Finite Group Databasebeta

← #11233579

Subgroups — #11233579

Order 768.

1638 conjugacy classes of subgroups.

The subgroup lattice has 564 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#76648C214yesmaximal, complemented
5#76538C214yesmaximal, complemented
6#76066C212yesmaximal, complemented
7#76007C212yesmaximal
8#75220C214yesmaximal, complemented
9#75037C214yesmaximal, complemented
10#74160C211yesmaximal, characteristic
11#74157C211yesmaximal, characteristic, complemented
12#73693C212yesmaximal
13#73641C212yesmaximal, complemented
14#73398C211yesmaximal, characteristic, complemented
15#73387C211yesmaximal, characteristic
16#45350—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#946C2214yes—
19C2×#889C2214yes—
20C2×#887C2212yes—
21C2×#876C2214yes—
22C2×#875C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#391C2214yes—
26C3×#389C2212yes—
27C3×#387C2214yes—
28C3×#382C2211yescharacteristic
29C3×C4×C4⋊C4C2212yes—
30C3×#376C2211yescharacteristic
31#5230C2212yes—
32#5230C2212yes—
33#4993C2212yes—
34#4993C2212yes—
35#4993C2214yes—
36#4990C2214yes—
37#4988C2212yes—
38#4988C2212yes—
39#4981C2214yes—
40#4975C2212yes—
41#4975C2212yes—
42Dic3×C4⋊C4C2212yes—
43#4973C2212yes—
44#4972C2214yes—
45#4972C2212yes—
46#4940C2214yes—
47#4939C2211yescharacteristic
48#4935C2211yescharacteristic
49C4×C4⋊Dic3C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51#4933C2214yes—
52C4×Dic3⋊C4C2214yes—
53C4×Dic3⋊C4C2212yes—
54C4×Dic3⋊C4C2212yes—
55#4929C2211yescharacteristic
56#4676C2214yes—
57#4675C2214yes—
58#4675C2214yes—
59#4670C2214yes—
60#4670C2214yes—
61#4669C2214yes—
62#4668C2214yes—
63#4666C2212yes—
64#4664C2212yes—
65#4660C2212yes—
66#4660C2212yes—
67#4657C2212yes—
68#4657C2214yes—
69#4657C2212yes—
70#4652C2214yes—
71#4650C2214yes—
72#4650C2212yes—
73#4649C2212yes—
74#4647C2214yes—
75#4647C2214yes—
76#2773—34no—
77#2698—34no—
78#2567—31no—
79#2556—34no—
80#2552—32no—
81#2542—31no—
82#2519—34no—
83#2512—32no—
84#2448—32no—
85#2402—31no—
86#2357—31no—
87#2355—32no—
88#2351—31no—
89#2331D311yescharacteristic, complemented
90#2326—31no—
91C22×C3×Q8C2312yes—
92C22×D12C2311yescharacteristic
93C22×C4×D3C2312yes—
94C22×Dic6C2311yescharacteristic
95C3×C4⋊Q8—24no—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2314yes—
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2314yes—
103C2×C3×C42C2311yescharacteristic
104#946—48no—
105C2×(C2×D3)⋊C4C2314yes—
106C2×(C2×D3)⋊C4C2314yes—
107C2×(C2×D3)⋊C4C2314yes—
108C2×(C2×D3)⋊C4C2312yes—
109C2×(C2×D3)⋊C4C2314yes—
110C2×(C2×D3)⋊C4C2314yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×C4⋊Dic3C2312yes—
113C2×C4⋊Dic3C2312yes—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2314yes—
