Finite Group Databasebeta

← #11235168

Subgroups — #11235168

Order 768.

1762 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2331C211yesmaximal, characteristic, complemented
2#83259C211yesmaximal, characteristic
3#83243C211yesmaximal, characteristic, complemented
4#76637C214yesmaximal, complemented
5#76566C214yesmaximal, complemented
6#76077C212yesmaximal, complemented
7#76059C212yesmaximal
8#75212C214yesmaximal, complemented
9#75075C214yesmaximal, complemented
10#74228C211yesmaximal, characteristic
11#74157C211yesmaximal, characteristic, complemented
12#73716C212yesmaximal, complemented
13#73654C212yesmaximal
14#73397C211yesmaximal, characteristic, complemented
15#73387C211yesmaximal, characteristic
16#45434—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#946C2214yes—
19C2×#896C2214yes—
20C2×#893C2212yes—
21C2×#887C2212yes—
22C2×#875C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×#870C2214yes—
25C2×C4×Dic6C2211yescharacteristic
26C2×C4×Dic6C2212yes—
27C3×#391C2214yes—
28C3×#389C2212yes—
29C3×#387C2214yes—
30C3×#382C2211yescharacteristic
31C3×C4×C4⋊C4C2212yes—
32C3×#376C2211yescharacteristic
33#5230C2212yes—
34#5230C2212yes—
35#4994C2214yes—
36#4993C2212yes—
37#4993C2212yes—
38#4988C2212yes—
39#4983C2214yes—
40#4976C2214yes—
41#4975C2212yes—
42#4975C2212yes—
43#4973C2212yes—
44#4972C2212yes—
45#4972C2214yes—
46#4972C2212yes—
47#4940C2214yes—
48#4939C2211yescharacteristic
49C4×(C2×D3)⋊C4C2214yes—
50#4935C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53C4×C4⋊Dic3C2212yes—
54#4929C2211yescharacteristic
55#4928C2214yes—
56#4676C2214yes—
57#4670C2214yes—
58#4670C2214yes—
59#4668C2214yes—
60#4668C2214yes—
61#4668C2214yes—
62#4667C2212yes—
63#4667C2214yes—
64D3×#95C2212yes—
65#4661C2212yes—
66#4661C2212yes—
67#4657C2214yes—
68#4657C2214yes—
69#4657C2214yes—
70#4656C2212yes—
71#4656C2212yes—
72#4650C2212yes—
73#4649C2212yes—
74#4647C2214yes—
75#4646C2214yes—
76#2768—34no—
77#2704—34no—
78#2582—32no—
79#2567—31no—
80#2556—34no—
81#2552—32no—
82#2549—34no—
83#2542—31no—
84#2538—32no—
85#2399—31no—
86#2367—32no—
87#2351—31no—
88#2349—31no—
89#2331D311yescharacteristic, complemented
90#2326—31no—
91C22×C3×Q8C2312yes—
92C22×D12C2311yescharacteristic
93C22×C4×D3C2314yes—
94C22×C4×D3C2312yes—
95C22×Dic6C2312yes—
96C22×Dic6C2311yescharacteristic
97C3×C4⋊Q8—24no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2314yes—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106#946—48no—
107C2×(C2×D3)⋊C4C2312yes—
108C2×(C2×D3)⋊C4C2314yes—
109C2×(C2×D3)⋊C4C2314yes—
110C2×(C2×D3)⋊C4C2314yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×C4⋊Dic3C2312yes—
114C2×C4⋊Dic3C2312yes—
115C2×C4⋊Dic3C2312yes—
116C2×C4⋊Dic3C2312yes—
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2314yes—
121C2×Dic3⋊C4C2314yes—
122C2×Dic3⋊C4C2312yes—
