Finite Group Databasebeta

← #11235113

Subgroups — #11235113

Order 768.

1726 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2331C211yesmaximal, characteristic, complemented
2#83260C211yesmaximal, characteristic, complemented
3#83259C211yesmaximal, characteristic, complemented
4#76650C214yesmaximal, complemented
5#76540C214yesmaximal, complemented
6#76048C212yesmaximal, complemented
7#76047C212yesmaximal, complemented
8#75409C214yesmaximal, complemented
9#75044C214yesmaximal, complemented
10#74160C211yesmaximal, characteristic, complemented
11#74160C211yesmaximal, characteristic, complemented
12#73712C212yesmaximal
13#73642C212yesmaximal, complemented
14#73397C211yesmaximal, characteristic, complemented
15#73389C211yesmaximal, characteristic
16#45430—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#946C2214yes—
19C2×#898C2214yes—
20C2×#875C2214yes—
21C2×#874C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2212yes—
25C3×#391C2214yes—
26C3×#389C2212yes—
27C3×#387C2214yes—
28C3×#382C2211yescharacteristic
29C3×C4×C4⋊C4C2212yes—
30C3×#376C2211yescharacteristic
31#5231C2212yes—
32#5231C2212yes—
33#4994C2214yes—
34#4993C2212yes—
35#4993C2212yes—
36#4992C2214yes—
37#4990C2214yes—
38#4988C2212yes—
39#4988C2212yes—
40#4988C2212yes—
41#4988C2212yes—
42#4987C2214yes—
43#4977C2212yes—
44#4975C2212yes—
45Dic3×C4⋊C4C2212yes—
46Dic3×C4⋊C4C2212yes—
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49#4939C2214yes—
50#4936C2211yescharacteristic
51C4×C4⋊Dic3C2212yes—
52C4×C4⋊Dic3C2211yescharacteristic
53C4×C4⋊Dic3C2211yescharacteristic
54#4933C2214yes—
55C4×Dic3⋊C4C2214yes—
56#4675C2214yes—
57#4674C2214yes—
58#4672C2214yes—
59#4672C2214yes—
60#4670C2214yes—
61#4670C2214yes—
62#4670C2214yes—
63#4669C2214yes—
64#4666C2212yes—
65#4666C2212yes—
66#4665C2214yes—
67#4665C2214yes—
68#4664C2212yes—
69#4664C2212yes—
70#4661C2212yes—
71#4661C2212yes—
72#4656C2212yes—
73#4656C2212yes—
74#4654C2214yes—
75#4646C2214yes—
76#2740—34no—
77#2700—34no—
78#2563—31no—
79#2560—34no—
80#2548—32no—
81#2542—31no—
82#2521—34no—
83#2513—32no—
84#2512—32no—
85#2399—31no—
86#2357—31no—
87#2357—31no—
88#2355—32no—
89#2331D311yescharacteristic, complemented
90#2328—31no—
91C22×C3×Q8C2312yes—
92C22×D12C2311yescharacteristic
93C22×D12C2311yescharacteristic
94C22×D12C2314yes—
95C22×C4×D3C2312yes—
96C22×C4×D3C2312yes—
97C22×Dic6C2312yes—
98C3×C4⋊Q8—24no—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2314yes—
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107#946—48no—
108C2×(C2×D3)⋊C4C2312yes—
109C2×(C2×D3)⋊C4C2312yes—
110C2×(C2×D3)⋊C4C2314yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×C4⋊Dic3C2312yes—
121C2×C4⋊Dic3C2312yes—
122C2×C4⋊Dic3C2312yes—
123C2×C4⋊Dic3C2311yescharacteristic
124C2×C4⋊Dic3C2311yescharacteristic
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2312yes—
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2312yes—