117C2×Dic3⋊C4C2312yes—
118C2×Dic3⋊C4C2312yes—
119C2×Dic3⋊C4C2314yes—
120C2×Dic3⋊C4C2314yes—
121C2×Dic3⋊C4C2312yes—
122C2×Dic3⋊C4C2314yes—
123C2×Dic3⋊C4C2312yes—
124C2×Dic3⋊C4C2312yes—
125C2×Dic3⋊C4C2314yes—
126C2×Dic3⋊C4C2314yes—
127C2×Dic3⋊C4C2314yes—
128C2×C4×Dic3C2314yes—
129C2×C4×Dic3C2312yes—
130C2×C4×Dic3C2312yes—
131C2×C4×Dic3C2312yes—
132C2×C4×Dic3C2314yes—
133#889—48no—
134#887—28no—
135#876—48no—
136#875—24no—
137C4×D12—24no—
138C4×Dic6—24no—
139C3×#95C2312yes—
140C3×#95C2314yes—
141C3×#95C2312yes—
142C3×#95C2314yes—
143C3×#95C2312yes—
144C3×#95C2312yes—
145C3⋊#95C2314yes—
146C3⋊#95C2311yescharacteristic
147C3⋊#95C2311yescharacteristic
148C3⋊#95C2314yes—
149C3⋊#95C2314yes—
150C3⋊#95C2314yes—
151C3⋊#95C2312yes—
152C3⋊#95C2314yes—
153C3⋊#95C2312yes—
154C3⋊#95C2311yescharacteristic
155C3⋊#95C2311yescharacteristic
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C2×C4⋊Q8C2×D311yescharacteristic
159C2×#126—34no—
160C2×#125—34no—
161C2×#124—31no—
162C2×C22⋊Q8—34no—
163C2×C4×Q8—32no—
164C2×C4×Q8—31no—
165C2×C4×D4—31no—
166#400—32no—
167#400—32no—
168#399—34no—
169#399—32no—
170#399—32no—
171#398—34no—
172#398—32no—
173#398—32no—
174#398—34no—
175#397—34no—
176#397—34no—
177#396—34no—
178#395—32no—
179#395—32no—
180#391—32no—
181#391—31no—
182#391C2×D314yes—
183#390—32no—
184#390—32no—
185#390—31no—
186#389—34no—
187#389—32no—
188#389—34no—
189#389—34no—
190#389C2×D312yes—
191#388—34no—
192#388—34no—
193#388—34no—
194#388—34no—
195#388—34no—
196#387—34no—
197#387—34no—
198#387C2×D314yes—
199#385—34no—
200#384—32no—
201#384—32no—
202#384—31no—
203#382—32no—
204#382—34no—
205#382C2×D311yescharacteristic
206#381—32no—
207C4×C4⋊C4—32no—
208C4×C4⋊C4—31no—
209C4×C4⋊C4—31no—
210C4×C4⋊C4—32no—
211C4×C4⋊C4—34no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4—32no—
214C4×C4⋊C4C2×D312yes—
215C4×C22⋊C4—32no—
216#376C2×D311yescharacteristic
217C23×D3C2412yes—
218C2×C3×Q8—24no—
219C2×C3×Q8—44no—
220C22×C3×C4C2412yes—
221C22×C3×C4C2414yes—
222C22×C3×C4C2412yes—
223C22×C3×C4C2414yes—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×Dic3C2412yes—
228C22×Dic3C2412yes—
229C22×Dic3C2412yes—
230C22×Dic3C2412yes—
231C22×Dic3C2412yes—
232C22×Dic3C2412yes—
233C22×Dic3C2412yes—
234C2×D12—22no—
235C2×D12—22no—
236C2×D12—22no—
237C2×C4×D3—24no—
238C2×C4×D3—44no—
239C2×Dic6—22no—
240C2×Dic6—22no—
241C2×Dic6#4114yes—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—44no—
246C3×C42—22no—
247C3×C42—22no—
248C3×C42—44no—
249C3×C42#4114yes—
250(C2×D3)⋊C4—24no—
251(C2×D3)⋊C4—28no—
252(C2×D3)⋊C4—28no—
253(C2×D3)⋊C4—28no—
254(C2×D3)⋊C4—44no—
255(C2×D3)⋊C4—44no—
256(C2×D3)⋊C4—44no—
257C4⋊Dic3—22no—
258C4⋊Dic3—22no—
259C4⋊Dic3—24no—
260C4⋊Dic3—24no—
