Finite Group Databasebeta

← #34021053

Subgroups — #34021053

Order 18688.

1602 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
1#22626403C212yesmaximal
2#22625829C212yesmaximal, complemented
3#22625828C212yesmaximal, complemented
4#22625827C212yesmaximal, complemented
5#22625826C212yesmaximal, complemented
6#22625628C212yesmaximal, complemented
7C2×#17989465C212yesmaximal, complemented
8C2×#17989059C211yesmaximal, characteristic
9C22×C22⋊#64408C2214yes—
10#17989535—28no—
11#17989530C2214yes—
12#17989528C2214yes—
13#17989465C2214yes—
14#17989465C2214yes—
15#17989465C2214yes—
16#17989465C2214yes—
17C22×D4×D73C2211yescharacteristic
18#17989059—28no—
19C2×C4×D292C411yescharacteristic, complemented
20C2×D4×Dic73C411yescharacteristic, complemented
21C2×#14556553C411yescharacteristic, complemented
22C2×C4×C22⋊D73C411yescharacteristic, complemented
23C2×C4×C22⋊D73C411yescharacteristic, complemented
24C2×#14556551C411yescharacteristic, complemented
25C2×#14556551C411yescharacteristic, complemented
26C2×#14556549C2211yescharacteristic
27C2×#14556548C2211yescharacteristic
28C22×#11924005C2211yescharacteristic
29C22×#11924005C2211yescharacteristic
30C2×C22⋊C4×D73C2211yescharacteristic
31C2×C22⋊C4×D73C2211yescharacteristic
32C2×C4×D4×C73C411yescharacteristic, complemented
33C23×#64408—24no—
34C2×C22⋊#64408—24no—
35C2×C22⋊#64408—24no—
36C2×C22⋊#64408—24no—
37C2×C22⋊#64408—216no—
38C2×C22⋊#64408C2318yes—
39(C2×#64408)⋊C4—28no—
40C24×D73C2311yescharacteristic
41C24×D73C2311yescharacteristic
42C22×D292C2×C411yescharacteristic
43C22×C22⋊D73C2×C411yescharacteristic
44C22×C22⋊D73C2×C411yescharacteristic
45C2×D4×D73—28no—
46C2×D4×D73D418yes—
47C23×Dic73C2×C411yescharacteristic
48C23×Dic73C2×C411yescharacteristic, complemented
49C22×C4×D73C2×C411yescharacteristic
50C22×C4×D73C2×C411yescharacteristic
51C22×C4×D73C2×C411yescharacteristic, complemented
52C22×C4×D73C2×C411yescharacteristic, complemented
53C22×C4×D73C2311yescharacteristic
54C4×D292—24no—
55D4×Dic73—24no—
56#14556553—24no—
57C4×C22⋊D73—24no—
58C4×C22⋊D73—24no—
59#14556551—24no—
60#14556551—24no—
61#14556549—24no—
62#14556548—24no—
63C2×C4⋊Dic73C2×C411yescharacteristic, complemented
64C2×#11924007C2×C411yescharacteristic, complemented
65C2×#11924007C2×C411yescharacteristic, complemented
66C2×C22⋊Dic73C2×C411yescharacteristic
67C2×C22⋊Dic73C2×C411yescharacteristic, complemented
68C2×#11924005—24no—
69C2×#11924005—24no—
70C2×#11924005C2×C411yescharacteristic
71C2×#11924005C2×C411yescharacteristic
72C2×#11924005C2×C411yescharacteristic, complemented
73C2×#11924005C2×C411yescharacteristic, complemented
74C2×#11924005C2311yescharacteristic
75C2×#11924005C2311yescharacteristic
76C2×#11924005C2311yescharacteristic
77C2×#11924005C2311yescharacteristic
