Finite Group Databasebeta

← #34022729

Subgroups — #34022729

Order 18688.

1162 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626408C211yesmaximal, characteristic, complemented
2#22626369C211yesmaximal, characteristic, complemented
3#22626121C212yesmaximal, complemented
4#22626072C212yesmaximal
5#22625955C212yesmaximal, complemented
6#22625954C212yesmaximal, complemented
7#22625678C212yesmaximal, complemented
8C4×#14556650C212yesmaximal, complemented
9C2×#17989178C211yesmaximal, characteristic
10C2×#14556670C2212yes—
11C2×#14556650C2214yes—
12C2×#14556649C2214yes—
13#17989536—24no—
14#17989530C2212yes—
15#17989529C2212yes—
16C4⋊C4×#64408—24no—
17C4×C22⋊#64408C2212yes—
18C4×C22⋊#64408C2212yes—
19C4×C22⋊#64408C2212yes—
20#17989465C2214yes—
21#17989465C2214yes—
22#17989178—24no—
23#17989178—24no—
24C2×#14556586C411yescharacteristic, complemented
25C2×#14556583C411yescharacteristic, complemented
26C2×#14556582C411yescharacteristic, complemented
27C2×#14556576C2211yescharacteristic
28C2×#14556575C411yescharacteristic, complemented
29C2×#14556573C411yescharacteristic, complemented
30C2×#14556573C2211yescharacteristic
31C2×#14556571C2211yescharacteristic
32C2×#14556569C411yescharacteristic, complemented
33C2×#14556568C411yescharacteristic, complemented
34C2×#14556564C2211yescharacteristic
35C2×C4×D292C2211yescharacteristic
36C2×#14556551C2211yescharacteristic
37C2×C4⋊C4×D73C2211yescharacteristic
38C2×#126×C73C411yescharacteristic, complemented
39#14556670—28no—
40C2×C4⋊#64408—22no—
41C2×C4⋊#64408—22no—
42C2×C22⋊#64408C2314yes—
43C2×C22⋊#64408C2314yes—
44C2×C4×#64408—22no—
45C2×C4×#64408—22no—
46C2×C4×#64408—22no—
47C2×C4×#64408—22no—
48#14556650—28no—
49#14556649—28no—
50(C2×#64408)⋊C4—22no—
51(C2×#64408)⋊C4—22no—
52(C2×#64408)⋊C4—22no—
53(C2×#64408)⋊C4—22no—
54(C2×#64408)⋊C4—24no—
55C22×Dic146C2×C411yescharacteristic
56C22×D292C2311yescharacteristic
57C22×C22⋊D73C2×C411yescharacteristic
58C22×C22⋊D73C2311yescharacteristic
59C23×Dic73C2×C411yescharacteristic
60C22×C4×D73C2×C411yescharacteristic
61C22×C4×D73C2311yescharacteristic
62C22×C4×D73C2311yescharacteristic
63C22×C4×D73C2311yescharacteristic
64#14556586—24no—
65#14556583—24no—
66#14556582—24no—
67#14556576—22no—
68#14556576—22no—
69#14556575—24no—
70#14556573—22no—
71#14556573—22no—
72#14556573—24no—
73#14556571—22no—
74#14556571—22no—
75#14556569—24no—
76#14556568—24no—
77#14556564—22no—
78#14556564—22no—
79C4×D292—22no—
80C4×D292—22no—
81#14556551—22no—
82#14556551—22no—
83C2×C4⋊Dic73C2×C411yescharacteristic
84C2×C4⋊Dic73C2×C411yescharacteristic
85C2×C4⋊Dic73C2×C411yescharacteristic
86C2×C4⋊Dic73C2×C411yescharacteristic
87C2×C4⋊Dic73C2×C411yescharacteristic
88C2×#11924007C2×C411yescharacteristic
89C2×#11924007C2×C411yescharacteristic
90C2×#11924007C2×C411yescharacteristic
91C2×#11924007C2×C411yescharacteristic
92C2×#11924007C2×C411yescharacteristic
93C2×#11924007C2×C411yescharacteristic
94C4⋊C4×D73D414yes—
95C4⋊C4×D73D414yes—
96C2×C22⋊Dic73C2×C411yescharacteristic
97C2×#11924005C2×C411yescharacteristic
98C2×#11924005C2×C411yescharacteristic
99C2×#11924005C2×C411yescharacteristic
100C2×#11924005C2×C411yescharacteristic
101C2×#11924005C2311yescharacteristic
102C2×#11924005C2311yescharacteristic
103C2×C4×Dic73C2×C411yescharacteristic
104#126×C73—24no—
105C2×C4⋊C4×C73C2×C411yescharacteristic