123C2×Dic3⋊C4C2314yes—
124C2×Dic3⋊C4C2314yes—
125C2×Dic3⋊C4C2312yes—
126C2×Dic3⋊C4C2312yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2314yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2314yes—
131C2×C4×Dic3C2312yes—
132C2×C4×Dic3C2312yes—
133C2×C4×Dic3C2312yes—
134C2×C4×Dic3C2314yes—
135#896—48no—
136#893—44no—
137#887—44no—
138#875Q818yescomplemented
139C4×D12—24no—
140#870—48no—
141C4×Dic6—24no—
142C4×Dic6—44no—
143C3×#95C2314yes—
144C3×#95C2312yes—
145C3×#95C2312yes—
146C3×#95C2312yes—
147C3×#95C2312yes—
148C3×#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2314yes—
153C3⋊#95C2311yescharacteristic
154C3⋊#95C2311yescharacteristic
155C3⋊#95C2312yes—
156C3⋊#95C2311yescharacteristic
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2311yescharacteristic
160C2×C4⋊Q8C2×D311yescharacteristic
161C2×#125—34no—
162C2×#124—31no—
163C2×C22⋊Q8—34no—
164C2×C22⋊Q8—34no—
165C2×C4×Q8—32no—
166C2×C4×Q8—31no—
167C2×C4×Q8—32no—
168C2×C4×D4—32no—
169C2×C4×D4—31no—
170#399—32no—
171#399—32no—
172#398—34no—
173#398—34no—
174#398—34no—
175#398—34no—
176#398—32no—
177#398—32no—
178#396—34no—
179#396—34no—
180#395—32no—
181#395—32no—
182#395—32no—
183#395—32no—
184#391—32no—
185#391—31no—
186#391—34no—
187#391C2×D314yes—
188#390—32no—
189#390—31no—
190#389—34no—
191#389—32no—
192#389—34no—
193#389—32no—
194#389—34no—
195#389—34no—
196#389C2×D312yes—
197#388—34no—
198#388—34no—
199#388—34no—
200#387C2×D314yes—
201#385—34no—
202#385—34no—
203#385—34no—
204#384—32no—
205#384—32no—
206#384—31no—
207#382—32no—
208#382C2×D311yescharacteristic
209#380—32no—
210#380—34no—
211C4×C4⋊C4—31no—
212C4×C4⋊C4—31no—
213C4×C4⋊C4—32no—
214C4×C4⋊C4—32no—
215C4×C4⋊C4C2×D312yes—
216C4×C22⋊C4—34no—
217#376C2×D311yescharacteristic
218C2×#95—32no—
219C23×D3C2412yes—
220C2×C3×Q8—24no—
221C2×C3×Q8—44no—
222C22×C3×C4C2414yes—
223C22×C3×C4C2412yes—
224C22×C3×C4C2414yes—
225C22×C3×C4C2412yes—
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2411yescharacteristic
229C22×Dic3C2412yes—
230C22×Dic3C2412yes—
231C22×Dic3C2412yes—
232C22×Dic3C2412yes—
233C22×Dic3C2412yes—
234C22×Dic3C2412yes—
235C22×Dic3C2412yes—
236C2×D12—22no—
237C2×D12—22no—
238C2×D12C2×Q814yes—
239C2×C4×D3—24no—
240C2×C4×D3—28no—
241C2×C4×D3—44no—
242C2×C4×D3—44no—
243C2×C4×D3—44no—
244C2×Dic6—22no—
245C2×Dic6—22no—
246C2×Dic6—24no—
247C2×Dic6—24no—
248C2×Dic6—42no—
249C2×Dic6C2×Q814yes—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—44no—
254C3×C42—22no—
255C3×C42—44no—
256C3×C42C2×Q814yes—
257C3×C42#4114yes—
258(C2×D3)⋊C4—24no—
259(C2×D3)⋊C4—28no—
260(C2×D3)⋊C4—44no—
261(C2×D3)⋊C4—44no—
262(C2×D3)⋊C4—44no—
263(C2×D3)⋊C4C2×Q8116yes—
264C4⋊Dic3—22no—
265C4⋊Dic3—22no—
266C4⋊Dic3—24no—
267C4⋊Dic3—24no—
268C4⋊Dic3—24no—
269C4⋊Dic3—24no—
270C4⋊Dic3—42no—
271C4⋊Dic3—44no—
272Dic3⋊C4—24no—
273Dic3⋊C4—28no—
