Finite Group Databasebeta

← #11024927

Subgroups — #11024927

Order 768.

2232 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
1C2×#5518C211yesmaximal, characteristic, complemented
2C22×#869C211yesmaximal, characteristic, complemented
3C3×#2290C211yesmaximal, characteristic
4#82349C212yesmaximal
5#82262C211yesmaximal, characteristic, complemented
6#82074C211yesmaximal, characteristic, complemented
7#75695C214yesmaximal, complemented
8#75159C214yesmaximal, complemented
9#74737C214yesmaximal
10#74441C214yesmaximal, complemented
11#73672C212yesmaximal, complemented
12#73488C212yesmaximal, complemented
13C22⋊C4×Dic6C212yesmaximal, complemented
14#73407C211yesmaximal, characteristic, complemented
15#73388C211yesmaximal, characteristic, complemented
16#37695—31nomaximal, Sylow
17C23×Dic6C2212yes—
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C22×C4⋊Dic3C2214yes—
22C2×#924C2212yes—
23C2×#924C2212yes—
24#5518—28no—
25C2×#889C2214yes—
26C2×#879C2214yes—
27C2×#870C2211yescharacteristic
28C2×#869—28no—
29C2×#869C2212yes—
30C2×#869C2211yescharacteristic
31C2×#869C2214yes—
32C2×#869C2211yescharacteristic
33C2×C4×Dic6C2212yes—
34C2×C4×Dic6C2212yes—
35C3×#388C2212yes—
36C3×#387C2212yes—
37C3×#386C2212yes—
38C3×#385C2212yes—
39C3×C4×C4⋊C4C2211yescharacteristic
40C3×C4×C22⋊C4C2211yescharacteristic
41C3×C4×C22⋊C4C2211yescharacteristic
42C3×#376C2211yescharacteristic
43#5220C2214yes—
44#5212C2212yes—
45#5210C2214yes—
46#4982C2214yes—
47#4981C2214yes—
48#4980C2214yes—
49#4979C2214yes—
50#4953C2214yes—
51#4951C2214yes—
52#4948C2214yes—
53#4947C2214yes—
54#4946C2214yes—
55#4942C2214yes—
56Dic3×C22⋊C4C2214yes—
57#4936C2211yescharacteristic
58#4933C2214yes—
59#4929C2212yes—
60#4929C2212yes—
61#4929C2212yes—
62#4929C2212yes—
63#4928C2212yes—
64#4660C2214yes—
65#4659C2214yes—
66#4659C2214yes—
67#4657C2214yes—
68#4655C2214yes—
69#4653C2214yes—
70#4653C2214yes—
71#4647C2214yes—
72#4646C2214yes—
73#4646C2214yes—
74C2×#541—31no—
75C22×C4⋊Q8—31no—
76#2851—31no—
77#2716—34no—
78#2713—34no—
79#2642—31no—
80#2621—31no—
81#2481—34no—
82#2473—34no—
83#2446—32no—
84#2429—32no—
85#2409—32no—
86Q8×C22⋊C4—32no—
87#2327—31no—
88#2290D311yescharacteristic
89C22×C3×D4C2311yescharacteristic
90C23×C3×C4C2312yes—
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2311yescharacteristic
93C23×Dic3C2314yes—
94C22×Dic6—28no—
95C22×Dic6—28no—
96C22×Dic6—44no—
97C22×Dic6C2312yes—
98C22×Dic6C2314yes—
99C22×Dic6C2312yes—
100C3×C4×D4—24no—
101C3×#117—24no—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C22⋊C4C2312yes—
106C2×C3×C22⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4C2311yescharacteristic
108C2×C3×C22⋊C4C2312yes—
109C2×C3×C22⋊C4C2312yes—
110C2×C3×C42—24no—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C2×C3×C42C2312yes—
116C2×C3×C42C2311yescharacteristic
117C2×#217C2314yes—
118C2×#217C2314yes—
119C2×#217C2314yes—
120C2×C4⋊Dic3—216no—
121C2×C4⋊Dic3C2314yes—
122C2×C4⋊Dic3C2314yes—
123C2×C4⋊Dic3C2314yes—
124C2×C4⋊Dic3C2312yes—
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2314yes—
