Finite Group Databasebeta

← #11235170

Subgroups — #11235170

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×#2567C211yesmaximal, characteristic, complemented
2#83259C211yesmaximal, characteristic
3#83228C211yesmaximal, characteristic, complemented
4#82460C211yesmaximal, characteristic, complemented
5#82455C211yesmaximal, characteristic
6#77253C214yesmaximal, complemented
7#77216C214yesmaximal, complemented
8#76548C212yesmaximal, complemented
9#76493C212yesmaximal
10#76143C214yesmaximal, complemented
11#76068C214yesmaximal, complemented
12#74369C212yesmaximal
13#74337C212yesmaximal, complemented
14#73770C211yesmaximal, characteristic
15#73669C211yesmaximal, characteristic, complemented
16#45434—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×#946C2214yes—
21C2×Q8×Dic3C2212yes—
22C2×#897C2214yes—
23C2×#889C2214yes—
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2212yes—
26C2×C4×Dic6C2212yes—
27C3×#395C2212yes—
28C3×#395C2212yes—
29C3×#391C2212yes—
30C3×#389C2214yes—
31C3×#384C2212yes—
32C3×C4×C4⋊C4C2211yescharacteristic
33#5229C2211yescharacteristic
34#5229C2214yes—
35#4994C2214yes—
36#4994C2214yes—
37#4993C2212yes—
38#4993C2212yes—
39#4989C2214yes—
40#4989C2214yes—
41#4988C2211yescharacteristic
42#4988C2212yes—
43#4987C2214yes—
44#4984C2212yes—
45#4984C2212yes—
46#4984C2214yes—
47#4983C2214yes—
48#4982C2214yes—
49#4982C2212yes—
50#4980C2214yes—
51#4979C2212yes—
52#4976C2214yes—
53#4975C2212yes—
54#4975C2211yescharacteristic
55Dic3×C4⋊C4C2212yes—
56#4940C2214yes—
57#4936C2211yescharacteristic
58#4936C2212yes—
59C4×C4⋊Dic3C2212yes—
60C4×C4⋊Dic3C2211yescharacteristic
61#4932C2211yescharacteristic
62C4×Dic3⋊C4C2212yes—
63#4928C2212yes—
64#4670C2214yes—
65#4669C2214yes—
66#4668C2214yes—
67#4667C2212yes—
68#4666C2214yes—
69D3×#95C2212yes—
70#4661C2214yes—
71#4657C2214yes—
72#4657C2212yes—
73#4656C2212yes—
74#4647C2214yes—
75#4646C2214yes—
76#2768—34no—
77#2704—34no—
78#2582—32no—
79#2567D311yescharacteristic, complemented
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#2331—31no—
90#2326—31no—
91C22×C3×Q8C2311yescharacteristic
92C22×C3×Q8C2312yes—
93C22×D12C2311yescharacteristic
94C22×C4×D3C2314yes—
95C22×C4×D3C2312yes—
96C22×Dic6C2312yes—
97C3×C4⋊Q8—24no—
98C3×C4×Q8—24no—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2312yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C42C2312yes—
107C2×C3×C42C2312yes—
108C2×C3×C42C2311yescharacteristic
109#947Q818yescomplemented
110#946—48no—
111Q8×Dic3—44no—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2312yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2312yes—
121C2×C4⋊Dic3C2314yes—
122C2×C4⋊Dic3C2314yes—
123C2×C4⋊Dic3C2312yes—
124C2×C4⋊Dic3C2314yes—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2312yes—
127C2×Dic3⋊C4C2312yes—
128C2×Dic3⋊C4C2312yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2312yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2314yes—
133C2×C4×Dic3C2312yes—
134C2×C4×Dic3C2311yescharacteristic
135C2×C4×Dic3C2314yes—
136C2×C4×Dic3C2314yes—
137C2×C4×Dic3C2311yescharacteristic
138#897—48no—
139#889—48no—
140C4×D12—24no—
141C4×D12—44no—
142C4×Dic6—44no—
143C3×#95C2314yes—
144C3×#95C2314yes—
145C3×#95C2311yescharacteristic
146C3×#95C2311yescharacteristic
147C3⋊#95C2314yes—
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2312yes—