78C22⋊C4×D73—24no—
79C22⋊C4×D73—24no—
80C2×C4×Dic73C2×C411yescharacteristic, complemented
81C2×C4×Dic73C2×C411yescharacteristic, complemented
82C2×C4×Dic73C2×C411yescharacteristic, complemented
83C22×D4×C73C2×C411yescharacteristic
84C23×C4×C73C2×C411yescharacteristic
85C23×C4×C73C2×C411yescharacteristic, complemented
86C4×D4×C73—24no—
87C2×C4⋊C4×C73C2×C411yescharacteristic, complemented
88C2×C22⋊C4×C73C2×C411yescharacteristic
89C2×C22⋊C4×C73C2×C411yescharacteristic, complemented
90C2×C42×C73C2×C411yescharacteristic, complemented
91C22×#64408—28no—
92C22×#64408—48no—
93C22×#64408—84no—
94C22×#64408—84no—
95C22⋊#64408—216no—
96C22⋊#64408—48no—
97C22⋊#64408—84no—
98C22⋊#64408—84no—
99C23×D73—24no—
100C23×D73—81no—
101C23×D73—81no—
102C23×D73—81no—
103C23×D73—81no—
104C23×D73C22×C411yescharacteristic
105C23×D73C22×C411yescharacteristic
106C23×D73C2×D414yes—
107C23×D73C2×D414yes—
108C23×D73#4118yes—
109C23×D73C2411yescharacteristic
110C2×D292—24no—
111C2×D292C22⋊C414yes—
112C2×C22⋊D73—24no—
113C2×C22⋊D73—24no—
114C2×C22⋊D73C22⋊C414yes—
115C2×C22⋊D73C22⋊C414yes—
116D4×D73—216no—
117D4×D73—216no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—81no—
121C22×Dic73—81no—
122C22×Dic73C22×C411yescharacteristic
123C22×Dic73C22×C411yescharacteristic
124C22×Dic73C22×C411yescharacteristic
125C22×Dic73C22×C411yescharacteristic
126C22×Dic73C22×C411yescharacteristic
127C2×C4×D73—24no—
128C2×C4×D73—22no—
129C2×C4×D73—22no—
130C2×C4×D73—22no—
131C2×C4×D73—22no—
132C2×C4×D73—81no—
133C2×C4×D73—81no—
134C2×C4×D73—81no—
135C2×C4×D73—81no—
136C2×C4×D73C2×D414yes—
137C4⋊Dic73—22no—
138#11924007—22no—
139#11924007—22no—
140C22⋊Dic73—22no—
141C22⋊Dic73—22no—
142#11924005—22no—
143#11924005—22no—
144#11924005—22no—
145#11924005—22no—
146#11924005—22no—
147#11924005—22no—
148#11924005—22no—
149#11924005—22no—
150C4×Dic73—22no—
151C4×Dic73—22no—
152C4×Dic73—22no—
153C24×C73C22×C411yescharacteristic
154C24×C73C22×C411yescharacteristic
155C2×D4×C73—24no—
156C2×D4×C73C22⋊C414yes—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—81no—
160C22×C4×C73—81no—
161C22×C4×C73C22×C411yescharacteristic
162C22×C4×C73C22×C411yescharacteristic
163C22×C4×C73C22×C411yescharacteristic
164C22×C4×C73C22×C411yescharacteristic
165C22×C4×C73C22×C411yescharacteristic
166C4⋊C4×C73—22no—
167C22⋊C4×C73—22no—
168C22⋊C4×C73—22no—
169C42×C73—22no—
170C23×C73—22no—
171C23×C73—41no—
172C23×C73—41no—
173C23×C73—41no—
174C23×C73—41no—
175C23×C73C2×C22⋊C412yes—
176C23×C73C2×C22⋊C412yes—
177C23×C73#11714yes—
178C23×C73C23×C411yescharacteristic
179C22×D73—22no—
180C22×D73—28no—
181C22×D73—41no—
182C22×D73—41no—
183C22×D73—41no—
184C22×D73—41no—
185C22×D73—41no—
186C22×D73—41no—
187C22×D73—44no—
188C22×D73—44no—
189C22×D73—44no—
