Finite Group Databasebeta

← #10877344

Subgroups — #10877344

Order 768.

732 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
1#82235C211yesmaximal, characteristic
2#82233C211yesmaximal, characteristic
3C3×#1798C211yesmaximal, characteristic
4#70486C211yesmaximal, characteristic, complemented
5#70485C211yesmaximal, characteristic, complemented
6#70455C211yesmaximal, characteristic, complemented
7#70453C211yesmaximal, characteristic, complemented
8#69931C211yesmaximal, characteristic
9#69930C211yesmaximal, characteristic
10#69929C211yesmaximal, characteristic
11#69927C211yesmaximal, characteristic
12#68693C211yesmaximal, characteristic, complemented
13#68691C211yesmaximal, characteristic, complemented
14#68644C211yesmaximal, characteristic, complemented
15#68642C211yesmaximal, characteristic, complemented
16#27690—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C22⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26#5212C2211yescharacteristic
27C2×#818C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35#4984—21no—
36#4984—21no—
37#4978—21no—
38#4976—21no—
39#4950—21no—
40#4950—21no—
41#4945—21no—
42#4944—21no—
43#4936C2211yescharacteristic
44#4936C2211yescharacteristic
45#4935C2211yescharacteristic
46#4935C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48#4929C2211yescharacteristic
49#4928C2211yescharacteristic
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2323—31no—
61#2317—31no—
62#2101—31no—
63#2099—31no—
64#2097—31no—
65#2095—31no—
66#1949—31no—
67#1947—31no—
68#1873—31no—
69#1872—31no—
70#1854—31no—
71#1852—31no—
72#1805—31no—
73#1803—31no—
74#1798D311yescharacteristic
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#117—22no—
83C3×#117—22no—
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93#926—41no—
94#926—41no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—41no—
101C2×C4⋊Dic3—41no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109#924—22no—
110#924Q814yes—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×C4⋊C8—22no—
120C3×C4⋊C8—22no—
121C3×C22⋊C8—22no—
122C3×C22⋊C8—22no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#818—41no—
133#818—41no—
134#818—41no—
135#817—41no—
136#816—41no—
137#816—41no—
138#816—41no—
139#816—41no—
140C2×C22⋊Q8—31no—
141C2×C4×Q8—31no—
142C2×#117—31no—
143C2×#117C2×D311yescharacteristic
144C22×C4⋊C4—31no—
145C2×#107—31no—
146C2×#107—31no—
147C2×#107—31no—
148C2×#106—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×C4⋊C8C2×D311yescharacteristic
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×C22⋊C8C2×D311yescharacteristic
156C2×C22⋊C8C2×D311yescharacteristic
157#391—31no—
158#389—31no—
159#389—61no—
160#389—61no—
161#389—61no—
162#388—61no—
163#388—61no—
164#387—61no—
165#385—61no—
166#385—61no—
167#384—31no—
168#384—31no—
169#382—31no—
170#382—31no—
171#380—31no—
172C4×C4⋊C4—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336C2×D311yescharacteristic
182#336C2×D311yescharacteristic
183C24×C3C2×D411yescharacteristic
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×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×Q812yes—
194C22×C3×C4#4112yes—
195C22×C3×C4C2411yescharacteristic
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—42no—
205C22×Dic3—81no—
206C22×Dic3—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—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42#4112yes—
232C3×C42#4112yes—
233#217—24no—
234#217—42no—
235#217—81no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—42no—
250C4⋊Dic3—81no—
251C4⋊Dic3—81no—
252C4⋊Dic3—81no—
253C4⋊Dic3—81no—
254C4⋊Dic3C2×Q812yes—
255Dic3⋊C4—24no—
256Dic3⋊C4—42no—
257Dic3⋊C4—81no—
258C4×Dic3—42no—
259C4×Dic3—42no—
260C4×Dic3—81no—
261C22×Q8—31no—
262C23×C4—61no—
263C23×C4C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
268C22⋊Q8—34no—
269C22⋊Q8—62no—
270C4×Q8—62no—
271C4×Q8—62no—
272#117—22no—
273#117—22no—
274#117—121no—
275#117—121no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—121no—
292C2×C4⋊C4—121no—
293C2×C4⋊C4D1211yescharacteristic
294C2×C4⋊C4D1211yescharacteristic
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4D1211yescharacteristic
299C2×C22⋊C4D1211yescharacteristic
300C2×C42—61no—
301C2×C42—61no—
302C2×C42—61no—
303C2×C42C22×D311yescharacteristic
304C2×C42C22×D311yescharacteristic
305#107—121no—
306#107—121no—