127#924—28no—
128#924—44no—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2314yes—
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2314yes—
137#889—216no—
138#879—216no—
139#870—24no—
140#869—28no—
141#869—24no—
142#869—216no—
143#869D418yescomplemented
144C4×Dic6—28no—
145C4×Dic6—44no—
146C3×#95C2312yes—
147C3×#95C2312yes—
148C3×#95C2312yes—
149C3×#95C2312yes—
150C3×#95C2312yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C23×Q8—32no—
163#541—68no—
164C2×C4⋊Q8—31no—
165C2×C4⋊Q8—31no—
166C2×C4⋊Q8—32no—
167C2×C4⋊Q8—34no—
168C2×C4⋊Q8—68no—
169C2×#126—34no—
170C2×#125—34no—
171C2×#125—31no—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8—32no—
174C2×C4×Q8—32no—
175C2×C4×Q8—32no—
176C2×C4×D4C2×D311yescharacteristic
177C2×#117C2×D311yescharacteristic
178C22×C4⋊C4—34no—
179C22×C42C2×D311yescharacteristic
180#400—34no—
181#400—34no—
182#400—34no—
183#398—34no—
184#398—34no—
185#398—34no—
186#397—34no—
187#396—34no—
188#395—34no—
189#393—34no—
190#391—32no—
191#391—32no—
192#391—32no—
193#391—32no—
194#391—34no—
195#391—34no—
196#389—32no—
197#389—34no—
198#388—34no—
199#388C2×D312yes—
200#387—34no—
201#387—34no—
202#387—34no—
203#387—34no—
204#387—34no—
205#387C2×D312yes—
206#386—34no—
207#386C2×D312yes—
208#385—34no—
209#385—34no—
210#385C2×D312yes—
211#382—31no—
212#380—32no—
213#380—34no—
214C4×C4⋊C4C2×D311yescharacteristic
215C4×C22⋊C4—34no—
216C4×C22⋊C4C2×D311yescharacteristic
217C4×C22⋊C4C2×D311yescharacteristic
218#376C2×D311yescharacteristic
219C24×C3C2411yescharacteristic
220C24×C3C2411yescharacteristic
221C2×C3×D4—22no—
222C2×C3×D4—24no—
223C22×C3×C4—24no—
224C22×C3×C4—24no—
225C22×C3×C4—22no—
226C22×C3×C4—22no—
227C22×C3×C4—24no—
228C22×C3×C4—22no—
229C22×C3×C4—22no—
230C22×C3×C4—22no—
231C22×C3×C4#4114yes—
232C22×C3×C4C2411yescharacteristic
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2412yes—
235C22×C3×C4C2412yes—
236C22×C3×C4C2411yescharacteristic
237C22×C3×C4C2412yes—
238C22×C3×C4C2411yescharacteristic
239C22×C3×C4C2412yes—
240C22×C3×C4C2411yescharacteristic
241C22×Dic3—28no—
242C22×Dic3—28no—
243C22×Dic3—44no—
244C22×Dic3C2414yes—
245C22×Dic3C2414yes—
246C22×Dic3C2414yes—
247C22×Dic3C2414yes—
248C2×Dic6—24no—
249C2×Dic6—24no—
250C2×Dic6—24no—
251C2×Dic6—28no—
252C2×Dic6—24no—
253C2×Dic6—24no—
254C2×Dic6—28no—
255C2×Dic6—432no—
256C2×Dic6—44no—
257C2×Dic6C2×D418yes—
258C3×C4⋊C4—24no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—24no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—24no—
263C3×C22⋊C4—24no—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—24no—
266C3×C42—22no—
267C3×C42—22no—
268C3×C42—24no—
269C3×C42—22no—
270C3×C42—22no—
271C3×C42C2×D414yes—
272#217—28no—
273#217—28no—
274#217—44no—
275C4⋊Dic3—24no—
276C4⋊Dic3—28no—
277C4⋊Dic3—24no—
278C4⋊Dic3—28no—
279C4⋊Dic3—28no—
280C4⋊Dic3C2×D4116yes—
281Dic3⋊C4—28no—
282Dic3⋊C4—28no—
283Dic3⋊C4—28no—
284Dic3⋊C4—28no—
285Dic3⋊C4—44no—
286Dic3⋊C4—44no—
287C4×Dic3—28no—
288C4×Dic3—44no—
289C22×Q8—32no—
290C22×Q8—32no—
291C22×Q8—34no—
292C22×Q8—68no—
293C22×Q8—68no—
294C22×Q8—124no—