190C22×D73—44no—
191C22×D73—88no—
192C22×D73—88no—
193C22×D73C2×C22⋊C412yes—
194C22×D73C2×C22⋊C412yes—
195C22×D73#11714yes—
196C22×D73#12318yes—
197C22×D73#12318yes—
198C22×D73C22×D412yes—
199C22×D73C2×#4114yes—
200C2×#64408—48no—
201C2×#64408—48no—
202C2×#64408—48no—
203C2×#64408—816no—
204D4×C73—28no—
205C2×C4×C73—22no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—41no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—82no—
215C2×C4×C73—82no—
216C2×C4×C73C2×C22⋊C412yes—
217C22⋊D73—28no—
218C22⋊D73—28no—
219C2×Dic73—22no—
220C2×Dic73—21no—
221C2×Dic73—21no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—41no—
225C2×Dic73—41no—
226C2×Dic73—41no—
227C2×Dic73—41no—
228C2×Dic73—82no—
229C2×Dic73—82no—
230C2×Dic73C2×C22⋊C412yes—
231D292—28no—
232C4×D73—28no—
233C4×D73—82no—
234C4×D73—82no—
235C4×D73—82no—
236C4×D73—82no—
237C22×C73—21no—
238C22×C73—21no—
239C22×C73—22no—
240C22×C73—22no—
241C22×C73—22no—
242C22×C73—22no—
243C22×C73—22no—
244C22×C73—44no—
245C22×C73—44no—
246C22×C73#38114yes—
247C22×C73C22×C22⋊C411yescharacteristic
248C22×C73C2×#11712yes—
249C2×D73—22no—
250C2×D73—22no—
251C2×D73—22no—
252C2×D73—22no—
253C2×D73—22no—
254C2×D73—22no—
255C2×D73—22no—
256C2×D73—48no—
257C2×D73—44no—
258C2×D73—44no—
259C2×D73—48no—
260C2×D73—44no—
261C2×D73—88no—
262C2×D73#38114yes—
263C2×D73C2×#12314yes—
264#64408—88no—
265C4×C73—22no—
266C4×C73—41no—
267C4×C73—41no—
268C4×C73—41no—
269C4×C73—41no—
270Dic73—22no—
271Dic73—41no—
272Dic73—41no—
273Dic73—41no—
274Dic73—41no—
275#20689—731nomaximal, Sylow
276C2×C73—22no—
277C2×C73—24no—
278C2×C73—42no—
279C2×C73C2×#38111yescharacteristic
280C2×C73#288311yescharacteristic
281C2×C73#288311yescharacteristic
282D73—24no—
283D73—24no—
284D73—42no—
285C2×#518—731no—
286#2308—732no—
287#1820—732no—
288#1820—732no—
289#1819—732no—
290#1819—732no—
291C2×#342—732no—
292C2×#342—732no—
293C73#2068911yescharacteristic, complemented, Sylow
294C23×D4—731no—
295#518—1468no—
296C2×C4×D4—731no—
297C2×C4×D4—731no—
298C2×C4×D4—731no—
299C2×C4×D4—731no—
300C2×C4×D4—731no—
301C2×C4×D4—731no—
302C2×C4×D4—731no—
303C2×C4×D4#6440811yescharacteristic, complemented
304C2×#117—731no—
305C2×#117—731no—
306C22×C22⋊C4—731no—
307C22×C22⋊C4—731no—
308C22×C22⋊C4—731no—
309C22×C22⋊C4—731no—
310C22×C22⋊C4—734no—
311#386—1468no—
312#381—734no—
313#379—734no—
314#342—734no—
315#342—734no—
316#342—734no—
317#342—734no—
318C25—731no—
319C25—731no—
320C22×D4—731no—
321C22×D4—731no—
322C22×D4—738no—
323C22×D4—731no—
324C22×D4—1468no—
325C22×D4C2×#6440811yescharacteristic
326C23×C4—731no—
327C23×C4—731no—
328C23×C4—731no—
329C23×C4—731no—
330C23×C4—731no—
331C23×C4—731no—
332C23×C4—731no—
333C23×C4—1464no—
334C23×C4C2×#6440811yescharacteristic
335C23×C4C2×#6440811yescharacteristic, complemented
