Finite Group Databasebeta

← #11235102

Subgroups — #11235102

Order 768.

2014 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C212yesmaximal
2C3×#2399C211yesmaximal, characteristic, complemented
3#82482C212yesmaximal, complemented
4#77864C214yesmaximal, complemented
5#77761C214yesmaximal, complemented
6#77266C214yesmaximal, complemented
7#77251C214yesmaximal, complemented
8#73802C211yesmaximal, characteristic
9#73781C212yesmaximal
10#73770C211yesmaximal, characteristic
11#73711C211yesmaximal, characteristic, complemented
12#73672C212yesmaximal, complemented
13#73637C211yesmaximal, characteristic, complemented
14#73388C211yesmaximal, characteristic, complemented
15#73387C211yesmaximal, characteristic
16#45429—31nomaximal, Sylow
17C2×C3×C4×Q8C2212yes—
18C2×#946C2214yes—
19#5584—216no—
20C2×#898C2214yes—
21C2×#898C2214yes—
22C2×#892C2214yes—
23C2×#875C2212yes—
24C2×#875C2211yescharacteristic
25C2×C4×D12C2212yes—
26C2×C4×D12C2211yescharacteristic
27C2×#869C2212yes—
28C2×#869C2211yescharacteristic
29C2×C4×Dic6C2212yes—
30C2×C4×Dic6C2212yes—
31C2×C4×Dic6C2211yescharacteristic
32C3×#398C2212yes—
33C3×#398C2212yes—
34C3×#395C2212yes—
35C3×#395C2212yes—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C4⋊C4C2212yes—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×#376C2211yescharacteristic
40#5229C2214yes—
41#4992C2214yes—
42#4989C2214yes—
43#4989C2214yes—
44#4987C2214yes—
45#4984C2214yes—
46#4982C2214yes—
47#4982C2214yes—
48#4982C2214yes—
49#4980C2214yes—
50#4976C2214yes—
51#4972C2214yes—
52#4939C2211yescharacteristic
53#4936C2211yescharacteristic
54#4935C2211yescharacteristic
55#4929C2212yes—
56#4929C2212yes—
57#4929C2212yes—
58#4929C2211yescharacteristic
59#4928C2212yes—
60#4928C2212yes—
61#4675C2214yes—
62#4674C2212yes—
63#4674C2212yes—
64#4670C2214yes—
65#4668C2214yes—
66#4667C2212yes—
67#4664C2214yes—
68D3×#95C2212yes—
69#4661C2212yes—
70#4661C2212yes—
71#4660C2214yes—
72#4655C2214yes—
73#4653C2214yes—
74#4653C2214yes—
75#4650C2212yes—
76#4649C2212yes—
77C2×#556—32no—
78#2722—34no—
79#2698—34no—
80#2582—32no—
81#2547—31no—
82#2546—31no—
83#2478—34no—
84#2447—32no—
85#2446—32no—
86#2438—34no—
87#2399D311yescharacteristic, complemented
88#2351—31no—
89#2349—31no—
90#2327—31no—
91#2326—31no—
92C22×C3×Q8C2312yes—
93C22×D12C2311yescharacteristic
94C22×C4×D3C2314yes—
95C22×C4×D3C2312yes—
96C22×Dic6C2312yes—
97C22×Dic6C2311yescharacteristic
98C3×C4×Q8—28no—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C42C2312yes—
107C2×C3×C42C2312yes—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111#946—216no—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×(C2×D3)⋊C4C2312yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2314yes—
121C2×C4⋊Dic3C2314yes—
122C2×C4⋊Dic3C2311yescharacteristic
123C2×C4⋊Dic3C2311yescharacteristic
124C2×C4⋊Dic3C2312yes—