106C2×C4⋊C4×C73C2×C411yescharacteristic
107C2×C4⋊C4×C73C2×C411yescharacteristic
108C2×C22⋊C4×C73C2×C411yescharacteristic
109C2×C22⋊C4×C73C2×C411yescharacteristic
110C2×C22⋊C4×C73C2×C411yescharacteristic
111C2×C42×C73C2×C411yescharacteristic
112C22×#64408—24no—
113C22×#64408—24no—
114C4⋊#64408—28no—
115C4⋊#64408—82no—
116C22⋊#64408—24no—
117C22⋊#64408—24no—
118C22⋊#64408—24no—
119C22⋊#64408#4118yes—
120C4×#64408—28no—
121C4×#64408—82no—
122C4×#64408—82no—
123C4×#64408—82no—
124C23×D73C22×C411yescharacteristic
125C23×D73C2411yescharacteristic
126C2×Dic146—22no—
127C2×Dic146—81no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292—24no—
131C2×C22⋊D73—21no—
132C2×C22⋊D73—21no—
133C2×C22⋊D73—24no—
134C2×C22⋊D73—22no—
135C2×C22⋊D73—81no—
136C22×Dic73—22no—
137C22×Dic73—81no—
138C22×Dic73C22×C411yescharacteristic
139C22×Dic73C22×C411yescharacteristic
140C22×Dic73C22×C411yescharacteristic
141C22×Dic73C22×C411yescharacteristic
142C22×Dic73C22×C411yescharacteristic
143C22×Dic73C22×C411yescharacteristic
144C2×C4×D73—24no—
145C2×C4×D73—22no—
146C2×C4×D73—81no—
147C2×C4×D73—81no—
148C2×C4×D73—81no—
149C2×C4×D73C2×D412yes—
150C2×C4×D73C2×D412yes—
151C2×C4×D73C2×D412yes—
152C2×C4×D73C2×D412yes—
153C2×C4×D73C2×D412yes—
154C2×C4×D73C2×D412yes—
155C4⋊Dic73—22no—
156C4⋊Dic73—22no—
157C4⋊Dic73—22no—
158C4⋊Dic73—22no—
159C4⋊Dic73C22⋊C414yes—
160#11924007—22no—
161#11924007—22no—
162#11924007—22no—
163#11924007—22no—
164#11924007C22⋊C414yes—
165#11924007C22⋊C414yes—
166C22⋊Dic73—22no—
167#11924005—21no—
168#11924005—21no—
169#11924005—21no—
170#11924005—21no—
171#11924005—22no—
172#11924005—22no—
173#11924005—22no—
174#11924005—22no—
175C4×Dic73—22no—
176C24×C73C22×C411yescharacteristic
177C22×C4×C73C22×C411yescharacteristic
178C22×C4×C73C22×C411yescharacteristic
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C22×C411yescharacteristic
181C22×C4×C73C22×C411yescharacteristic
182C22×C4×C73C22×C411yescharacteristic
183C4⋊C4×C73—22no—
184C4⋊C4×C73—22no—
185C4⋊C4×C73C22⋊C414yes—
186C22⋊C4×C73—22no—
187C22⋊C4×C73—22no—
188C22⋊C4×C73—22no—
189C42×C73—22no—
190C23×C73—21no—
191C23×C73—22no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73C23×C411yescharacteristic
195C22×D73—21no—
196C22×D73—22no—
197C22×D73—41no—
198C22×D73—41no—
199C22×D73—41no—
200C22×D73—41no—
201C22×D73C22×D411yescharacteristic
202C22×D73C22×D411yescharacteristic
203C22×D73C2×#4114yes—
204C2×#64408—28no—
205C2×#64408—44no—
206C2×#64408—44no—
207C2×#64408—44no—
208C2×#64408—44no—
209C2×#64408—44no—
210C2×C4×C73—22no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—22no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73C2×C22⋊C412yes—
220C2×C4×C73C2×C22⋊C412yes—
221C2×C4×C73C2×C22⋊C412yes—
222C22⋊D73—42no—
223C22⋊D73—42no—
224C22⋊D73—82no—
225C2×Dic73—22no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—22no—
230C2×Dic73—41no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—82no—
235C2×Dic73C2×C22⋊C412yes—
236C2×Dic73C2×C22⋊C412yes—
237C2×Dic73C2×C22⋊C412yes—
238D292—42no—
239D292—42no—
240C4×D73—22no—
241C4×D73—22no—
242C4×D73—24no—
243C4×D73—42no—
244C4×D73—42no—
245C4×D73—42no—
246C4×D73—42no—
247C4×D73—82no—
248Dic146—82no—
249C22×C73—21no—
250C22×C73—21no—
251C22×C73—41no—
252C22×C73—41no—
253C22×C73—41no—
254C22×C73—82no—
255C22×C73C22×C22⋊C411yescharacteristic
256C22×C73C4×#4112yes—