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2314yes—
131C2×C4×Dic3C2312yes—
132C2×C4×Dic3C2312yes—
133C2×C4×Dic3C2314yes—
134C2×C4×Dic3C2314yes—
135#898—48no—
136#875—48no—
137#874—24no—
138C4×D12—24no—
139C4×D12—24no—
140C4×Dic6—28no—
141C3×#95C2314yes—
142C3×#95C2312yes—
143C3×#95C2312yes—
144C3×#95C2312yes—
145C3×#95C2312yes—
146C3×#95C2314yes—
147C3⋊#95C2314yes—
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2314yes—
152C3⋊#95C2311yescharacteristic
153C3⋊#95C2311yescharacteristic
154C3⋊#95C2312yes—
155C3⋊#95C2311yescharacteristic
156C3⋊#95C2311yescharacteristic
157C3⋊#95C2312yes—
158C2×C4⋊Q8C2×D311yescharacteristic
159C2×#127—31no—
160C2×#126—34no—
161C2×#124—34no—
162C2×C22⋊Q8—34no—
163C2×C4×Q8—32no—
164C2×C4×D4—31no—
165C2×C4×D4—31no—
166#399—32no—
167#399—32no—
168#398—32no—
169#398—32no—
170#397—34no—
171#397—34no—
172#397—34no—
173#396—34no—
174#395—32no—
175#395—32no—
176#394—34no—
177#394—34no—
178#393—32no—
179#393—32no—
180#391—34no—
181#391C2×D314yes—
182#390—32no—
183#390—32no—
184#390—32no—
185#390—32no—
186#390—34no—
187#390—31no—
188#390—31no—
189#389C2×D312yes—
190#388—34no—
191#388—34no—
192#388—34no—
193#388—34no—
194#388—34no—
195#387—34no—
196#387—34no—
197#387C2×D314yes—
198#386—34no—
199#386—34no—
200#384—32no—
201#382—32no—
202#382—31no—
203#382C2×D311yescharacteristic
204#381—32no—
205#381—32no—
206#380—34no—
207C4×C4⋊C4—32no—
208C4×C4⋊C4—32no—
209C4×C4⋊C4—34no—
210C4×C4⋊C4—31no—
211C4×C4⋊C4—31no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4C2×D312yes—
214C4×C22⋊C4—32no—
215C4×C22⋊C4—32no—
216#376C2×D311yescharacteristic
217C23×D3C2412yes—
218C23×D3C2412yes—
219C2×C3×Q8—24no—
220C2×C3×Q8—44no—
221C22×C3×C4C2414yes—
222C22×C3×C4C2412yes—
223C22×C3×C4C2414yes—
224C22×C3×C4C2412yes—
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2411yescharacteristic
228C22×Dic3C2412yes—
229C22×Dic3C2412yes—
230C22×Dic3C2412yes—
231C22×Dic3C2412yes—
232C22×Dic3C2412yes—
233C22×Dic3C2412yes—
234C2×D12—22no—
235C2×D12—22no—
236C2×D12—22no—
237C2×D12—22no—
238C2×D12—22no—
239C2×D12—22no—
240C2×D12—28no—
241C2×D12—44no—
242C2×D12—44no—
243C2×C4×D3—24no—
244C2×C4×D3—24no—
245C2×C4×D3—44no—
246C2×C4×D3—44no—
247C2×Dic6—24no—
248C2×Dic6—24no—
249C2×Dic6—42no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—44no—
254C3×C42—22no—
255C3×C42—44no—
256C3×C42#4114yes—
257C3×C42#4114yes—
258(C2×D3)⋊C4—24no—
259(C2×D3)⋊C4—24no—
260(C2×D3)⋊C4—28no—
261(C2×D3)⋊C4—28no—
262(C2×D3)⋊C4—44no—
263(C2×D3)⋊C4—44no—
264(C2×D3)⋊C4—44no—
265(C2×D3)⋊C4—44no—
266(C2×D3)⋊C4—44no—
267(C2×D3)⋊C4—44no—
268(C2×D3)⋊C4—44no—
269(C2×D3)⋊C4—44no—
270C4⋊Dic3—24no—
271C4⋊Dic3—22no—
272C4⋊Dic3—22no—
273C4⋊Dic3—24no—
274C4⋊Dic3—24no—
275C4⋊Dic3—42no—
276C4⋊Dic3—42no—
277Dic3⋊C4—24no—
278Dic3⋊C4—24no—
279Dic3⋊C4—28no—
280Dic3⋊C4—44no—
281C4×Dic3—24no—
282C4×Dic3—24no—
283C4×Dic3—44no—
284C4×Dic3—44no—
285C22×Q8—32no—
286C22×Q8C22×D312yes—
287C22×D4—31no—