274Dic3⋊C4—28no—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—44no—
278Dic3⋊C4—44no—
279Dic3⋊C4—44no—
280Dic3⋊C4—44no—
281Dic3⋊C4—44no—
282C4×Dic3—24no—
283C4×Dic3—24no—
284C4×Dic3—24no—
285C4×Dic3—44no—
286C22×Q8—31no—
287C22×Q8—32no—
288C22×Q8C22×D312yes—
289C22×D4—31no—
290C23×C4—32no—
291C23×C4—34no—
292C4⋊Q8—24no—
293#125—128no—
294#124—38no—
295C22⋊Q8—128no—
296C22⋊Q8—128no—
297C4×Q8—64no—
298C4×Q8—124no—
299C4×Q8—124no—
300C4×D4—64no—
301C4×D4—124no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—34no—
307C2×C4⋊C4—34no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—32no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—34no—
317C2×C4⋊C4—34no—
318C2×C4⋊C4—34no—
319C2×C4⋊C4—34no—
320C2×C4⋊C4C22×D312yes—
321C2×C4⋊C4C22×D312yes—
322C2×C4⋊C4C22×D312yes—
323C2×C4⋊C4C22×D314yes—
324C2×C22⋊C4—34no—
325C2×C22⋊C4—32no—
326C2×C22⋊C4—34no—
327C2×C22⋊C4—34no—
328C2×C22⋊C4—34no—
329C2×C22⋊C4—34no—
330C2×C42—32no—
331C2×C42—32no—
332C2×C42—32no—
333C2×C42—34no—
334C2×C42C22×D311yescharacteristic
335C2×C42C22×D314yes—
336C2×C42C22×D311yescharacteristic
337C2×C42C22×D311yescharacteristic
338#95—31no—
339#95—34no—
340#95—32no—
341#95—34no—
342#95—31no—
343#95—31no—
344#95—31no—
345#95—32no—
346#95—34no—
347#95—34no—
348#95—34no—
349#95C22×D312yes—
350#95C22×D312yes—
351#95C22×D312yes—
352#95C22×D312yes—
353#95C22×D314yes—
354#95C22×D314yes—
355C23×C3C2511yescharacteristic
356C22×D3—22no—
357C22×D3—24no—
358C22×D3—22no—
359C22×D3—24no—
360C22×D3C22×Q814yes—
361C3×Q8—84no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—24no—
370C2×C3×C4—22no—
371C2×C3×C4—24no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—24no—
375C2×C3×C4C22×Q814yes—
376C2×C3×C4C22×Q812yes—
377C2×C3×C4C2×#4112yes—
378C2×C3×C4C2×#4112yes—
379C2×C3×C4C2×#4112yes—
380C2×C3×C4H(5,𝔽2)12yes—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3C22×Q814yes—
398C2×Dic3H(5,𝔽2)14yes—
399C2×Dic32−1+414yes—
400D12—28no—
401C4×D3—88no—
402C4×D3—84no—
403Dic6—28no—
404Dic6—48no—
405C24—32no—
406C2×Q8—24no—
407C2×Q8—34no—
408C2×Q8—44no—
409C2×Q8—64no—
410C2×Q8—64no—
411C2×Q8—62no—
412C2×Q8—62no—
413C2×Q8—122no—
414C2×D4—34no—
415C2×D4—62no—
416C2×D4—62no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—32no—
421C22×C4—32no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—68no—
425C22×C4—64no—
426C22×C4—124no—
427C22×C4—124no—
428C22×C4—124no—
429C22×C4C23×D314yes—
430C22×C4C23×D312yes—
431C22×C4C23×D314yes—
432C22×C4C23×D311yescharacteristic
433C22×C4C23×D311yescharacteristic
434C22×C4C23×D312yes—
435C22×C4C23×D311yescharacteristic
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—44no—
440C4⋊C4—64no—
441C4⋊C4—68no—
442C4⋊C4—68no—
443C4⋊C4—64no—
444C4⋊C4—64no—
445C4⋊C4—64no—
446C4⋊C4—62no—
447C4⋊C4—64no—
448C4⋊C4—62no—
449C4⋊C4—122no—
450C4⋊C4—122no—
451C4⋊C4—124no—
452C4⋊C4—122no—