261C4⋊Dic3—44no—
262Dic3⋊C4—24no—
263Dic3⋊C4—28no—
264Dic3⋊C4—24no—
265Dic3⋊C4—28no—
266Dic3⋊C4—24no—
267Dic3⋊C4—28no—
268Dic3⋊C4—24no—
269Dic3⋊C4—24no—
270Dic3⋊C4—44no—
271Dic3⋊C4—44no—
272Dic3⋊C4—44no—
273C4×Dic3—24no—
274C4×Dic3—24no—
275C4×Dic3—24no—
276C4×Dic3—28no—
277C4×Dic3—44no—
278C22×Q8—31no—
279C22×Q8C22×D312yes—
280C22×D4—31no—
281C23×C4—32no—
282C4⋊Q8—24no—
283#126—128no—
284#125—128no—
285#124—64no—
286C22⋊Q8—128no—
287C4×Q8—68no—
288C4×Q8—64no—
289C4×D4—64no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—34no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—32no—
294C2×C4⋊C4—32no—
295C2×C4⋊C4—34no—
296C2×C4⋊C4—32no—
297C2×C4⋊C4—34no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4—32no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—34no—
303C2×C4⋊C4—34no—
304C2×C4⋊C4—34no—
305C2×C4⋊C4—34no—
306C2×C4⋊C4C22×D312yes—
307C2×C4⋊C4C22×D312yes—
308C2×C4⋊C4C22×D312yes—
309C2×C4⋊C4C22×D314yes—
310C2×C22⋊C4—32no—
311C2×C22⋊C4—34no—
312C2×C22⋊C4—34no—
313C2×C22⋊C4—34no—
314C2×C22⋊C4—34no—
315C2×C22⋊C4—34no—
316C2×C22⋊C4—34no—
317C2×C42—32no—
318C2×C42—34no—
319C2×C42—32no—
320C2×C42—32no—
321C2×C42—34no—
322C2×C42C22×D311yescharacteristic
323C2×C42C22×D311yescharacteristic
324C2×C42C22×D314yes—
325C2×C42C22×D311yescharacteristic
326#95—31no—
327#95—31no—
328#95—31no—
329#95—34no—
330#95—31no—
331#95—34no—
332#95—32no—
333#95—32no—
334#95—34no—
335#95—32no—
336#95—32no—
337#95—34no—
338#95—34no—
339#95C22×D312yes—
340#95C22×D312yes—
341#95C22×D312yes—
342#95C22×D312yes—
343#95C22×D314yes—
344#95C22×D314yes—
345C23×C3C2511yescharacteristic
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—24no—
349C22×D3—24no—
350C22×D3—22no—
351C3×Q8—84no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—22no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—24no—
365C2×C3×C4—22no—
366C2×C3×C4C2×#4112yes—
367C2×C3×C4C2×#4112yes—
368C2×C3×C4C2×#4112yes—
369C2×C3×C4C2×#4112yes—
370C2×C3×C42−1+412yes—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3C2×#4114yes—
388C2×Dic3H(5,𝔽2)14yes—
389C2×Dic32−1+414yes—
390D12—44no—
391C4×D3—84no—
392Dic6—28no—
393C24—32no—
394C2×Q8—24no—
395C2×Q8—34no—
396C2×Q8—44no—
397C2×Q8—62no—
398C2×Q8—62no—
399C2×D4—62no—
400C2×D4—62no—
401C2×D4—62no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—64no—
410C22×C4—124no—
411C22×C4C23×D312yes—
412C22×C4C23×D314yes—
413C22×C4C23×D312yes—
414C22×C4C23×D314yes—
415C22×C4C23×D311yescharacteristic
416C22×C4C23×D311yescharacteristic
417C22×C4C23×D311yescharacteristic
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—44no—
422C4⋊C4—68no—
423C4⋊C4—64no—
424C4⋊C4—68no—
425C4⋊C4—64no—
426C4⋊C4—68no—
427C4⋊C4—64no—
428C4⋊C4—64no—
429C4⋊C4—64no—