151C3⋊#95C2312yes—
152C3⋊#95C2312yes—
153C3⋊#95C2312yes—
154C3⋊#95C2314yes—
155C3⋊#95C2312yes—
156C3⋊#95C2311yescharacteristic
157C3⋊#95C2312yes—
158C3⋊#95C2311yescharacteristic
159C3⋊#95C2314yes—
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—32no—
167C2×C4×Q8C2×D311yescharacteristic
168C2×C4×D4—31no—
169C2×C4×D4—32no—
170#399—32no—
171#399—32no—
172#398—32no—
173#398—34no—
174#398—34no—
175#398—34no—
176#398—34no—
177#398—32no—
178#396—34no—
179#396—34no—
180#395—32no—
181#395—32no—
182#395C2×D312yes—
183#395C2×D312yes—
184#391—31no—
185#391—34no—
186#391—34no—
187#391C2×D312yes—
188#390—31no—
189#390—32no—
190#389—32no—
191#389—34no—
192#389—32no—
193#389—34no—
194#389—32no—
195#389—34no—
196#389C2×D314yes—
197#388—34no—
198#388—34no—
199#388—34no—
200#387—34no—
201#385—34no—
202#385—34no—
203#385—34no—
204#384—31no—
205#384—32no—
206#384C2×D312yes—
207#382—31no—
208#382—32no—
209#380—34no—
210#380—32no—
211C4×C4⋊C4—32no—
212C4×C4⋊C4—31no—
213C4×C4⋊C4—32no—
214C4×C4⋊C4—32no—
215C4×C4⋊C4C2×D311yescharacteristic
216C4×C22⋊C4—34no—
217#376—31no—
218C2×#95—32no—
219C23×D3C2412yes—
220C2×C3×Q8—22no—
221C2×C3×Q8—24no—
222C2×C3×Q8—22no—
223C2×C3×Q8—44no—
224C2×C3×Q8C2×Q814yes—
225C22×C3×C4C2414yes—
226C22×C3×C4C2412yes—
227C22×C3×C4C2412yes—
228C22×C3×C4C2412yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2412yes—
232C22×C3×C4C2411yescharacteristic
233C22×Dic3C2414yes—
234C22×Dic3C2412yes—
235C22×Dic3C2412yes—
236C22×Dic3C2412yes—
237C22×Dic3C2412yes—
238C22×Dic3C2412yes—
239C2×D12—22no—
240C2×D12—22no—
241C2×D12C2×Q814yes—
242C2×C4×D3—24no—
243C2×C4×D3—28no—
244C2×C4×D3—44no—
245C2×C4×D3—44no—
246C2×C4×D3—44no—
247C2×Dic6—24no—
248C2×Dic6—24no—
249C2×Dic6—42no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—44no—
256C3×C4⋊C4—44no—
257C3×C42—24no—
258C3×C42—24no—
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—22no—
267C4⋊Dic3—24no—
268C4⋊Dic3—28no—
269C4⋊Dic3—24no—
270C4⋊Dic3—42no—
271C4⋊Dic3—44no—
272C4⋊Dic3—44no—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—28no—
276Dic3⋊C4—42no—
277Dic3⋊C4—42no—
278Dic3⋊C4—44no—
279Dic3⋊C4—44no—
280Dic3⋊C4—44no—
281C4×Dic3—22no—
282C4×Dic3—24no—
283C4×Dic3—44no—
284C4×Dic3—44no—
285C4×Dic3C2×Q814yes—
286C22×Q8—32no—
287C22×Q8C22×D311yescharacteristic
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—24no—
298C4×Q8—124no—
299C4×Q8—124no—
300C4×D4—64no—
301C4×D4—124no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—34no—
306C2×C4⋊C4—34no—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—34no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—32no—
313C2×C4⋊C4—34no—
314C2×C4⋊C4—34no—
315C2×C4⋊C4—34no—
316C2×C4⋊C4—34no—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D312yes—
319C2×C4⋊C4C22×D312yes—
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—31no—
332C2×C42—32no—
333C2×C42—34no—
334C2×C42—34no—
335C2×C42C22×D312yes—
336C2×C42C22×D312yes—
337C2×C42C22×D311yescharacteristic
338#95—32no—
339#95—34no—
340#95—32no—
341#95—32no—
342#95—32no—
343#95—31no—
344#95—32no—
345#95—31no—
346#95—34no—
347#95—32no—
348#95—34no—
349#95—34no—
350#95—34no—
351#95C22×D311yescharacteristic
352#95C22×D314yes—
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—
362C3×Q8—84no—
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—22no—
370C2×C3×C4—24no—