125C2×C4⋊Dic3C2314yes—
126C2×Dic3⋊C4C2314yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2312yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2314yes—
131C2×C4×Dic3C2312yes—
132C2×C4×Dic3C2314yes—
133C2×C4×Dic3C2314yes—
134#898—216no—
135#898—216no—
136#892—216no—
137#875—28no—
138#875D418yes—
139C4×D12—28no—
140C4×D12—24no—
141#869—28no—
142#869—24no—
143C4×Dic6—28no—
144C4×Dic6—28no—
145C4×Dic6—24no—
146C3×#95C2311yescharacteristic
147C3×#95C2311yescharacteristic
148C3×#95C2312yes—
149C3×#95C2312yes—
150C3×#95C2312yes—
151C3×#95C2312yes—
152C3×#95C2311yescharacteristic
153C3×#95C2311yescharacteristic
154C3⋊#95C2312yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C3⋊#95C2314yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165#556—616no—
166C2×C4⋊Q8—31no—
167C2×C4⋊Q8—32no—
168C2×#126—34no—
169C2×#126—34no—
170C2×#124—31no—
171C2×#124—32no—
172C2×C22⋊Q8—34no—
173C2×C4×Q8—31no—
174C2×C4×Q8—32no—
175C2×C4×Q8—32no—
176C2×C4×Q8C2×D312yes—
177C2×C4×D4—31no—
178C2×C4×D4—32no—
179C2×#117—34no—
180#400—34no—
181#398—34no—
182#398—34no—
183#398—34no—
184#398—32no—
185#398—32no—
186#398C2×D312yes—
187#398C2×D312yes—
188#397—34no—
189#397—32no—
190#397—32no—
191#397—34no—
192#395C2×D312yes—
193#395C2×D312yes—
194#391—34no—
195#391—31no—
196#391—32no—
197#391—32no—
198#391—32no—
199#390—31no—
200#389—34no—
201#389—34no—
202#389—34no—
203#389—32no—
204#389—32no—
205#388—34no—
206#387—34no—
207#387—34no—
208#387—34no—
209#387—34no—
210#385—34no—
211#385—34no—
212#385—34no—
213#384—31no—
214#382—31no—
215#382—32no—
216#381—34no—
217#380—34no—
218#380—32no—
219C4×C4⋊C4—32no—
220C4×C4⋊C4C2×D311yescharacteristic
221C4×C4⋊C4C2×D311yescharacteristic
222C4×C4⋊C4C2×D312yes—
223#376C2×D311yescharacteristic
224C2×#95—32no—
225C23×D3C2412yes—
226C2×C3×Q8—24no—
227C2×C3×Q8—24no—
228C2×C3×Q8—42no—
229C22×C3×C4C2412yes—
230C22×C3×C4C2412yes—
231C22×C3×C4C2412yes—
232C22×C3×C4C2412yes—
233C22×C3×C4C2412yes—
234C22×C3×C4C2411yescharacteristic
235C22×C3×C4C2411yescharacteristic
236C22×C3×C4C2412yes—
237C22×C3×C4C2411yescharacteristic
238C22×Dic3C2412yes—
239C22×Dic3C2414yes—
240C22×Dic3C2412yes—
241C22×Dic3C2414yes—
242C22×Dic3C2412yes—
243C2×D12—22no—
244C2×D12—22no—
245C2×D12C2×D414yes—
246C2×C4×D3—28no—
247C2×C4×D3—28no—
248C2×C4×D3—24no—
249C2×C4×D3—44no—
250C2×C4×D3—44no—
251C2×Dic6—22no—
252C2×Dic6—22no—
253C2×Dic6—24no—
254C2×Dic6—24no—
255C2×Dic6—24no—
256C2×Dic6C2×D414yes—
257C3×C4⋊C4—24no—
258C3×C4⋊C4—24no—
259C3×C4⋊C4—24no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4—24no—
262C3×C4⋊C4—42no—
263C3×C4⋊C4—42no—
264C3×C42—24no—
265C3×C42—24no—
266C3×C42—22no—
267C3×C42C2×D414yes—
268C3×C42#4114yes—
269(C2×D3)⋊C4—28no—
270(C2×D3)⋊C4—28no—
271(C2×D3)⋊C4—28no—
272(C2×D3)⋊C4—24no—
273(C2×D3)⋊C4—44no—
274(C2×D3)⋊C4C2×D4116yes—
275C4⋊Dic3—24no—
276C4⋊Dic3—22no—