257C2×D73—21no—
258C2×D73—21no—
259C2×D73—21no—
260C2×D73—21no—
261C2×D73—44no—
262C2×D73—41no—
263C2×D73—41no—
264C2×D73—41no—
265C2×D73—82no—
266C2×D73#52512yes—
267C2×D73#53212yes—
268#64408—48no—
269#64408—84no—
270C4×C73—22no—
271C4×C73—22no—
272C4×C73—22no—
273C4×C73—41no—
274C4×C73—41no—
275C4×C73—41no—
276Dic73—22no—
277Dic73—22no—
278Dic73—22no—
279Dic73—41no—
280Dic73—41no—
281Dic73—41no—
282#21423—731nomaximal, Sylow
283C2×C73—22no—
284C2×C73—81no—
285C2×C73#229711yescharacteristic
286C2×C73#288311yescharacteristic
287C2×C73#288411yescharacteristic
288D73—22no—
289D73—22no—
290D73—81no—
291C2×#539—731no—
292#2313—731no—
293#2274—731no—
294#1996—732no—
295#1964—732no—
296#1899—732no—
297#1899—732no—
298C4×#99—732no—
299C4×#99—732no—
300C73#2142311yescharacteristic, complemented, Sylow
301#539—1464no—
302#539—1464no—
303C2×#126—731no—
304C2×#126#6440811yescharacteristic, complemented
305C2×#125—731no—
306C2×#125—731no—
307C2×#123—731no—
308C2×#123—731no—
309C2×#123—731no—
310C2×#123—731no—
311C2×C22⋊Q8—731no—
312C2×C22⋊Q8—731no—
313C2×C22⋊Q8—731no—
314C2×#121—731no—
315C2×C4×D4—731no—
316C2×C4×D4—731no—
317C2×#117—732no—
318C22×C4⋊C4—731no—
319C2×#99—734no—
320C2×#99—734no—
321#387—1464no—
322#381—732no—
323#380—732no—
324C4×C4⋊C4—1464no—
325C4×C22⋊C4—732no—
326C4×C22⋊C4—732no—
327C4×C22⋊C4—732no—
328#342—734no—
329#342—734no—
330C22×Q8—731no—
331C22×D4—731no—
332C22×D4—731no—
333C22×D4—731no—
334C23×C4—731no—
335C23×C4—731no—
336C23×C4—731no—
337C23×C4—731no—
338C23×C4—731no—
339#126—24no—
340#126—1464no—
341#125—1464no—
342#125—1464no—
343#123—1464no—
344#123—1464no—
345#123—1462no—
346#123—1462no—
347#123—1462no—
348#123—1462no—
349C22⋊Q8—1464no—
350C22⋊Q8—1462no—
351C22⋊Q8—1462no—
352C22⋊Q8—1464no—
353#121—1462no—
354#121—1462no—
355C4×D4—1462no—
356C4×D4—1462no—
357C4×D4—1462no—
358C4×D4—1462no—
359#117—1468no—
360C2×C4⋊C4—731no—
361C2×C4⋊C4—731no—
362C2×C4⋊C4—731no—
363C2×C4⋊C4—734no—
364C2×C4⋊C4—734no—
365C2×C4⋊C4—731no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—1462no—
374C2×C4⋊C4—1462no—
375C2×C4⋊C4C2×#6440811yescharacteristic
376C2×C4⋊C4C2×#6440811yescharacteristic
377C2×C4⋊C4C2×#6440811yescharacteristic
378C2×C22⋊C4—734no—
379C2×C22⋊C4—731no—
380C2×C22⋊C4—731no—
381C2×C22⋊C4—731no—
382C2×C22⋊C4—731no—
383C2×C22⋊C4—731no—
384C2×C22⋊C4—731no—
385C2×C22⋊C4—731no—
386C2×C22⋊C4—734no—
387C2×C22⋊C4C2×#6440811yescharacteristic
388C2×C22⋊C4C2×#6440811yescharacteristic
389C2×C22⋊C4C2×#6440811yescharacteristic
390C2×C42—731no—
391C2×C42—1462no—
392C2×C42—1462no—
393C2×C42—1462no—
394C2×C42—1462no—
395C2×C42C2×#6440811yescharacteristic
396#99—1468no—
397#99—1468no—
398#95—1462no—
399#95—1462no—
400#95—1462no—
401#95—1462no—
402#95—1464no—
403C24—731no—
404C24—731no—
405C24C22×#6440811yescharacteristic
406C2×Q8—5841no—
407C2×Q8—1462no—
408C2×D4—5841no—
409C2×D4—1464no—
410C2×D4—1461no—
411C2×D4—1461no—
412C2×D4—1462no—
413C2×D4—1464no—
414C2×D4—1461no—
415C2×D4—1461no—
416C22×C4—5841no—
417C22×C4—5841no—
418C22×C4—5841no—
419C22×C4—5841no—
420C22×C4—731no—
421C22×C4—731no—
422C22×C4—732no—
423C22×C4—732no—
424C22×C4—732no—
425C22×C4—732no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—732no—
430C22×C4—732no—
431C22×C4—731no—
432C22×C4—1464no—
433C22×C4—1464no—
434C22×C4—1464no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4C22×#6440811yescharacteristic
