Finite Group Databasebeta

← #10801721

Subgroups — #10801721

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#466C211yesmaximal, characteristic
2C2×Dic3×D8C211yesmaximal, characteristic, complemented
3C2×#5101C211yesmaximal, characteristic, complemented
4#82633C212yesmaximal, complemented
5#72038C211yesmaximal, characteristic, complemented
6#71971C211yesmaximal, characteristic, complemented
7#71682C212yesmaximal, complemented
8#70612C211yesmaximal, characteristic, complemented
9Dic3×#107C211yesmaximal, characteristic, complemented
10#70187C212yesmaximal, complemented
11#70152C212yesmaximal, complemented
12#21524—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×D4×Dic3C2212yes—
17C2×#926C2211yescharacteristic
18C22×#208C2211yescharacteristic
19C3×#466C418yescomplemented
20C2×C3×#107C2211yescharacteristic
21C2×C3×#102C2212yes—
22#5221—22no—
23#5221C2212yes—
24#5220—22no—
25C2×#832C2212yes—
26C2×#832C2212yes—
27#5212C2212yes—
28Dic3×D8—24no—
29C2×#818C2211yescharacteristic
30C2×#818C2211yescharacteristic
31#5101—24no—
32C2×#815C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4984C2212yes—
37Dic3×C4⋊C4C2212yes—
38#4951—22no—
39#4950—22no—
40#4551C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#466D311yescharacteristic
44C2×C4×D8—31no—
45C2×#405—31no—
46#2324—32no—
47#1970—31no—
48#1942—31no—
49#1936—32no—
50#1895—32no—
51#1808—31no—
52C4×#107—31no—
53#1762—32no—
54C22×C3×D4—22no—
55C22×C3×D4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3—22no—
58C2×C3×D8—22no—
59C2×C3×D8—22no—
60C2×C3×D8C2×C414yes—
61C22×C3×C8—22no—
62C22×C3×C8—22no—
63C22×C3×C8C2×C414yescomplemented
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×#121—24no—
67C3×#121C2×C418yescomplemented
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C22⋊C4—22no—
70C2×#217—22no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217D411yescharacteristic
75C2×#217D411yescharacteristic
76D4×Dic3—24no—
77D4×Dic3—24no—
78#926—22no—
79#926—22no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3D411yescharacteristic
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2312yes—
85C2×C4×Dic3—22no—
86C2×C4×Dic3C2311yescharacteristic
87C2×C4×Dic3C2311yescharacteristic
88C2×#208—22no—
89C2×#208—22no—
90C2×#208—22no—
91C22×C3⋊C8C2311yescharacteristic
92C22×C3⋊C8C2311yescharacteristic
93C3×#107C2×C414yescomplemented
94C3×#102C2×C418yescomplemented
95#832—24no—
96#832—24no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100#818—22no—
101#818—22no—
102#815—22no—
103#815—22no—
104C8×Dic3—22no—
105C8×Dic3D414yescomplemented
106C22×D8C2×D311yescharacteristic
107C22×M16—31no—
108C23×C8C2×D311yescharacteristic
109C2×#121C2×D312yes—
110C2×C4×D4—32no—
111C2×#117—31no—
112#466Dic318yescomplemented
113C4×D8—64no—
114C2×#107—31no—
115C2×#107—31no—
116C2×#107C2×D311yescharacteristic
117C2×#102—32no—
118C2×#102—32no—
119C2×#102C2×D312yes—
120#405—64no—
121C2×#97—31no—
122C2×#97—31no—
123C2×C4×C8—31no—
124C2×C4×C8—31no—
125#390—32no—
126#390—62no—
127#389—32no—
128#388—62no—
129#386—62no—
130#385—62no—
131#380—32no—
132C4×C4⋊C4—32no—
133#340—32no—
134#340—32no—
135#340—32no—
136C24×C3—22no—
137C24×C3C2×D411yescharacteristic
138C2×C3×D4—28no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—82no—
145C2×C3×D4—82no—
146C2×C3×D4C22×C414yes—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4C22×C412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4#4112yes—
