Finite Group Databasebeta

← #10896409

Subgroups — #10896409

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2331C211yesmaximal, characteristic, complemented
2#73387C211yesmaximal, characteristic
3#71794C211yesmaximal, characteristic, complemented
4#71792C211yesmaximal, characteristic
5#69632C211yesmaximal, characteristic, complemented
6#69631C211yesmaximal, characteristic, complemented
7#69626C211yesmaximal, characteristic
8#69625C211yesmaximal, characteristic
9#69561C211yesmaximal, characteristic, complemented
10#69558C211yesmaximal, characteristic
11#69488C211yesmaximal, characteristic
12#69488C211yesmaximal, characteristic
13#69467C211yesmaximal, characteristic, complemented
14#69466C211yesmaximal, characteristic, complemented
15#69437C211yesmaximal, characteristic, complemented
16#29267—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#875C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C3×#391—21no—
22C3×#391—21no—
23C3×#389C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#387—21no—
26C3×#387—21no—
27C3×#382C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C3×#376C2211yescharacteristic
31C2×#810C2211yescharacteristic
32C2×#810C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#810C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4939C2211yescharacteristic
44#4935C2211yescharacteristic
45#4929C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48#4670—21no—
49#4670—21no—
50#4668—21no—
51#4668—21no—
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2331D311yescharacteristic, complemented
61#2326—31no—
62#2097—31no—
63#2092—31no—
64#2024—31no—
65#2023—31no—
66#1983—31no—
67#1983—31no—
68#1948—31no—
69#1947—31no—
70#1865—31no—
71#1864—31no—
72#1863—31no—
73#1862—31no—
74#1832—31no—
75C22×C3×Q8—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C3×C4⋊Q8—22no—
80C3×C4⋊Q8—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C22×C3⋊C8C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108#875—22no—
109#875Q814yescomplemented
110C4×D12—41no—
111C4×D12—41no—
112C4×Dic6—41no—
113C4×Dic6—41no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95D411yescharacteristic
121C3×#95D411yescharacteristic
122C3×#95D411yescharacteristic
123C3×#95D411yescharacteristic
124C3⋊#95—21no—
125C3⋊#95—21no—
126#810—22no—
127#810—22no—
128#810—41no—
129#810—41no—
130#809—22no—
131#809—22no—
132#809—41no—
133#809—41no—
134#808—41no—
135#808—41no—
136#807—41no—
137#807—41no—
138C4⋊(C3⋊C8)—22no—
139C4⋊(C3⋊C8)—22no—
140C2×C4⋊Q8C2×D311yescharacteristic
141C2×#124—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×#107—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
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—21no—
159#391—21no—
160#391—31no—
161#390—31no—
162#389C2×D311yescharacteristic
163#389C2×D311yescharacteristic
164#388—61no—
165#388—61no—
166#387—21no—
167#387—21no—
168#385—61no—
169#385—61no—
170#384—31no—
171#382C2×D311yescharacteristic
172C4×C4⋊C4C2×D311yescharacteristic
173C4×C4⋊C4C2×D311yescharacteristic
174#376C2×D311yescharacteristic
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#336—31no—
180#336—31no—
181#336—31no—
182#336—31no—
183C23×D3—21no—
184C2×C3×Q8—24no—
185C2×C3×Q8—81no—
186C2×C3×Q8—81no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12C2×Q812yes—
207C2×C4×D3—42no—
208C2×C4×D3—81no—
209C2×C4×D3—81no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6C2×Q812yes—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—81no—
224C3×C4⋊C4—81no—
225C3×C4⋊C4#4112yes—
226C3×C4⋊C4#4112yes—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—81no—
232C3×C42—81no—
233C3×C42C2×Q812yes—
234C3×C42#4112yes—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—81no—
239(C2×D3)⋊C4—81no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—81no—
245C4⋊Dic3—81no—
246Dic3⋊C4—42no—
247C4×Dic3—42no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
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—21no—
261C22×Q8—31no—
262C22×D4—31no—
263C23×C4—61no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—31no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270#124—34no—
271#124—62no—
272C4×Q8—121no—
273C4×Q8—121no—
274C4×D4—121no—
275C4×D4—121no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—61no—
296C2×C42C22×D311yescharacteristic
297C2×C42C22×D311yescharacteristic
298C2×C42C22×D311yescharacteristic
299#107—121no—
300#107—121no—
301#106—121no—
302#106—121no—
303C4⋊C8—62no—
304C4⋊C8—62no—
305#103—62no—
306#103—62no—
307#103—121no—
308#103—121no—
309#102—62no—
310#102—62no—
311#102—121no—