277C4⋊Dic3—24no—
278C4⋊Dic3—24no—
279C4⋊Dic3—22no—
280C4⋊Dic3—28no—
281C4⋊Dic3—28no—
282C4⋊Dic3—44no—
283Dic3⋊C4—28no—
284Dic3⋊C4—24no—
285Dic3⋊C4—28no—
286Dic3⋊C4—44no—
287Dic3⋊C4—44no—
288C4×Dic3—28no—
289C4×Dic3—24no—
290C4×Dic3—28no—
291C22×Q8—31no—
292C22×Q8—32no—
293C22×Q8C22×D312yes—
294C22×D4—31no—
295C23×C4—34no—
296C23×C4—32no—
297C4⋊Q8—68no—
298C4⋊Q8—64no—
299#126—616no—
300#126—616no—
301#124—38no—
302#124—68no—
303C22⋊Q8—616no—
304C4×Q8—28no—
305C4×Q8—68no—
306C4×Q8—68no—
307C4×Q8—64no—
308C4×D4—68no—
309C4×D4—64no—
310#117—616no—
311C2×C4⋊C4—34no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—34no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—32no—
321C2×C4⋊C4—34no—
322C2×C4⋊C4—34no—
323C2×C4⋊C4—34no—
324C2×C4⋊C4C22×D312yes—
325C2×C4⋊C4C22×D312yes—
326C2×C4⋊C4C22×D312yes—
327C2×C4⋊C4C22×D312yes—
328C2×C4⋊C4C22×D312yes—
329C2×C4⋊C4C22×D312yes—
330C2×C4⋊C4C22×D312yes—
331C2×C22⋊C4—34no—
332C2×C22⋊C4—34no—
333C2×C22⋊C4—34no—
334C2×C22⋊C4—32no—
335C2×C22⋊C4—34no—
336C2×C22⋊C4—34no—
337C2×C42—34no—
338C2×C42—34no—
339C2×C42—32no—
340C2×C42C22×D312yes—
341C2×C42C22×D312yes—
342C2×C42C22×D311yescharacteristic
343C2×C42C22×D311yescharacteristic
344C2×C42C22×D311yescharacteristic
345#95—32no—
346#95—34no—
347#95—32no—
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#95C22×D311yescharacteristic
357#95C22×D312yes—
358#95C22×D312yes—
359#95C22×D311yescharacteristic
360#95C22×D311yescharacteristic
361#95C22×D312yes—
362#95C22×D311yescharacteristic
363#95C22×D312yes—
364C23×C3C2511yescharacteristic
365C22×D3—22no—
366C22×D3—24no—
367C22×D3—22no—
368C22×D3—24no—
369C22×D3C22×D414yes—
370C3×Q8—48no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4C22×D414yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C2×#4112yes—
392C2×C3×C4C2×#4112yes—
393C2×C3×C4C2×#4112yes—
394C2×C3×C42−1+412yes—
395C2×C3×C42−1+414yes—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—24no—
399C2×Dic3—24no—
400C2×Dic3—22no—
401C2×Dic3—24no—
402C2×Dic3—22no—
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3—24no—
406C2×Dic3—24no—
407C2×Dic3—24no—
408C2×Dic3C22×D414yes—
409D12—28no—
410C4×D3—416no—
411C4×D3—84no—
412Dic6—28no—
413Dic6—48no—
414C24—32no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—34no—
418C2×Q8—42no—
419C2×Q8—64no—
420C2×Q8—62no—
421C2×Q8—64no—
422C2×Q8—62no—
423C2×Q8—64no—
424C2×D4—34no—
425C2×D4—62no—
426C2×D4—62no—
427C22×C4—32no—
428C22×C4—32no—
429C22×C4—34no—
430C22×C4—32no—
431C22×C4—34no—
432C22×C4—68no—
433C22×C4—68no—
434C22×C4—64no—
435C22×C4—124no—
436C22×C4—124no—
437C22×C4C23×D312yes—
438C22×C4C23×D312yes—
439C22×C4C23×D312yes—
440C22×C4C23×D312yes—
441C22×C4C23×D312yes—
442C22×C4C23×D311yescharacteristic
443C22×C4C23×D311yescharacteristic
444C22×C4C23×D312yes—
445C22×C4C23×D311yescharacteristic
