Finite Group Databasebeta

← #11235176

Subgroups — #11235176

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×#2351C211yesmaximal, characteristic, complemented
2#82482C212yesmaximal, complemented
3#82478C211yesmaximal, characteristic, complemented
4#82466C212yesmaximal
5#82455C211yesmaximal, characteristic
6#76244C214yesmaximal, complemented
7#76112C214yesmaximal, complemented
8#76066C212yesmaximal, complemented
9#76059C212yesmaximal
10#74228C211yesmaximal, characteristic
11#74208C214yesmaximal, complemented
12#74154C211yesmaximal, characteristic, complemented
13#73800C211yesmaximal, characteristic
14#73718C214yesmaximal, complemented
15#73669C211yesmaximal, characteristic, complemented
16#45434—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2212yes—
19C2×#947C2211yescharacteristic
20C2×#897C2214yes—
21C2×#896C2214yes—
22C2×#893C2212yes—
23C2×#889C2214yes—
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C4×Dic6C2212yes—
27C3×#389C2212yes—
28C3×#389C2214yes—
29C3×#382C2211yescharacteristic
30C3×#382C2212yes—
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×C4×C4⋊C4C2212yes—
33#5229C2211yescharacteristic
34#5229C2212yes—
35#4994C2214yes—
36#4990C2214yes—
37#4989C2214yes—
38#4989C2214yes—
39#4988C2211yescharacteristic
40#4982C2214yes—
41#4982C2214yes—
42#4982C2212yes—
43#4981C2214yes—
44#4979C2212yes—
45#4979C2212yes—
46#4979C2212yes—
47#4975C2211yescharacteristic
48#4975C2212yes—
49Dic3×C4⋊C4C2212yes—
50#4972C2212yes—
51#4972C2212yes—
52#4940C2214yes—
53#4940C2214yes—
54#4939C2212yes—
55C4×(C2×D3)⋊C4C2214yes—
56#4935C2212yes—
57C4×C4⋊Dic3C2211yescharacteristic
58#4932C2211yescharacteristic
59C4×Dic3⋊C4C2212yes—
60#4929C2214yes—
61#4928C2214yes—
62#4928C2214yes—
63#4677C2212yes—
64#4677C2212yes—
65#4676C2214yes—
66#4668C2214yes—
67#4667C2214yes—
68#4667C2212yes—
69D3×#95C2212yes—
70#4657C2214yes—
71#4657C2212yes—
72#4656C2212yes—
73#4653C2214yes—
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#2351D311yescharacteristic, complemented
88#2349—31no—
89#2331—31no—
90#2326—31no—
91C22×C3×Q8C2311yescharacteristic
92C22×D12C2311yescharacteristic
93C22×C4×D3C2314yes—
94C22×C4×D3C2312yes—
95C22×Dic6C2312yes—
96C22×Dic6C2312yes—
97C3×C4×Q8—24no—
98C3×C4×Q8—44no—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2314yes—
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C42C2312yes—
106C2×C3×C42C2312yes—
107C2×C3×C42C2314yes—
108C2×C3×C42C2311yescharacteristic
109#947Q818yescomplemented
110C2×(C2×D3)⋊C4C2312yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×C4⋊Dic3C2312yes—
117C2×C4⋊Dic3C2312yes—
118C2×C4⋊Dic3C2314yes—
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2312yes—
121C2×C4⋊Dic3C2314yes—
122C2×C4⋊Dic3C2312yes—
123C2×Dic3⋊C4C2312yes—
124C2×Dic3⋊C4C2312yes—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2314yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2312yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2312yes—
132C2×C4×Dic3C2312yes—
133C2×C4×Dic3C2311yescharacteristic
134C2×C4×Dic3C2314yes—
135C2×C4×Dic3C2311yescharacteristic
136#897—48no—
137#896—48no—
138#893—44no—
139#889—48no—
140C4×D12—24no—
141#869—24no—
142C4×Dic6—44no—
143C3×#95C2312yes—
144C3×#95C2314yes—
145C3×#95C2312yes—
146C3×#95C2311yescharacteristic
147C3×#95C2311yescharacteristic
148C3⋊#95C2312yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2312yes—
153C3⋊#95C2312yes—
154C3⋊#95C2312yes—
155C3⋊#95C2311yescharacteristic
156C3⋊#95C2311yescharacteristic
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C2×C4⋊Q8—31no—
161C2×#125—34no—
162C2×#124—31no—
163C2×C22⋊Q8—34no—
164C2×C22⋊Q8—34no—
165C2×C4×Q8—32no—