371C2×C3×C4—24no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—22no—
377C2×C3×C4C22×Q814yes—
378C2×C3×C4C22×Q812yes—
379C2×C3×C4C2×#4112yes—
380C2×C3×C4C2×#4112yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C4H(5,𝔽2)12yes—
383C2×C3×C42−1+414yes—
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—24no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—24no—
399C2×Dic3C22×Q814yes—
400C2×Dic3H(5,𝔽2)14yes—
401D12—28no—
402C4×D3—84no—
403C4×D3—88no—
404Dic6—48no—
405C24—32no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—44no—
410C2×Q8—64no—
411C2×Q8—64no—
412C2×Q8—122no—
413C2×Q8D3×Q814yes—
414C2×D4—34no—
415C2×D4—62no—
416C2×D4—62no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—34no—
420C22×C4—32no—
421C22×C4—32no—
422C22×C4—32no—
423C22×C4—64no—
424C22×C4—68no—
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—24no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—64no—
444C4⋊C4—68no—
445C4⋊C4—68no—
446C4⋊C4—64no—
447C4⋊C4—62no—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—122no—
451C4⋊C4—122no—
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—24no—
465C42—24no—
466C42—34no—
467C42—64no—
468C42—62no—
469C42—124no—
470C42—124no—
471C42#23714yes—
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—42no—
493Dic3—42no—
494Dic3—44no—
495Dic3—42no—
496C23—34no—
497C23—62no—
498C23—64no—
499C23—62no—
500C23—64no—
501C23C24×D311yescharacteristic
502Q8—28no—
503Q8—84no—
504Q8—128no—
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—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
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—64no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—64no—
537C2×C4—244no—
538C2×C4—248no—
539C2×C4C2×#23712yes—
540C2×C4C2×#23712yes—
541C2×C4C2×#23712yes—
542C2×C4C2×D3×Q814yes—
543C2×C4C2×D3×Q812yes—
544C2×C4#100714yes—
545C2×C4#100912yes—
546C2×C3C2×#55411yescharacteristic
547C2×C3#350312yes—
548C2×C3#350811yescharacteristic
549C2×C3#350911yescharacteristic
550C2×C3#353612yes—
551D3—44no—
552C22—68no—
553C22—64no—
554C22—122no—
555C22—124no—
556C22—122no—
557C22C22×#23711yescharacteristic
558C22C22×D3×Q811yescharacteristic
559C22C2×#100712yes—
560C22C2×#100911yescharacteristic
561C22D3×H(5,𝔽2)12yes—
562C4—22no—
563C4—24no—
564C4—22no—
565C4—24no—
566C4—44no—
567C4—42no—
568C4—42no—
569C4—64no—
570C4—64no—
571C4—122no—
572C4—124no—
573C4—122no—
574C4—122no—
575C4#563514yes—
576C3#4543411yescharacteristic, complemented, Sylow
577C2—124no—
578C2C2×#563511yescharacteristic
579C2#8620411yescharacteristic
580C2#8631911yescharacteristic
581C2#8695212yes—
582C2#8704112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2567C2
prime radicalC22×D12C23
hypercenter#2567D3
hypocenterC3#45434
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2567

Complements

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

NH ≅ G/Ntype
1. C3×#2567C2semidirect
3. #83228C2semidirect
4. #82460C2semidirect
6. #77253C2semidirect
7. #77216C2semidirect
8. #76548C2semidirect
10. #76143C2semidirect
11. #76068C2semidirect
13. #74337C2semidirect
15. #73669C2semidirect
79. #2567D3semidirect
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×#25672
#832592
#832282
#824602
#824552
#772532
#772162
#765482
#764932
#761432
#760682
#743692
#743372
#737702
#736692
#454343