312#102—121no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—61no—
320#95—61no—
321#95#6711yescharacteristic
322#95#6711yescharacteristic
323#95#6711yescharacteristic
324#95#6711yescharacteristic
325C23×C3C22×D411yescharacteristic
326C22×D3—22no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C3×Q8—84no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4C22⋊Q812yes—
363C2×C3×C4C22⋊Q812yes—
364C2×C3×C4C22×Q811yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×Dic3—22no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379D12—22no—
380D12—22no—
381D12—41no—
382D12—41no—
383C4×D3—84no—
384Dic6—22no—
385Dic6—22no—
386Dic6—41no—
387Dic6—41no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C24—61no—
393C2×Q8—24no—
394C2×Q8—32no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×D4—32no—
403C2×D4—61no—
404C2×D4—61no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—122no—
416C22×C4—241no—
417C22×C4—241no—
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C2×#6711yescharacteristic
423C22×C4C2×#6711yescharacteristic
424C22×C4C23×D311yescharacteristic
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C4⋊C4—24no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—122no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4#23512yes—
454C4⋊C4#23512yes—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—122no—
458C22⋊C4—241no—
459C22⋊C4—241no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—81no—
465C42—81no—
466C42—122no—
467C42D3×Q812yes—
468C42#23912yes—
469C22×C3C2×C22⋊Q811yescharacteristic
470C22×C3#52511yescharacteristic
471C22×C3#53511yescharacteristic
472C22×C3#57611yescharacteristic
473C22×C3#57711yescharacteristic
474C22×C3#57711yescharacteristic
475C22×C3#57811yescharacteristic
476C2×D3—42no—
477C2×D3—42no—
478C2×D3—42no—
479C2×D3—81no—
480C2×D3—81no—
481C2×D3—81no—
482C2×D3—81no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—41no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4#55411yescharacteristic
494C3×C4#55411yescharacteristic
495C3×C4#55911yescharacteristic
496C3×C4#55911yescharacteristic
497Dic3—42no—
498Dic3—81no—
499Dic3—81no—
500C23—62no—
501C23—121no—
502C23—121no—
503C23—121no—
504C23—121no—
505C23—121no—
506C23—121no—
507C23C22×#6711yescharacteristic
508Q8—62no—
509Q8—62no—
510Q8—84no—
511Q8—121no—
512Q8—121no—
513D4—62no—
514D4—62no—
515D4—121no—
516D4—121no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—62no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—244no—
555C2×C4#94612yes—
556C2×C4#94612yes—
557C2×C4C2×#23511yescharacteristic
558C2×C4C2×#23511yescharacteristic
559C2×C4C2×D3×Q811yescharacteristic
560C2×C4C2×#23911yescharacteristic
561C2×C4#100911yescharacteristic
562C2×C4#100911yescharacteristic
563C8—121no—
564C8—121no—
565C8—121no—
566C8—121no—
567C2×C3#258011yescharacteristic
568C2×C3#308311yescharacteristic
569C2×C3#308411yescharacteristic
570C2×C3#311811yescharacteristic
571C2×C3#312111yescharacteristic
572C2×C3#322411yescharacteristic
573C2×C3#322511yescharacteristic
574D3—82no—
575C22—122no—
576C22—122no—
577C22—122no—
578C22—241no—
579C22—241no—
580C22—241no—
581C22—241no—
582C22#579111yescharacteristic
583C22C2×#94611yescharacteristic
584C22#583011yescharacteristic
585C22#583511yescharacteristic
586C22#583611yescharacteristic
587C22#583711yescharacteristic
588C22#583811yescharacteristic
589C4—22no—
590C4—22no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—42no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—122no—
600C4—241no—
601C4—241no—
602C4#557911yescharacteristic
603C4#557911yescharacteristic
604C4#558511yescharacteristic
605C4#558511yescharacteristic
606C3#2926711yescharacteristic, complemented, Sylow
607C2—242no—
608C2#8250111yescharacteristic
609C2#8254111yescharacteristic
610C2#8255211yescharacteristic
611C2#8346311yescharacteristic
612C2#8346411yescharacteristic
613C2#8347511yescharacteristic
614C2#8347611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2331C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2331D3
hypocenterC3#29267
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
1. C3×#2331C2semidirect
3. #71794C2semidirect
5. #69632C2semidirect
6. #69631C2semidirect
9. #69561C2semidirect
13. #69467C2semidirect
14. #69466C2semidirect
15. #69437C2semidirect
60. #2331D3semidirect
109. #875Q8semidirect
606. C3#29267semidirect

Maximal subgroups

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

subgroupindex
C3×#23312
#733872
#717942
#717922
#696322
#696312
#696262
#696252
#695612
#695582
#694882
#694672
#694662
#694372
#292673