288C22×D4—31no—
289C22×D4—34no—
290C23×C4—32no—
291C23×C4—32no—
292C4⋊Q8—24no—
293#127—64no—
294#126—128no—
295#124—128no—
296C22⋊Q8—128no—
297C4×Q8—68no—
298C4×D4—64no—
299C4×D4—64no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—34no—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—34no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4C22×D312yes—
312C2×C4⋊C4C22×D312yes—
313C2×C4⋊C4C22×D312yes—
314C2×C4⋊C4C22×D314yes—
315C2×C22⋊C4—32no—
316C2×C22⋊C4—32no—
317C2×C22⋊C4—34no—
318C2×C22⋊C4—34no—
319C2×C22⋊C4—34no—
320C2×C22⋊C4—34no—
321C2×C22⋊C4—34no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—34no—
324C2×C22⋊C4—34no—
325C2×C22⋊C4—34no—
326C2×C22⋊C4—34no—
327C2×C42—32no—
328C2×C42—32no—
329C2×C42—34no—
330C2×C42—34no—
331C2×C42C22×D311yescharacteristic
332C2×C42C22×D314yes—
333C2×C42C22×D311yescharacteristic
334C2×C42C22×D311yescharacteristic
335#95—31no—
336#95—31no—
337#95—34no—
338#95—31no—
339#95—31no—
340#95—34no—
341#95—34no—
342#95—34no—
343#95—32no—
344#95—32no—
345#95—34no—
346#95C22×D312yes—
347#95C22×D312yes—
348#95C22×D312yes—
349#95C22×D312yes—
350#95C22×D314yes—
351#95C22×D314yes—
352C23×C3C2511yescharacteristic
353C22×D3—22no—
354C22×D3—22no—
355C22×D3—22no—
356C22×D3—22no—
357C22×D3—24no—
358C22×D3—24no—
359C22×D3—22no—
360C22×D3—22no—
361C22×D3—24no—
362C22×D3—24no—
363C3×Q8—84no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—24no—
369C2×C3×C4—24no—
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×C4C2×#4112yes—
378C2×C3×C4C2×#4112yes—
379C2×C3×C4C2×#4112yes—
380C2×C3×C4C2×#4114yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C42−1+412yes—
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—22no—
393C2×Dic3—22no—
394C2×Dic3—24no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3H(5,𝔽2)14yes—
399D12—44no—
400D12—44no—
401D12—88no—
402C4×D3—84no—
403C4×D3—84no—
404Dic6—48no—
405C24—32no—
406C24—32no—
407C2×Q8—24no—
408C2×Q8—44no—
409C2×Q8—64no—
410C2×Q8—64no—
411C2×Q8—122no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—68no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—124no—
420C2×D4—124no—
421C22×C4—32no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—32no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—124no—
430C22×C4—124no—
431C22×C4C23×D314yes—
432C22×C4C23×D312yes—
433C22×C4C23×D314yes—
434C22×C4C23×D311yescharacteristic
435C22×C4C23×D311yescharacteristic
436C22×C4C23×D312yes—
437C22×C4C23×D311yescharacteristic
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—44no—
442C4⋊C4—64no—
443C4⋊C4—68no—
444C4⋊C4—64no—
445C4⋊C4—62no—
446C4⋊C4—62no—
447C4⋊C4—64no—
448C4⋊C4—64no—
449C4⋊C4—64no—
450C4⋊C4—124no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C22⋊C4—68no—
454C22⋊C4—68no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—124no—
458C22⋊C4—124no—
459C22⋊C4—124no—
460C22⋊C4—124no—
461C22⋊C4—124no—
462C22⋊C4—124no—
463C22⋊C4—124no—