453C4⋊C4—124no—
454C4⋊C4—124no—
455C4⋊C4—124no—
456C4⋊C4—124no—
457C4⋊C4—124no—
458C22⋊C4—316no—
459C22⋊C4—68no—
460C22⋊C4—64no—
461C22⋊C4—124no—
462C22⋊C4—124no—
463C22⋊C4—124no—
464C42—22no—
465C42—44no—
466C42—64no—
467C42—64no—
468C42—64no—
469C42—124no—
470C42#23514yes—
471C42D3×Q814yes—
472C22×C3C23×Q811yescharacteristic
473C22×C3C22×#4111yescharacteristic
474C22×C3C2×H(5,𝔽2)11yescharacteristic
475C22×C3C2×H(5,𝔽2)12yes—
476C22×C3C2×2−1+412yes—
477C2×D3—24no—
478C2×D3—28no—
479C2×D3—42no—
480C2×D3—42no—
481C2×D3—44no—
482C3×C4—24no—
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—44no—
486C3×C4—42no—
487C3×C4—44no—
488C3×C4#55414yes—
489Dic3—24no—
490Dic3—24no—
491Dic3—24no—
492Dic3—42no—
493Dic3—42no—
494Dic3—42no—
495Dic3—42no—
496C23—34no—
497C23—62no—
498C23—62no—
499C23—64no—
500C23—64no—
501C23C24×D311yescharacteristic
502Q8—68no—
503Q8—84no—
504Q8—128no—
505D4—68no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—34no—
520C2×C4—34no—
521C2×C4—34no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—64no—
534C2×C4—62no—
535C2×C4—64no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—244no—
539C2×C4—248no—
540C2×C4C2×#23512yes—
541C2×C4C2×#23512yes—
542C2×C4C2×#23512yes—
543C2×C4C2×D3×Q814yes—
544C2×C4C2×D3×Q812yes—
545C2×C4#100912yes—
546C2×C3C2×#55411yescharacteristic
547C2×C3#350312yes—
548C2×C3#350811yescharacteristic
549C2×C3#350911yescharacteristic
550C2×C3#353612yes—
551D3—44no—
552C22—64no—
553C22—68no—
554C22—124no—
555C22—122no—
556C22—122no—
557C22C22×#23511yescharacteristic
558C22C22×D3×Q811yescharacteristic
559C22C2×#100911yescharacteristic
560C22D3×H(5,𝔽2)12yes—
561C22D3×2−1+412yes—
562C4—24no—
563C4—22no—
564C4—22no—
565C4—44no—
566C4—42no—
567C4—44no—
568C4—64no—
569C4—64no—
570C4—64no—
571C4—122no—
572C4—122no—
573C4—122no—
574C4—122no—
575C4#557914yes—
576C3#4543411yescharacteristic, complemented, Sylow
577C2—124no—
578C2C2×#557911yescharacteristic
579C2#8586211yescharacteristic
580C2#8587611yescharacteristic
581C2#8656112yes—
582C2#8658612yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2331C2
prime radicalC22×D12C23
hypercenter#2331D3
hypocenterC3#45434
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2331

Complements

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

NH ≅ G/Ntype
1. C3×#2331C2semidirect
3. #83243C2semidirect
4. #76637C2semidirect
5. #76566C2semidirect
6. #76077C2semidirect
8. #75212C2semidirect
9. #75075C2semidirect
11. #74157C2semidirect
12. #73716C2semidirect
14. #73397C2semidirect
89. #2331D3semidirect
138. #875Q8semidirect
576. C3#45434semidirect

Maximal subgroups

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

subgroupindex
C3×#23312
#832592
#832432
#766372
#765662
#760772
#760592
#752122
#750752
#742282
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