Finite Group Databasebeta

← #10801713

Subgroups — #10801713

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×#5115C211yesmaximal, characteristic
2C2×#5101C211yesmaximal, characteristic, complemented
3C2×#4872C211yesmaximal, characteristic, complemented
4#73872C212yesmaximal, complemented
5C3×#1808C211yesmaximal, characteristic, complemented
6#70628C211yesmaximal, characteristic, complemented
7#70611C211yesmaximal, characteristic, complemented
8Dic3×#107C211yesmaximal, characteristic, complemented
9#69517C212yesmaximal, complemented
10#68856C212yesmaximal, complemented
11#68782C212yesmaximal, complemented
12#21524—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14C2×#930C2212yes—
15C2×#926C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C22×D24C2211yescharacteristic
18C2×#893C2212yes—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C3×#380C2212yes—
22#5115C418yescomplemented
23C2×#821C2212yes—
24C2×#818C2211yescharacteristic
25#5101—24no—
26C2×#815C2211yescharacteristic
27C2×#815C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30#4988C2212yes—
31Dic3×C4⋊C4C2212yes—
32#4972C2212yes—
33C2×#809C2212yes—
34C2×#809C2212yes—
35#4957—22no—
36#4956—22no—
37#4872—24no—
38#4670—22no—
39#4668—22no—
40C3×#340C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#466—31no—
44C2×C4×D8—31no—
45C2×#405—31no—
46#2324—32no—
47#1970—31no—
48#1942—31no—
49#1936—32no—
50#1895—32no—
51#1808D311yescharacteristic, complemented
52C4×#107—31no—
53#1762—32no—
54C23×C3×C4C2311yescharacteristic
55C22×#67—22no—
56C22×D12C2312yes—
57C22×C4×D3—22no—
58C22×C3×C8—22no—
59C22×C3×C8—22no—
60C22×C3×C8C2×C414yescomplemented
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—22no—
66C2×#217D411yescharacteristic
67C2×#217D411yescharacteristic
68#930—24no—
69#930C2×C418yescomplemented
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74#926—22no—
75#926—22no—
76C2×C4⋊Dic3D411yescharacteristic
77C2×C4⋊Dic3D411yescharacteristic
78C2×C4⋊Dic3C2312yes—
79C2×Dic3⋊C4—22no—
80C2×C4×Dic3—22no—
81C2×C4×Dic3C2311yescharacteristic
82C2×C4×Dic3C2311yescharacteristic
83C2×#208—22no—
84C2×#208—22no—
85C2×#208—22no—
86C22×C3⋊C8C2311yescharacteristic
87C22×C3⋊C8C2311yescharacteristic
88C2×D24—22no—
89C2×D24—22no—
90C2×D24C2×C414yes—
91#893—24no—
92#893—24no—
93C3×#107—22no—
94C3×#107—22no—
95C3×#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98#821C2×C418yescomplemented
99#818C2×C414yescomplemented
100#815—22no—
101#815—22no—
102C8×Dic3—22no—
103C8×Dic3D414yescomplemented
104#809—24no—
105#809—24no—
106C22×D8—31no—
107C22×M16—31no—
108C23×C8C2×D311yescharacteristic
109C2×#121—32no—
110C2×C4×D4—32no—
111C2×#117—31no—
112#466—38no—
113C4×D8—64no—
114C2×#107—31no—
115C2×#107C2×D311yescharacteristic
116C2×#107C2×D311yescharacteristic
117C2×#102—32no—
118C2×#102—32no—
119C2×#102—32no—
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#380C2×D312yes—
132C4×C4⋊C4—32no—
133#340—32no—
134#340—32no—
135#340C2×D312yes—
136C24×C3C2×D411yescharacteristic
137C23×D3—22no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4C22×C412yes—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4#4112yes—
147C22×C3×C4C2411yescharacteristic
148C2×#67—28no—
149C2×#67—82no—
150C2×#67—82no—
151C22×Dic3—22no—
152C22×Dic3C2×D411yescharacteristic
153C22×Dic3C2×D411yescharacteristic
154C22×Dic3C2×D411yescharacteristic
155C22×Dic3C2×D411yescharacteristic
156C2×D12—22no—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12C22×C414yes—
162C2×C4×D3—44no—
163C2×C4×D3—82no—
164C2×C4×D3—82no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—28no—
168C2×C3×C8C22×C412yes—
169C2×C3×C8C22×C412yes—
170C2×C3×C8C2×D412yes—
171C2×C3×C8#4112yes—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C22⋊C4—82no—