307#107—121no—
308#106—121no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311#103—121no—
312#103—121no—
313#103—121no—
314#103—121no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3C22⋊Q812yes—
333C23×C3C22×D411yescharacteristic
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—22no—
343C2×C3×C4—22no—
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×C4C22⋊Q812yes—
351C2×C3×C4#12312yes—
352C2×C3×C4#12312yes—
353C2×C3×C4C22×Q811yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4H(5,𝔽2)11yescharacteristic
358C2×C3×C4H(5,𝔽2)11yescharacteristic
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—84no—
384Dic6—22no—
385Dic6—22no—
386Dic6—41no—
387Dic6—41no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C24C2×D1211yescharacteristic
393C2×Q8—32no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—122no—
412C22×C4—241no—
413C22×C4—241no—
414C22×C4C2×D1211yescharacteristic
415C22×C4C2×D1211yescharacteristic
416C22×C4C2×D1211yescharacteristic
417C22×C4C2×D1211yescharacteristic
418C22×C4C2×D1211yescharacteristic
419C22×C4#23712yes—
420C22×C4D3×Q812yes—
421C22×C4C23×D311yescharacteristic
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—32no—
437C4⋊C4—64no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C4⋊C4—241no—
454C4⋊C4—241no—
455C4⋊C4—241no—
456C4⋊C4—241no—
457C4⋊C4—241no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—64no—
461C22⋊C4—122no—
462C22⋊C4—241no—
463C42—21no—
464C42—21no—
465C42—122no—
466C42—122no—
467C42—241no—
468C42#23712yes—
469C42#23712yes—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3C2×C22⋊Q811yescharacteristic
478C22×C3C2×#12311yescharacteristic
479C22×C3#53511yescharacteristic
480C22×C3#57611yescharacteristic
481C22×C3#57711yescharacteristic
482C22×C3#57811yescharacteristic
483C22×C3#57811yescharacteristic
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4#55311yescharacteristic
490C3×C4#55311yescharacteristic
491C3×C4#55411yescharacteristic
492C3×C4#55411yescharacteristic
493Dic3—42no—
494Dic3—42no—
495Dic3—81no—
496Dic3—81no—
497Dic3—81no—
498Dic3—81no—
499Dic3—81no—
500Dic3—81no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23#89712yes—
508C23C22×D1211yescharacteristic
509Q8—62no—
510Q8—62no—
511Q8—121no—
512Q8—121no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—244no—
554C2×C4#88612yes—
555C2×C4#88612yes—
556C2×C4#89712yes—
557C2×C4C2×#23711yescharacteristic
558C2×C4C2×#23711yescharacteristic
559C2×C4C2×#23711yescharacteristic
560C2×C4C2×D3×Q811yescharacteristic
561C2×C4#100911yescharacteristic
562C2×C4#100911yescharacteristic
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C2×C3—42no—
568C2×C3#248711yescharacteristic
569C2×C3#308211yescharacteristic
570C2×C3#308311yescharacteristic
571C2×C3#309711yescharacteristic
572C2×C3#309811yescharacteristic
573C2×C3#322311yescharacteristic
574C2×C3#322411yescharacteristic
575C22—22no—
576C22—22no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22C2×#88611yescharacteristic
583C22C2×#89711yescharacteristic
584C22#552511yescharacteristic
585C22#575111yescharacteristic
586C22#575211yescharacteristic
587C22#575311yescharacteristic
588C22#575311yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—122no—
595C4—122no—
596C4—241no—
597C4—241no—
598C4—241no—
599C4—241no—
600C4—241no—
601C4—241no—
602C4#561811yescharacteristic
603C4#561811yescharacteristic
604C4#563511yescharacteristic
605C4#563511yescharacteristic
606C3#2769011yescharacteristic, complemented, Sylow
607C2—42no—
608C2#7402811yescharacteristic
609C2#7865211yescharacteristic
610C2#7865311yescharacteristic
611C2#7909111yescharacteristic
612C2#7910111yescharacteristic
613C2#7939211yescharacteristic
614C2#7939811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1798C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1798D3
hypocenterC3#27690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
4. #70486C2semidirect
5. #70485C2semidirect
6. #70455C2semidirect
7. #70453C2semidirect
12. #68693C2semidirect
13. #68691C2semidirect
14. #68644C2semidirect
15. #68642C2semidirect
606. C3#27690semidirect

Maximal subgroups

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

subgroupindex
#822352
#822332
C3×#17982
#704862
#704852
#704552
#704532
#699312
#699302
#699292
#699272
#686932
#686912
#686442
#686422
#276903