155C22×C3×C4C2411yescharacteristic
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—44no—
159C22×Dic3—82no—
160C22×Dic3—82no—
161C22×Dic3C2×D411yescharacteristic
162C22×Dic3C2×D411yescharacteristic
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165C3×D8—28no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—28no—
169C2×C3×C8C22×C412yes—
170C2×C3×C8C22×C412yes—
171C2×C3×C8C2×D412yes—
172C2×C3×C8#4112yes—
173C3×C4⋊C4—22no—
174C3×C4⋊C4C22×C414yes—
175C3×C22⋊C4—28no—
176#217—22no—
177#217—22no—
178#217—44no—
179#217—82no—
180#217—82no—
181#217—82no—
182C4⋊Dic3—22no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3—22no—
187C4⋊Dic3—22no—
188C4⋊Dic3—82no—
189C4×Dic3—21no—
190C4×Dic3—21no—
191C4×Dic3—82no—
192C4×Dic3C2×D412yes—
193C4×Dic3#4112yes—
194#208—28no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8C2×D412yes—
200C2×C3⋊C8#4112yes—
201C22×D4—22no—
202C22×D4C22×D312yes—
203C23×C4—62no—
204C23×C4C22×D311yescharacteristic
205C2×D8—22no—
206C2×D8—22no—
207C2×D8C2×Dic314yes—
208C2×M16—62no—
209C2×M16—62no—
210C2×M16—62no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—31no—
214C22×C8—31no—
215C22×C8C2×Dic314yescomplemented
216C22×C8C22×D311yescharacteristic
217C22×C8C22×D311yescharacteristic
218#121—24no—
219#121C2×Dic318yescomplemented
220C4×D4—64no—
221C4×D4—64no—
222#117—62no—
223#117—62no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—32no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4C22×D312yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—31no—
232C2×C22⋊C4—31no—
233C2×C22⋊C4—62no—
234C2×C22⋊C4—62no—
235C2×C22⋊C4—62no—
236C2×C22⋊C4—62no—
237C2×C42—31no—
238C2×C42—31no—
239C2×C42—62no—
240#107—62no—
241#107—62no—
242#107C2×Dic314yescomplemented
243#102—64no—
244#102—64no—
245#102C2×Dic318yescomplemented
246#97—62no—
247#97—62no—
248C4×C8—34no—
249C4×C8—62no—
250#95—62no—
251#95—62no—
252#95—62no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—24no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3C4×D412yes—
267C23×C3C22×D411yescharacteristic
268C3×D4—24no—
269C3×D4—24no—
270C3×D4—42no—
271C3×D4—42no—
272C3×D4—88no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4C4×D412yes—
283C2×C3×C4C23×C411yescharacteristic
284C2×C3×C4C22×D411yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—88no—
306C2×Dic3#12112yes—
307C2×Dic3#12112yes—
308C3×C8—22no—
309C3×C8C4×D414yes—
310C3⋊C8—22no—
311C3⋊C8—22no—
312C24—22no—
313C24D3×D411yescharacteristic
314C2×D4—28no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4C22×Dic314yes—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—31no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—124no—
335C22×C4—242no—
336C22×C4—242no—
337C22×C4D3×D411yescharacteristic
338C22×C4#23712yes—
339C22×C4C22×Dic312yes—
340C22×C4C23×D311yescharacteristic
341D8—28no—
342M16—68no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—28no—
346C2×C8—32no—
347C2×C8—32no—
348C2×C8—61no—
349C2×C8—61no—
350C2×C8—61no—
351C2×C8—61no—
352C2×C8D3×D412yes—
353C2×C8#23712yes—
354C2×C8C22×Dic312yes—
355C2×C8C22×Dic312yes—
356C4⋊C4—22no—
357C4⋊C4—62no—
358C4⋊C4—62no—
359C4⋊C4—62no—
360C4⋊C4—62no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—242no—
364C4⋊C4C22×Dic314yes—
365C22⋊C4—28no—
366C22⋊C4—62no—
367C22⋊C4—62no—
368C22⋊C4—124no—
369C22⋊C4—242no—
370C22⋊C4—242no—
371C22⋊C4—242no—
372C42—32no—