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—68no—
454C4⋊C4—68no—
455C4⋊C4—68no—
456C4⋊C4—64no—
457C4⋊C4—68no—
458C4⋊C4—62no—
459C4⋊C4—64no—
460C4⋊C4—62no—
461C4⋊C4—64no—
462C4⋊C4—64no—
463C4⋊C4—124no—
464C4⋊C4—124no—
465C4⋊C4—124no—
466C22⋊C4—316no—
467C22⋊C4—68no—
468C22⋊C4—68no—
469C22⋊C4—68no—
470C22⋊C4—64no—
471C22⋊C4—124no—
472C42—24no—
473C42—24no—
474C42—22no—
475C42—68no—
476C42—68no—
477C42—64no—
478C42D3×D414yes—
479C42#23914yes—
480C22×C3C23×D411yescharacteristic
481C22×C3C22×#4111yescharacteristic
482C22×C3C2×2−1+411yescharacteristic
483C22×C3C2×2−1+412yes—
484C22×C3#58512yes—
485C2×D3—24no—
486C2×D3—28no—
487C2×D3—42no—
488C2×D3—42no—
489C2×D3—44no—
490C3×C4—24no—
491C3×C4—24no—
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—42no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4#54714yes—
499Dic3—24no—
500Dic3—44no—
501Dic3—44no—
502Dic3—42no—
503Dic3—42no—
504C23—34no—
505C23—62no—
506C23—62no—
507C23—64no—
508C23—64no—
509C23C24×D311yescharacteristic
510Q8—48no—
511Q8—68no—
512Q8—128no—
513D4—68no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—34no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—64no—
536C2×C4—64no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—1216no—
546C2×C4—244no—
547C2×C4C2×D3×D414yes—
548C2×C4C2×D3×D412yes—
549C2×C4C2×#23912yes—
550C2×C4C2×#23912yes—
551C2×C4C2×#23912yes—
552C2×C4#100712yes—
553C2×C4#101014yes—
554C2×C3C2×#54711yescharacteristic
555C2×C3#348912yes—
556C2×C3#350911yescharacteristic
557C2×C3#351011yescharacteristic
558C2×C3#355512yes—
559D3—44no—
560C22—68no—
561C22—64no—
562C22—122no—
563C22—122no—
564C22—124no—
565C22C22×D3×D411yescharacteristic
566C22C22×#23911yescharacteristic
567C22C2×#100711yescharacteristic
568C22C2×#101012yes—
569C22#596412yes—
570C4—24no—
571C4—24no—
572C4—22no—
573C4—22no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—64no—
579C4—124no—
580C4—124no—
581C4—122no—
582C4—122no—
583C4#572614yes—
584C3#4542911yescharacteristic, complemented, Sylow
585C2—124no—
586C2#8604212yes—
587C2C2×#572611yescharacteristic
588C2#8626211yescharacteristic
589C2#8626311yescharacteristic
590C2#8626812yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2399C2
prime radicalC22×D12C23
hypercenter#2399D3
hypocenterC3#45429
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2399

Complements

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

NH ≅ G/Ntype
2. C3×#2399C2semidirect
3. #82482C2semidirect
4. #77864C2semidirect
5. #77761C2semidirect
6. #77266C2semidirect
7. #77251C2semidirect
11. #73711C2semidirect
12. #73672C2semidirect
13. #73637C2semidirect
14. #73388C2semidirect
87. #2399D3semidirect
584. C3#45429semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
C3×#23992
#824822
#778642
#777612
#772662
#772512
#738022
#737812
#737702
#737112
#736722
#736372
#733882
#733872
#454293