336C4×D4—24no—
337C4×D4—1464no—
338C4×D4—1464no—
339C4×D4—1464no—
340C4×D4—1464no—
341C4×D4—1464no—
342C4×D4—1464no—
343C4×D4—1464no—
344#117—1464no—
345#117—1464no—
346C2×C4⋊C4—731no—
347C2×C4⋊C4—731no—
348C2×C4⋊C4—731no—
349C2×C4⋊C4C2×#6440811yescharacteristic, complemented
350C2×C22⋊C4—731no—
351C2×C22⋊C4—731no—
352C2×C22⋊C4—731no—
353C2×C22⋊C4—731no—
354C2×C22⋊C4—731no—
355C2×C22⋊C4—731no—
356C2×C22⋊C4—731no—
357C2×C22⋊C4—731no—
358C2×C22⋊C4—731no—
359C2×C22⋊C4—731no—
360C2×C22⋊C4—738no—
361C2×C22⋊C4—1464no—
362C2×C22⋊C4—1464no—
363C2×C22⋊C4—1464no—
364C2×C22⋊C4—14616no—
365C2×C22⋊C4—1464no—
366C2×C22⋊C4—1464no—
367C2×C22⋊C4—1464no—
368C2×C22⋊C4—1464no—
369C2×C22⋊C4C2×#6440811yescharacteristic
370C2×C22⋊C4C2×#6440811yescharacteristic, complemented
371C2×C42—731no—
372C2×C42—731no—
373C2×C42—731no—
374C2×C42C2×#6440811yescharacteristic, complemented
375#95—1468no—
376C24—5841no—
377C24—5841no—
378C24—5841no—
379C24—5841no—
380C24—731no—
381C24—731no—
382C24—734no—
383C24—731no—
384C24—734no—
385C24—738no—
386C24—1464no—
387C24C22×#6440811yescharacteristic
388C24C22×#6440811yescharacteristic
389C2×D4—24no—
390C2×D4—734no—
391C2×D4—734no—
392C2×D4—734no—
393C2×D4—1464no—
394C2×D4—14616no—
395C2×D4—14616no—
396C2×D4—1464no—
397C2×D4—1464no—
398C2×D4C22⋊#6440814yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—2928no—
404C22×C4—5841no—
405C22×C4—5841no—
406C22×C4—5841no—
407C22×C4—5841no—
408C22×C4—5844no—
409C22×C4—5844no—
410C22×C4—5841no—
411C22×C4—5841no—
412C22×C4—731no—
413C22×C4—731no—
414C22×C4—731no—
415C22×C4—731no—
416C22×C4—731no—
417C22×C4—734no—
418C22×C4—1462no—
419C22×C4—1462no—
420C22×C4—1462no—
421C22×C4—1462no—
422C22×C4—1464no—
423C22×C4—1468no—
424C22×C4—1462no—
425C22×C4—1462no—
426C22×C4C22×#6440811yescharacteristic
427C22×C4C22×#6440811yescharacteristic
428C22×C4C22×#6440811yescharacteristic
429C22×C4C22×#6440811yescharacteristic
430C22×C4C22×#6440811yescharacteristic
431C4⋊C4—22no—
432C4⋊C4—1462no—
433C4⋊C4—1462no—
434C4⋊C4—1462no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—2928no—
438C22⋊C4—5844no—
439C22⋊C4—5844no—
440C22⋊C4—1462no—
441C22⋊C4—1462no—
442C22⋊C4—1462no—
443C22⋊C4—1462no—
444C22⋊C4—1462no—
445C22⋊C4—1462no—
446C22⋊C4—1462no—
447C22⋊C4—1462no—
448C22⋊C4—14616no—
449C22⋊C4—1462no—
450C22⋊C4—1462no—
451C42—22no—
452C42—1462no—
453C42—1462no—
454C42—1462no—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—2921no—
461C23—2921no—
462C23—2924no—
463C23—2921no—
464C23—2921no—
465C23—2924no—
466C23—2921no—
467C23—2921no—
468C23—2924no—
469C23—2924no—
470C23—5848no—
471C23—5848no—
472C23—734no—
473C23—734no—
474C23—738no—
475C23—738no—
476C23—732no—
477C23—732no—
478C23—732no—
479C23—1468no—
480C23—1462no—
