Finite Group Databasebeta

← #10896437

Subgroups — #10896437

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74321C211yesmaximal, characteristic
2#73626C211yesmaximal, characteristic, complemented
3C3×#1948C211yesmaximal, characteristic, complemented
4#70590C211yesmaximal, characteristic, complemented
5#70588C211yesmaximal, characteristic, complemented
6#70587C211yesmaximal, characteristic
7#70584C211yesmaximal, characteristic
8#70458C211yesmaximal, characteristic
9#70444C211yesmaximal, characteristic, complemented
10#69568C211yesmaximal, characteristic, complemented
11#69488C211yesmaximal, characteristic
12#68911C211yesmaximal, characteristic, complemented
13#68907C211yesmaximal, characteristic, complemented
14#68899C211yesmaximal, characteristic
15#68895C211yesmaximal, characteristic
16#29269—31nomaximal, Sylow
17C2×#893C2211yescharacteristic
18C2×#890C2211yescharacteristic
19C2×#887C2211yescharacteristic
20C2×#875C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C3×#384C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30#4990—21no—
31#4990—21no—
32#4989—21no—
33#4989—21no—
34#4988C2211yescharacteristic
35#4984C2211yescharacteristic
36#4975C2211yescharacteristic
37#4973C2211yescharacteristic
38#4972C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4933—21no—
47#4932C2211yescharacteristic
48#4928—21no—
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4654—21no—
51#4647—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2330—31no—
61#2326—31no—
62#2096—31no—
63#2094—31no—
64#2026—31no—
65#2025—31no—
66#1983—31no—
67#1983—31no—
68#1948—31no—
69#1948D311yescharacteristic, complemented
70#1865—31no—
71#1865—31no—
72#1864—31no—
73#1864—31no—
74#1832—31no—
75C22×D12C2311yescharacteristic
76C22×C4×D3—21no—
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108#893—41no—
109#893—41no—
110#890—22no—
111#890—22no—
112#887—41no—
113#887—41no—
114#875—22no—
115#875Q814yes—
116C3×#106—41no—
117C3×#106—41no—
118C3×C4⋊C8—22no—
119C3×#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129#821—22no—
130#821—41no—
131#816—22no—
132#816—41no—
133#810—22no—
134#810—41no—
135#809—22no—
136#809—41no—
137#808—41no—
138#808—41no—
139C4⋊(C3⋊C8)—22no—
140C2×#125—31no—
141C2×#124—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×#106—31no—
145C2×#106—31no—
146C2×#106C2×D311yescharacteristic
147C2×#106C2×D311yescharacteristic
148C2×C4⋊C8—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158#391—31no—
159#390—31no—
160#389—31no—
161#389—31no—
162#389—61no—
163#389—61no—
164#388—61no—
165#388—61no—
166#387—61no—
167#387—61no—
168#385—61no—
169#385—61no—
170#384—31no—
171#384C2×D311yescharacteristic
172C4×C4⋊C4—31no—
173C4×C4⋊C4C2×D311yescharacteristic
174#376—31no—
175#340—31no—
176#340—31no—
177#340C2×D311yescharacteristic
178#340C2×D311yescharacteristic
179#336—31no—
180#336—31no—
181#336—31no—
182#336—31no—
183C23×D3—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3C2×D411yescharacteristic
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12C2×Q812yes—
204C2×C4×D3—42no—
205C2×C4×D3—81no—
206C2×C4×D3—81no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C2×Q812yes—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—81no—
227C3×C4⋊C4—81no—
228C3×C4⋊C4#4112yes—
229C3×C42—81no—
230C3×C42C2×Q812yes—
231C3×C42#4112yes—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—24no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—81no—
236(C2×D3)⋊C4—81no—
237C4⋊Dic3—24no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—81no—
242C4⋊Dic3#4112yes—
243Dic3⋊C4—24no—
244Dic3⋊C4—42no—
245Dic3⋊C4—81no—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—42no—
253C4×Dic3—81no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C22×Q8—31no—
261C22×D4—31no—
262C23×C4—61no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#125—62no—
268#125—62no—
269#124—34no—
270#124—62no—
271C4×Q8—121no—
272C4×Q8—121no—
273C4×D4—121no—
274C4×D4—121no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C42—21no—
296C2×C42—31no—
297C2×C42—31no—
298C2×C42—61no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301#106—41no—
302#106—41no—
303#106—121no—
304#106—121no—
305C4⋊C8—22no—
306C4⋊C8—62no—
307#103—62no—
308#103—62no—
309#103—121no—
310#103—121no—
311#102—62no—
312#102—62no—
313#102—121no—
314#102—121no—
315#95—21no—
316#95—31no—
317#95—31no—
318#95—31no—
319#95—31no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325C23×C3C22×D411yescharacteristic
