#2133373