481C23C2×C22⋊#6440812yes—
482C23C2×C22⋊#6440812yes—
483C23#1455667014yes—
484C23C23×#6440811yescharacteristic
485D4—28no—
486D4—1468no—
487D4—1468no—
488D4—1468no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—2921no—
501C2×C4—2921no—
502C2×C4—2921no—
503C2×C4—2921no—
504C2×C4—2928no—
505C2×C4—2928no—
506C2×C4—2928no—
507C2×C4—5842no—
508C2×C4—5842no—
509C2×C4—5842no—
510C2×C4—5842no—
511C2×C4—5842no—
512C2×C4—5842no—
513C2×C4—58416no—
514C2×C4—732no—
515C2×C4—1468no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4—1462no—
521C2×C4C2×C22⋊#6440812yes—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—44no—
530C22—44no—
531C22—2924no—
532C22—2928no—
533C22—2924no—
534C22—2924no—
535C22—2928no—
536C22—5848no—
537C22—734no—
538C22—734no—
539C22—1462no—
540C22—1462no—
541C22—1462no—
542C22—1462no—
543C22—1462no—
544C22—1462no—
545C22—1462no—
546C22#1798953014yes—
547C22C22×C22⋊#6440811yescharacteristic
548C22C2×#1455667012yes—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—2921no—
555C4—2921no—
556C4—2921no—
557C4—2921no—
558C4—5848no—
559C4—1462no—
560C2—22no—
561C2—24no—
562C2—42no—
563C2—2922no—
564C2—1464no—
565C2—1464no—
566C2C2×#1798953011yescharacteristic
567C2#2262653011yescharacteristic
568C2#2262653111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×C4×D4×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC22×D4×D73C22
cosocleC23×D73C24
hypercenterC2×C4×D4#64408
hypocenterC73#20689
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #22625829C2semidirect
3. #22625828C2semidirect
4. #22625827C2semidirect
5. #22625826C2semidirect
6. #22625628C2semidirect
7. C2×#17989465C2semidirect
19. C2×C4×D292C4semidirect
20. C2×D4×Dic73C4semidirect
21. C2×#14556553C4semidirect
22. C2×C4×C22⋊D73C4semidirect
23. C2×C4×C22⋊D73C4semidirect
24. C2×#14556551C4semidirect
25. C2×#14556551C4semidirect
32. C2×C4×D4×C73C4semidirect
48. C23×Dic73C2×C4semidirect
51. C22×C4×D73C2×C4semidirect
52. C22×C4×D73C2×C4semidirect
63. C2×C4⋊Dic73C2×C4semidirect
64. C2×#11924007C2×C4semidirect
65. C2×#11924007C2×C4semidirect
67. C2×C22⋊Dic73C2×C4semidirect
72. C2×#11924005C2×C4semidirect
73. C2×#11924005C2×C4semidirect
80. C2×C4×Dic73C2×C4semidirect
81. C2×C4×Dic73C2×C4semidirect
82. C2×C4×Dic73C2×C4semidirect
85. C23×C4×C73C2×C4semidirect
87. C2×C4⋊C4×C73C2×C4semidirect
89. C2×C22⋊C4×C73C2×C4semidirect
90. C2×C42×C73C2×C4semidirect
293. C73#20689semidirect
303. C2×C4×D4#64408semidirect
335. C23×C4C2×#64408semidirect
349. C2×C4⋊C4C2×#64408semidirect
370. C2×C22⋊C4C2×#64408semidirect
374. C2×C42C2×#64408semidirect

Maximal subgroups

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

subgroupindex
#226264032
#226258292
#226258282
#226258272
#226258262
#226256282
C2×#179894652
C2×#179890592
#2068973