166C2×C4×Q8C2×D311yescharacteristic
167C2×C4×Q8C2×D312yes—
168C2×C4×D4—32no—
169C2×C4×D4—31no—
170#399—32no—
171#399—32no—
172#398—34no—
173#398—34no—
174#398—32no—
175#398—34no—
176#398—32no—
177#398—34no—
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#391—34no—
188#390—31no—
189#390—32no—
190#389—32no—
191#389—32no—
192#389—34no—
193#389—34no—
194#389—34no—
195#389C2×D314yes—
196#389C2×D312yes—
197#388—34no—
198#388—34no—
199#388—34no—
200#387—34no—
201#385—34no—
202#385—34no—
203#385—34no—
204#384—32no—
205#384—31no—
206#384—32no—
207#382C2×D311yescharacteristic
208#382C2×D312yes—
209#380—32no—
210#380—34no—
211C4×C4⋊C4—32no—
212C4×C4⋊C4—31no—
213C4×C4⋊C4—32no—
214C4×C4⋊C4C2×D311yescharacteristic
215C4×C4⋊C4C2×D312yes—
216C4×C22⋊C4—34no—
217#376—31no—
218C2×#95—32no—
219C23×D3C2412yes—
220C2×C3×Q8—22no—
221C2×C3×Q8—22no—
222C2×C3×Q8C2×Q814yes—
223C22×C3×C4C2414yes—
224C22×C3×C4C2412yes—
225C22×C3×C4C2412yes—
226C22×C3×C4C2412yes—
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2412yes—
230C22×C3×C4C2411yescharacteristic
231C22×Dic3C2412yes—
232C22×Dic3C2412yes—
233C22×Dic3C2414yes—
234C22×Dic3C2412yes—
235C22×Dic3C2412yes—
236C22×Dic3C2412yes—
237C2×D12—22no—
238C2×D12—22no—
239C2×D12C2×Q814yes—
240C2×C4×D3—24no—
241C2×C4×D3—28no—
242C2×C4×D3—44no—
243C2×C4×D3—44no—
244C2×C4×D3—44no—
245C2×Dic6—24no—
246C2×Dic6—24no—
247C2×Dic6—24no—
248C2×Dic6—42no—
249C2×Dic6—44no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—28no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—44no—
256C3×C42—24no—
257C3×C42—24no—
258C3×C42—44no—
259C3×C42#4114yes—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4—28no—
262(C2×D3)⋊C4—44no—
263(C2×D3)⋊C4—44no—
264(C2×D3)⋊C4—44no—
265(C2×D3)⋊C4C2×Q8116yes—
266C4⋊Dic3—24no—
267C4⋊Dic3—24no—
268C4⋊Dic3—22no—
269C4⋊Dic3—28no—
270C4⋊Dic3—24no—
271C4⋊Dic3—42no—
272C4⋊Dic3—44no—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—44no—
278Dic3⋊C4—44no—
279Dic3⋊C4—44no—
280Dic3⋊C4—44no—
281Dic3⋊C4—44no—
282C4×Dic3—22no—
283C4×Dic3—24no—
284C4×Dic3—44no—
285C4×Dic3C2×Q814yes—
286C22×Q8—32no—
287C22×Q8—32no—
288C22×Q8C22×D311yescharacteristic
289C22×D4—31no—
290C23×C4—32no—
291C23×C4—34no—
292C4⋊Q8—64no—
293#125—128no—
294#124—38no—
295C22⋊Q8—128no—
296C22⋊Q8—128no—
297C4×Q8—24no—
298C4×Q8—44no—
299C4×Q8—124no—
300C4×D4—64no—
301C4×D4—124no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—34no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—34no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—34no—
317C2×C4⋊C4—34no—
318C2×C4⋊C4C22×D312yes—
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C4⋊C4C22×D312yes—
321C2×C4⋊C4C22×D312yes—
322C2×C4⋊C4C22×D314yes—
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—31no—
331C2×C42—32no—
332C2×C42—31no—
333C2×C42—34no—
334C2×C42C22×D312yes—
335C2×C42C22×D312yes—
336C2×C42C22×D314yes—
337C2×C42C22×D311yescharacteristic
338#95—31no—
339#95—34no—
340#95—32no—
341#95—32no—
342#95—32no—
343#95—34no—
344#95—31no—
345#95—34no—
346#95—32no—
347#95—34no—
348#95—34no—
349#95—34no—
350#95C22×D312yes—
351#95C22×D311yescharacteristic
352#95C22×D312yes—
353#95C22×D311yescharacteristic
354#95C22×D314yes—
355C23×C3C2511yescharacteristic
356C22×D3—22no—
357C22×D3—24no—
358C22×D3—22no—
359C22×D3—24no—
360C22×D3C22×Q814yes—
361C3×Q8—28no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—24no—
370C2×C3×C4—24no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—24no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4C22×Q814yes—
377C2×C3×C4C22×Q812yes—
378C2×C3×C4C2×#4112yes—
379C2×C3×C4C2×#4112yes—
380C2×C3×C4C2×#4112yes—