295C22×D4C22×D311yescharacteristic
296C23×C4—34no—
297C23×C4C22×D312yes—
298C23×C4C22×D311yescharacteristic
299C23×C4C22×D311yescharacteristic
300C4⋊Q8—38no—
301C4⋊Q8—616no—
302C4⋊Q8—68no—
303C4⋊Q8—64no—
304#126—616no—
305#125—616no—
306#125—64no—
307C22⋊Q8—68no—
308C22⋊Q8—124no—
309C4×Q8—68no—
310C4×Q8—124no—
311C4×D4—24no—
312#117—24no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—34no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—34no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—32no—
319C2×C4⋊C4—34no—
320C2×C4⋊C4—34no—
321C2×C4⋊C4—34no—
322C2×C4⋊C4—34no—
323C2×C4⋊C4—34no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—616no—
326C2×C4⋊C4C22×D312yes—
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D312yes—
329C2×C22⋊C4—34no—
330C2×C22⋊C4—34no—
331C2×C22⋊C4—34no—
332C2×C22⋊C4C22×D312yes—
333C2×C22⋊C4C22×D311yescharacteristic
334C2×C22⋊C4C22×D311yescharacteristic
335C2×C22⋊C4C22×D312yes—
336C2×C22⋊C4C22×D312yes—
337C2×C42—24no—
338C2×C42—34no—
339C2×C42—34no—
340C2×C42C22×D311yescharacteristic
341C2×C42C22×D311yescharacteristic
342C2×C42C22×D311yescharacteristic
343C2×C42C22×D311yescharacteristic
344C2×C42C22×D312yes—
345C2×C42C22×D311yescharacteristic
346#95—34no—
347#95—34no—
348#95—34no—
349#95—34no—
350#95—34no—
351#95—34no—
352#95—34no—
353#95—34no—
354#95—34no—
355#95—34no—
356#95—34no—
357#95C22×D312yes—
358#95C22×D312yes—
359#95C22×D312yes—
360#95C22×D312yes—
361#95C22×D312yes—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—22no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—22no—
369C23×C3—22no—
370C23×C3C2×#4112yes—
371C23×C32−1+414yes—
372C23×C3C2511yescharacteristic
373C3×D4—44no—
374C2×C3×C4—216no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—28no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—44no—
396C2×C3×C4C22×D414yes—
397C2×C3×C4C22×D412yes—
398C2×C3×C4C2×#4112yes—
399C2×C3×C4C2×#4112yes—
400C2×C3×C4H(5,𝔽2)12yes—
401C2×C3×C42−1+414yes—
402C2×Dic3—24no—
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3—24no—
406C2×Dic3—24no—
407C2×Dic3—24no—
408C2×Dic3—24no—
409C2×Dic3—24no—
410C2×Dic3—24no—
411C2×Dic3—24no—
412C2×Dic3—24no—
413C2×Dic3—416no—
414C2×Dic3C22×D418yes—
415Dic6—216no—
416Dic6—416no—
417Dic6—48no—
418C24C23×D311yescharacteristic
419C24C23×D311yescharacteristic
420C2×Q8—38no—
421C2×Q8—68no—
422C2×Q8—64no—
423C2×Q8—68no—
424C2×Q8—64no—
425C2×Q8—64no—
426C2×Q8—64no—
427C2×Q8—64no—
428C2×Q8—124no—
429C2×Q8—1232no—
430C2×D4—22no—
431C2×D4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—34no—
441C22×C4—34no—
442C22×C4—34no—
443C22×C4—34no—
444C22×C4—68no—
445C22×C4—68no—
446C22×C4—124no—
447C22×C4#23514yes—
448C22×C4C23×D311yescharacteristic
449C22×C4C23×D311yescharacteristic
450C22×C4C23×D312yes—
451C22×C4C23×D312yes—
452C22×C4C23×D311yescharacteristic
453C22×C4C23×D311yescharacteristic
454C22×C4C23×D312yes—
455C22×C4C23×D312yes—
456C22×C4C23×D311yescharacteristic
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—316no—
461C4⋊C4—68no—
462C4⋊C4—68no—
463C4⋊C4—68no—
464C4⋊C4—68no—
465C4⋊C4—68no—