464C22⋊C4—124no—
465C42—22no—
466C42—44no—
467C42—64no—
468C42—64no—
469C42—124no—
470C42—124no—
471C42#23514yes—
472C42#23914yes—
473C22×C3C22×#4111yescharacteristic
474C22×C3C22×#4111yescharacteristic
475C22×C3C2×H(5,𝔽2)12yes—
476C22×C3C2×2−1+411yescharacteristic
477C22×C3#58512yes—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—44no—
481C2×D3—44no—
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—42no—
486C2×D3—44no—
487C2×D3—44no—
488C3×C4—24no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—44no—
492C3×C4—44no—
493C3×C4—42no—
494C3×C4#55614yes—
495Dic3—24no—
496Dic3—42no—
497Dic3—42no—
498Dic3—42no—
499Dic3—42no—
500Dic3—42no—
501C23—62no—
502C23—62no—
503C23—62no—
504C23—62no—
505C23—62no—
506C23—62no—
507C23—64no—
508C23—64no—
509C23—64no—
510C23—64no—
511C23C24×D311yescharacteristic
512Q8—84no—
513Q8—128no—
514D4—124no—
515D4—124no—
516D4—248no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—34no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—64no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—64no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—244no—
547C2×C4—244no—
548C2×C4C2×#23512yes—
549C2×C4C2×#23512yes—
550C2×C4C2×#23512yes—
551C2×C4C2×#23914yes—
552C2×C4C2×#23912yes—
553C2×C4#101012yes—
554C2×C3C2×#55611yescharacteristic
555C2×C3#349412yes—
556C2×C3#350811yescharacteristic
557C2×C3#351011yescharacteristic
558C2×C3#355412yes—
559D3—82no—
560D3—82no—
561C22—124no—
562C22—124no—
563C22—122no—
564C22—122no—
565C22—124no—
566C22—124no—
567C22—122no—
568C22—122no—
569C22—122no—
570C22—122no—
571C22C22×#23511yescharacteristic
572C22C22×#23911yescharacteristic
573C22C2×#101011yescharacteristic
574C22D3×H(5,𝔽2)12yes—
575C22#596812yes—
576C4—24no—
577C4—22no—
578C4—22no—
579C4—42no—
580C4—44no—
581C4—44no—
582C4—64no—
583C4—122no—
584C4—122no—
585C4—122no—
586C4—122no—
587C4—122no—
588C4#558414yes—
589C3#4543011yescharacteristic, complemented, Sylow
590C2—242no—
591C2—242no—
592C2C2×#558411yescharacteristic
593C2#8586211yescharacteristic
594C2#8587711yescharacteristic
595C2#8653512yes—
596C2#8659712yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2331C2
prime radicalC2×#874C22
hypercenter#2331D3
hypocenterC3#45430
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2331

Complements

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

NH ≅ G/Ntype
1. C3×#2331C2semidirect
2. #83260C2semidirect
3. #83259C2semidirect
4. #76650C2semidirect
5. #76540C2semidirect
6. #76048C2semidirect
7. #76047C2semidirect
8. #75409C2semidirect
9. #75044C2semidirect
10. #74160C2semidirect
11. #74160C2semidirect
13. #73642C2semidirect
14. #73397C2semidirect
89. #2331D3semidirect
589. C3#45430semidirect

Maximal subgroups

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

subgroupindex
C3×#23312
#832602
#832592
#766502
#765402
#760482
#760472
#754092
#750442
#741602
#737122
#736422
#733972
#733892
#454303