430C4⋊C4—64no—
431C4⋊C4—62no—
432C4⋊C4—64no—
433C4⋊C4—62no—
434C4⋊C4—124no—
435C4⋊C4—124no—
436C4⋊C4—124no—
437C4⋊C4—124no—
438C22⋊C4—68no—
439C22⋊C4—68no—
440C22⋊C4—68no—
441C22⋊C4—64no—
442C22⋊C4—124no—
443C22⋊C4—124no—
444C22⋊C4—124no—
445C42—22no—
446C42—22no—
447C42—44no—
448C42—68no—
449C42—64no—
450C42—64no—
451C42—64no—
452C42—124no—
453C42#23514yes—
454C22×C3C22×#4111yescharacteristic
455C22×C3C22×#4111yescharacteristic
456C22×C3C2×H(5,𝔽2)12yes—
457C22×C3C2×2−1+411yescharacteristic
458C22×C3C2×2−1+412yes—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—44no—
463C2×D3—44no—
464C3×C4—22no—
465C3×C4—22no—
466C3×C4—44no—
467C3×C4—44no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4#55214yes—
471Dic3—24no—
472Dic3—24no—
473Dic3—24no—
474Dic3—42no—
475Dic3—42no—
476Dic3—42no—
477Dic3—42no—
478C23—62no—
479C23—62no—
480C23—62no—
481C23—64no—
482C23—64no—
483C23C24×D311yescharacteristic
484Q8—68no—
485Q8—84no—
486D4—124no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—34no—
502C2×C4—34no—
503C2×C4—34no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—64no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—64no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—244no—
521C2×C4C2×#23512yes—
522C2×C4C2×#23512yes—
523C2×C4C2×#23512yes—
524C2×C4#100712yes—
525C2×C4D3×#4112yes—
526C2×C3C2×#55211yescharacteristic
527C2×C3#350811yescharacteristic
528C2×C3#350911yescharacteristic
529C2×C3#352212yes—
530C2×C3#355212yes—
531D3—82no—
532C22—124no—
533C22—122no—
534C22—124no—
535C22—122no—
536C22—122no—
537C22C22×#23511yescharacteristic
538C22C2×#100711yescharacteristic
539C22C2×D3×#4111yescharacteristic
540C22D3×H(5,𝔽2)12yes—
541C22D3×2−1+412yes—
542C4—22no—
543C4—22no—
544C4—42no—
545C4—44no—
546C4—42no—
547C4—44no—
548C4—64no—
549C4—64no—
550C4—64no—
551C4—122no—
552C4—122no—
553C4—122no—
554C4—122no—
555C4#558114yes—
556C3#4535011yescharacteristic, complemented, Sylow
557C2—242no—
558C2C2×#558111yescharacteristic
559C2#8586211yescharacteristic
560C2#8587611yescharacteristic
561C2#8655812yes—
562C2#8660212yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2331C2
prime radicalC22×D12C23
hypercenter#2331D3
hypocenterC3#45350
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. #76648C2semidirect
5. #76538C2semidirect
6. #76066C2semidirect
8. #75220C2semidirect
9. #75037C2semidirect
11. #74157C2semidirect
13. #73641C2semidirect
14. #73398C2semidirect
89. #2331D3semidirect
556. C3#45350semidirect

Maximal subgroups

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

subgroupindex
C3×#23312
#832592
#832432
#766482
#765382
#760662
#760072
#752202
#750372
#741602
#741572
#736932
#736412
#733982
#733872
#453503