381C2×C3×C4H(5,𝔽2)12yes—
382C2×C3×C4H(5,𝔽2)14yes—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—24no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—24no—
392C2×Dic3—22no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3C22×Q814yes—
399C2×Dic32−1+414yes—
400D12—28no—
401C4×D3—88no—
402C4×D3—84no—
403Dic6—48no—
404Dic6—84no—
405C24—32no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—64no—
409C2×Q8—64no—
410C2×Q8—64no—
411C2×Q8—122no—
412C2×Q8—124no—
413C2×Q8D3×Q814yes—
414C2×D4—34no—
415C2×D4—62no—
416C2×D4—62no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—32no—
421C22×C4—34no—
422C22×C4—32no—
423C22×C4—68no—
424C22×C4—64no—
425C22×C4—124no—
426C22×C4—124no—
427C22×C4—124no—
428C22×C4C23×D314yes—
429C22×C4C23×D312yes—
430C22×C4C23×D312yes—
431C22×C4C23×D311yescharacteristic
432C22×C4C23×D311yescharacteristic
433C22×C4C23×D312yes—
434C22×C4C23×D312yes—
435C22×C4C23×D311yescharacteristic
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—28no—
441C4⋊C4—44no—
442C4⋊C4—64no—
443C4⋊C4—64no—
444C4⋊C4—62no—
445C4⋊C4—64no—
446C4⋊C4—64no—
447C4⋊C4—68no—
448C4⋊C4—64no—
449C4⋊C4—122no—
450C4⋊C4—122no—
451C4⋊C4—122no—
452C4⋊C4—124no—
453C4⋊C4—124no—
454C4⋊C4—124no—
455C4⋊C4—124no—
456C4⋊C4—124no—
457C4⋊C4—124no—
458C22⋊C4—316no—
459C22⋊C4—64no—
460C22⋊C4—68no—
461C22⋊C4—124no—
462C22⋊C4—124no—
463C22⋊C4—124no—
464C42—24no—
465C42—24no—
466C42—34no—
467C42—44no—
468C42—64no—
469C42—62no—
470C42—124no—
471C42#23514yes—
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—24no—
486C3×C4—42no—
487C3×C4—44no—
488C3×C4—42no—
489C3×C4#55414yes—
490Dic3—24no—
491Dic3—24no—
492Dic3—44no—
493Dic3—42no—
494Dic3—42no—
495Dic3—42no—
496C23—34no—
497C23—62no—
498C23—62no—
499C23—64no—
500C23—64no—
501C23C24×D311yescharacteristic
502Q8—28no—
503Q8—128no—
504Q8—244no—
505D4—68no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—34no—
521C2×C4—34no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—64no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—64no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—244no—
538C2×C4—248no—
539C2×C4C2×#23512yes—
540C2×C4C2×#23512yes—
541C2×C4C2×#23512yes—
542C2×C4#100412yes—
543C2×C4C2×D3×Q814yes—
544C2×C4C2×D3×Q812yes—
545C2×C4#100914yes—
546C2×C3C2×#55411yescharacteristic
547C2×C3#350312yes—
548C2×C3#350811yescharacteristic
549C2×C3#350911yescharacteristic
550C2×C3#353612yes—
551D3—44no—
552C22—64no—
553C22—68no—
554C22—122no—
555C22—122no—
556C22—124no—
557C22C22×#23511yescharacteristic
558C22C2×#100411yescharacteristic
559C22C22×D3×Q811yescharacteristic
560C22C2×#100912yes—
561C22D3×2−1+412yes—
562C4—22no—
563C4—22no—
564C4—24no—
565C4—24no—
566C4—42no—
567C4—44no—
568C4—42no—
569C4—64no—
570C4—64no—
571C4—122no—
572C4—124no—
573C4—122no—
574C4—122no—
575C4#551114yes—
576C3#4543411yescharacteristic, complemented, Sylow
577C2—124no—
578C2C2×#551111yescharacteristic
579C2#8584811yescharacteristic
580C2#8587611yescharacteristic
581C2#8673912yes—
582C2#8701812yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2351C2
prime radicalC22×D12C23
hypercenter#2351D3
hypocenterC3#45434
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C3×#2351C2semidirect
2. #82482C2semidirect
3. #82478C2semidirect
6. #76244C2semidirect
7. #76112C2semidirect
8. #76066C2semidirect
11. #74208C2semidirect
12. #74154C2semidirect
14. #73718C2semidirect
15. #73669C2semidirect
87. #2351D3semidirect
109. #947Q8semidirect
576. C3#45434semidirect

Maximal subgroups

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

subgroupindex
C3×#23512
#824822
#824782
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