466C4⋊C4—64no—
467C4⋊C4—68no—
468C4⋊C4—64no—
469C4⋊C4—68no—
470C4⋊C4—124no—
471C4⋊C4—124no—
472C22⋊C4—22no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—68no—
478C22⋊C4—68no—
479C22⋊C4—124no—
480C42—22no—
481C42—22no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—68no—
486C42—124no—
487C42D3×D414yes—
488C22×C3—22no—
489C22×C3—24no—
490C22×C3—22no—
491C22×C3—24no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—42no—
495C22×C3—41no—
496C22×C3—42no—
497C22×C3#54114yescomplemented
498C22×C3C23×D411yescharacteristic
499C22×C3C22×#4111yescharacteristic
500C22×C3C2×H(5,𝔽2)11yescharacteristic
501C22×C3C2×2−1+412yes—
502C22×C3C2×2−1+412yes—
503C3×C4—22no—
504C3×C4—24no—
505C3×C4—24no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—42no—
509C3×C4—41no—
510C3×C4—42no—
511C3×C4#54614yes—
512Dic3—28no—
513Dic3—44no—
514Dic3—44no—
515Dic3—44no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23C2×#23512yes—
525C23#101014yes—
526C23C24×D311yescharacteristic
527Q8—616no—
528Q8—1216no—
529Q8—128no—
530D4—44no—
531C2×C4—216no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—38no—
553C2×C4—44no—
554C2×C4—64no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—1216no—
566C2×C4C2×#23512yes—
567C2×C4C2×#23512yes—
568C2×C4C2×D3×D414yes—
569C2×C4C2×D3×D412yes—
570C2×C4#100912yes—
571C2×C4#101014yes—
572C2×C3—24no—
573C2×C3—81no—
574C2×C3C2×#54111yescharacteristic
575C2×C3C2×#54611yescharacteristic
576C2×C3#348612yes—
577C2×C3#350911yescharacteristic
578C2×C3#355212yes—
579C22—24no—
580C22—22no—
581C22—22no—
582C22—24no—
583C22—42no—
584C22—41no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22#551814yescomplemented
589C22C22×#23511yescharacteristic
590C22C22×D3×D411yescharacteristic
591C22C2×#100911yescharacteristic
592C22C2×#101012yes—
593C22C2×#101012yes—
594C4—24no—
595C4—24no—
596C4—22no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—41no—
602C4—68no—
603C4—124no—
604C4—124no—
605C4—124no—
606C4#555014yes—
607C3#3769511yescharacteristic, complemented, Sylow
608C2—24no—
609C2—81no—
610C2C2×#551811yescharacteristic
611C2C2×#555011yescharacteristic
612C2#8583011yescharacteristic
613C2#8602212yes—
614C2#8613512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2290C2
prime radicalC22×C3×D4C23
hypercenter#2290D3
hypocenterC3#37695
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2290

Complements

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

NH ≅ G/Ntype
1. C2×#5518C2semidirect
2. C22×#869C2semidirect
5. #82262C2semidirect
6. #82074C2semidirect
7. #75695C2semidirect
8. #75159C2semidirect
10. #74441C2semidirect
11. #73672C2semidirect
12. #73488C2semidirect
13. C22⋊C4×Dic6C2semidirect
14. #73407C2semidirect
15. #73388C2semidirect
143. #869D4semidirect
497. C22×C3#541semidirect
588. C22#5518semidirect
607. C3#37695semidirect

Maximal subgroups

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

subgroupindex
C2×#55182
C22×#8692
C3×#22902
#823492
#822622
#820742
#756952
#751592
#747372
#744412
#736722
#734882
C22⋊C4×Dic62
#734072
#733882
#376953