326C22×D3—22no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4C22⋊Q812yes—
350C2×C3×C4C22⋊Q812yes—
351C2×C3×C4C22×Q811yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4H(5,𝔽2)11yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×Dic3—22no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378D12—22no—
379D12—22no—
380D12—41no—
381D12—41no—
382C4×D3—84no—
383Dic6—22no—
384Dic6—22no—
385Dic6—41no—
386Dic6—41no—
387C3×C8—41no—
388C3×C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C24—61no—
392C2×Q8—32no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×D4—32no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—61no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—122no—
416C22×C4—241no—
417C22×C4—241no—
418C22×C4D3×D411yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4C23×D311yescharacteristic
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—32no—
441C4⋊C4—64no—
442C4⋊C4—64no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—122no—
449C4⋊C4—241no—
450C4⋊C4—241no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4#23912yes—
454C22⋊C4—64no—
455C22⋊C4—64no—
456C22⋊C4—122no—
457C22⋊C4—241no—
458C22⋊C4—241no—
459C42—61no—
460C42—61no—
461C42—61no—
462C42—61no—
463C42—81no—
464C42—122no—
465C42—241no—
466C42#23512yes—
467C42D3×Q812yes—
468C22×C3C2×C22⋊Q811yescharacteristic
469C22×C3#52511yescharacteristic
470C22×C3#53511yescharacteristic
471C22×C3#57711yescharacteristic
472C22×C3#57711yescharacteristic
473C22×C3#57711yescharacteristic
474C22×C3#57711yescharacteristic
475C2×D3—42no—
476C2×D3—42no—
477C2×D3—42no—
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—81no—
481C2×D3—81no—
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—42no—
485C3×C4—81no—
486C3×C4—81no—
487C3×C4—81no—
488C3×C4#55411yescharacteristic
489C3×C4#55411yescharacteristic
490C3×C4#55911yescharacteristic
491C3×C4#55911yescharacteristic
492Dic3—42no—
493Dic3—41no—
494Dic3—41no—
495Dic3—42no—
496Dic3—41no—
497Dic3—41no—
498Dic3—81no—
499C23—62no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23—121no—
504C23—121no—
505C23—121no—
506C23C2×D3×D411yescharacteristic
507Q8—62no—
508Q8—62no—
509Q8—121no—
510Q8—121no—
511D4—62no—
512D4—62no—
513D4—121no—
514D4—121no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—62no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—62no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—244no—
553C2×C4#89612yes—
554C2×C4#89612yes—
555C2×C4C2×#23511yescharacteristic
556C2×C4#100411yescharacteristic
557C2×C4C2×D3×Q811yescharacteristic
558C2×C4C2×#23911yescharacteristic
559C2×C4D3×#4111yescharacteristic
560C2×C4#100911yescharacteristic
561C8—41no—
562C8—41no—
563C8—121no—
564C8—121no—
565C2×C3#258011yescharacteristic
566C2×C3#308411yescharacteristic
567C2×C3#308411yescharacteristic
568C2×C3#312511yescharacteristic
569C2×C3#312511yescharacteristic
570C2×C3#322111yescharacteristic
571C2×C3#322211yescharacteristic
572D3—82no—
573C22—122no—
574C22—122no—
575C22—122no—
576C22—241no—
577C22—241no—
578C22—241no—
579C22—241no—
580C22C2×#89611yescharacteristic
581C22#560711yescharacteristic
582C22#562711yescharacteristic
583C22#576211yescharacteristic
584C22#576211yescharacteristic
585C22#577511yescharacteristic
586C22#577811yescharacteristic
587C4—22no—
588C4—22no—
589C4—42no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—121no—
594C4—122no—
595C4—121no—
596C4—121no—
597C4—122no—
598C4—121no—
599C4—241no—
600C4#555911yescharacteristic
601C4#558511yescharacteristic
602C4#569011yescharacteristic
603C4#573011yescharacteristic
604C3#2926911yescharacteristic, complemented, Sylow
605C2—242no—
606C2#7607511yescharacteristic
607C2#7913011yescharacteristic
608C2#7924411yescharacteristic
609C2#8039111yescharacteristic
610C2#8043611yescharacteristic
611C2#8100111yescharacteristic
612C2#8104911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1948C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1948D3
hypocenterC3#29269
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. #73626C2semidirect
3. C3×#1948C2semidirect
4. #70590C2semidirect
5. #70588C2semidirect
9. #70444C2semidirect
10. #69568C2semidirect
12. #68911C2semidirect
13. #68907C2semidirect
69. #1948D3semidirect
604. C3#29269semidirect

Maximal subgroups

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

subgroupindex
#743212
#736262
C3×#19482
#705902
#705882
#705872
#705842
#704582
#704442
#695682
#694882
#689112
#689072
#688992
#688952
#292693