373C42—32no—
374C42—61no—
375C42—61no—
376C42—242no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—44no—
385C22×C3—44no—
386C22×C3—44no—
387C22×C3—82no—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3C2×C4×D411yescharacteristic
392C22×C3C2×#12111yescharacteristic
393C22×C3#53211yescharacteristic
394C22×C3#57611yescharacteristic
395C22×C3#57611yescharacteristic
396C22×C3#57611yescharacteristic
397C22×C3#57811yescharacteristic
398C3×C4—22no—
399C3×C4—44no—
400C3×C4C2×C4×D411yescharacteristic
401C3×C4C4×#4111yescharacteristic
402C3×C4#52511yescharacteristic
403C3×C4#52911yescharacteristic
404Dic3—22no—
405Dic3—22no—
406Dic3—41no—
407Dic3—41no—
408Dic3—82no—
409Dic3—82no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—24no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23D4×Dic312yes—
424C23C2×D3×D411yescharacteristic
425D4—24no—
426D4—24no—
427D4—42no—
428D4—42no—
429D4—88no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—32no—
438C2×C4—32no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—248no—
458C2×C4D4×Dic312yes—
459C2×C4C2×D3×D411yescharacteristic
460C2×C4C2×#23711yescharacteristic
461C2×C4C2×#23711yescharacteristic
462C2×C4D3×#4111yescharacteristic
463C2×C4D3×#4111yescharacteristic
464C2×C4C23×Dic311yescharacteristic
465C8—22no—
466C8—62no—
467C8—62no—
468C8D4×Dic314yes—
469C2×C3—42no—
470C2×C3—84no—
471C2×C3C4×#12111yescharacteristic
472C2×C3#294811yescharacteristic
473C2×C3#295111yescharacteristic
474C2×C3#305811yescharacteristic
475C2×C3#306111yescharacteristic
476C2×C3#317511yescharacteristic
477C2×C3#317611yescharacteristic
478C22—22no—
479C22—22no—
480C22—22no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—44no—
486C22—44no—
487C22—44no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22#559611yescharacteristic
493C22D3×#12111yescharacteristic
494C22#575711yescharacteristic
495C22#575711yescharacteristic
496C22#576911yescharacteristic
497C22#577111yescharacteristic
498C22C2×D4×Dic311yescharacteristic
499C4—22no—
500C4—44no—
501C4—62no—
502C4—62no—
503C4—121no—
504C4—121no—
505C4—242no—
506C4—242no—
507C4#560011yescharacteristic
508C4#560611yescharacteristic
509C4C2×D4×Dic311yescharacteristic
510C4Dic3×#4111yescharacteristic
511C3#2152411yescharacteristic, complemented, Sylow
512C2—42no—
513C2—84no—
514C2#8069811yescharacteristic
515C2#8069911yescharacteristic
516C2#8071611yescharacteristic
517C2#8071711yescharacteristic
518C2Dic3×#12111yescharacteristic
519C2#8382311yescharacteristic
520C2#8389811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#466C2
socleC23×C3C22×D4
cosocleC22×C3×C4C24
hypercenterC2×#466D3
hypocenterC3#21524
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×D8C2semidirect
3. C2×#5101C2semidirect
4. #82633C2semidirect
5. #72038C2semidirect
6. #71971C2semidirect
7. #71682C2semidirect
8. #70612C2semidirect
9. Dic3×#107C2semidirect
10. #70187C2semidirect
11. #70152C2semidirect
19. C3×#466C4semidirect
63. C22×C3×C8C2×C4semidirect
67. C3×#121C2×C4semidirect
93. C3×#107C2×C4semidirect
94. C3×#102C2×C4semidirect
105. C8×Dic3D4semidirect
112. #466Dic3semidirect
215. C22×C8C2×Dic3semidirect
219. #121C2×Dic3semidirect
242. #107C2×Dic3semidirect
245. #102C2×Dic3semidirect
511. C3#21524semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4662
C2×Dic3×D82
C2×#51012
#826332
#720382
#719712
#716822
#706122
Dic3×#1072
#701872
#701522
#215243