438C22×C4C22×#6440811yescharacteristic
439C22×C4C22×#6440811yescharacteristic
440C22×C4C22×#6440811yescharacteristic
441C22×C4C22×#6440811yescharacteristic
442C22×C4C22×#6440811yescharacteristic
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—5842no—
446C4⋊C4—734no—
447C4⋊C4—734no—
448C4⋊C4—734no—
449C4⋊C4—1462no—
450C4⋊C4—1462no—
451C4⋊C4—1462no—
452C4⋊C4—1462no—
453C4⋊C4—1462no—
454C4⋊C4—1462no—
455C4⋊C4—1468no—
456C4⋊C4—1462no—
457C4⋊C4—1462no—
458C4⋊C4C22⋊#6440814yes—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—738no—
463C22⋊C4—1462no—
464C22⋊C4—1462no—
465C22⋊C4—1461no—
466C22⋊C4—1461no—
467C22⋊C4—1461no—
468C22⋊C4—1461no—
469C22⋊C4—1464no—
470C22⋊C4—1464no—
471C22⋊C4—1462no—
472C22⋊C4—1464no—
473C22⋊C4—1462no—
474C22⋊C4—1462no—
475C42—22no—
476C42—5842no—
477C42—5842no—
478C42—5842no—
479C42—1462no—
480C42—1468no—
481C23—21no—
482C23—22no—
483C23—41no—
484C23—41no—
485C23—2921no—
486C23—2921no—
487C23—2921no—
488C23—2921no—
489C23—734no—
490C23—731no—
491C23—731no—
492C23—1461no—
493C23—1462no—
494C23C23×#6440811yescharacteristic
495Q8—5842no—
496D4—2922no—
497D4—2922no—
498D4—2922no—
499D4—2922no—
500D4—5842no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—2922no—
511C2×C4—2922no—
512C2×C4—2922no—
513C2×C4—2922no—
514C2×C4—2924no—
515C2×C4—2924no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2924no—
519C2×C4—2924no—
520C2×C4—2924no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—5842no—
524C2×C4—5842no—
525C2×C4—732no—
526C2×C4—732no—
527C2×C4—732no—
528C2×C4—1462no—
529C2×C4—1462no—
530C2×C4—1464no—
531C2×C4—1468no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1462no—
536C2×C4—1462no—
537C2×C4C2×C22⋊#6440812yes—
538C2×C4C2×C22⋊#6440812yes—
539C2×C4C2×C22⋊#6440812yes—
540C22—21no—
541C22—21no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—82no—
546C22—2924no—
547C22—2921no—
548C22—2921no—
549C22—2921no—
550C22—5842no—
551C22—732no—
552C22—732no—
553C22—1461no—
554C22—1461no—
555C22—1461no—
556C22—1461no—
557C22C22×C22⋊#6440811yescharacteristic
558C22#41×#6440812yes—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—2928no—
566C4—2921no—
567C4—2921no—
568C4—2921no—
569C4—5844no—
570C4—1462no—
571C4—1462no—
572C4—1462no—
573C2—22no—
574C2—81no—
575C2—5841no—
576C2—1462no—
577C2—1462no—
578C2#2262639211yescharacteristic
579C2#2262653111yescharacteristic
580C2#2262653311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×#126×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC2×#14556564C22
cosocleC23×D73C24
hypercenterC2×#126#64408
hypocenterC73#21423
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #22626408C2semidirect
2. #22626369C2semidirect
3. #22626121C2semidirect
5. #22625955C2semidirect
6. #22625954C2semidirect
7. #22625678C2semidirect
8. C4×#14556650C2semidirect
24. C2×#14556586C4semidirect
25. C2×#14556583C4semidirect
26. C2×#14556582C4semidirect
28. C2×#14556575C4semidirect
29. C2×#14556573C4semidirect
32. C2×#14556569C4semidirect
33. C2×#14556568C4semidirect
38. C2×#126×C73C4semidirect
300. C73#21423semidirect
304. C2×#126#64408semidirect

Maximal subgroups

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

subgroupindex
#226264082
#226263692
#226261212
#226260722
#226259552
#226259542
#226256782
C4×#145566502
C2×#179891782
#2142373