177#217—22no—
178#217—22no—
179(C2×D3)⋊C4—28no—
180(C2×D3)⋊C4—44no—
181(C2×D3)⋊C4—82no—
182(C2×D3)⋊C4—82no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3C22×C414yes—
187Dic3⋊C4—82no—
188C4×Dic3—21no—
189C4×Dic3—21no—
190C4×Dic3—82no—
191C4×Dic3C2×D412yes—
192C4×Dic3#4112yes—
193#208—28no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8C2×D412yes—
199C2×C3⋊C8#4112yes—
200D24—28no—
201C22×D4—32no—
202C22×D4—62no—
203C23×C4—62no—
204C23×C4C22×D311yescharacteristic
205C2×D8—34no—
206C2×D8—62no—
207C2×D8—62no—
208C2×M16—62no—
209C2×M16—62no—
210C2×M16—62no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—31no—
214C22×C8—31no—
215C22×C8C4×D314yescomplemented
216C22×C8C22×D311yescharacteristic
217C22×C8C22×D311yescharacteristic
218#121—38no—
219#121—64no—
220C4×D4—64no—
221C4×D4—64no—
222#117—62no—
223#117—62no—
224C2×C4⋊C4—32no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4C22×D312yes—
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—22no—
241#107—22no—
242#107—34no—
243#102—38no—
244#102—64no—
245#102—64no—
246#97—62no—
247#97—62no—
248C4×C8—34no—
249C4×C8—62no—
250#95—22no—
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×C3C4×D412yes—
260C23×C3C22×D411yescharacteristic
261C22×D3—24no—
262C22×D3—42no—
263C22×D3—42no—
264C22×D3—42no—
265C22×D3—42no—
266C22×D3—42no—
267C22×D3—42no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4C4×D412yes—
281C2×C3×C4C23×C411yescharacteristic
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287#67—88no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—24no—
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#12112yes—
302C2×Dic3#12112yes—
303D12—24no—
304D12—24no—
305D12—42no—
306D12—42no—
307C4×D3—88no—
308C3×C8—22no—
309C3×C8C4×D414yes—
310C3⋊C8—22no—
311C3⋊C8—22no—
312C24—62no—
313C24D3×D411yescharacteristic
314C2×D4—34no—
315C2×D4—68no—
316C2×D4—62no—
317C2×D4—62no—
318C2×D4—62no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4—242no—
322C2×D4—242no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—62no—
334C22×C4—124no—
335C22×C4—242no—
336C22×C4—242no—
337C22×C4C2×C4×D312yes—
338C22×C4D3×D411yescharacteristic
339C22×C4#23912yes—
340C22×C4C23×D311yescharacteristic
341D8—68no—
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×C8C2×C4×D312yes—
353C2×C8C2×C4×D312yes—
354C2×C8D3×D412yes—
355C2×C8#23712yes—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—34no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—62no—
364C4⋊C4—242no—
365C22⋊C4—62no—
366C22⋊C4—62no—
367C22⋊C4—68no—
368C22⋊C4—82no—
369C22⋊C4—124no—
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×C3C2×C4×D411yescharacteristic
385C22×C3C2×#12111yescharacteristic
386C22×C3#53211yescharacteristic
387C22×C3#57611yescharacteristic
388C22×C3#57611yescharacteristic
389C22×C3#57611yescharacteristic
390C22×C3#57811yescharacteristic
391C2×D3—44no—
392C2×D3—44no—
393C2×D3—44no—
394C2×D3—82no—
395C2×D3—82no—
396C2×D3—82no—
397C2×D3—82no—
398C3×C4—22no—
399C3×C4—82no—
400C3×C4—82no—
401C3×C4C2×C4×D411yescharacteristic
402C3×C4C4×#4111yescharacteristic
403C3×C4#52511yescharacteristic
404C3×C4#52911yescharacteristic
405Dic3—22no—
406Dic3—22no—
407Dic3—44no—
408Dic3—41no—
409Dic3—41no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—64no—
417C23—122no—
418C23—122no—
419C23—122no—
420C23—122no—
421C23—122no—
422C23—122no—
423C23#89312yes—
424C23C2×D3×D411yescharacteristic
425D4—64no—
426D4—64no—
427D4—122no—
428D4—122no—
429D4—248no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—32no—
437C2×C4—32no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—64no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—248no—
458C2×C4#89312yes—
459C2×C4C22×C4×D311yescharacteristic
460C2×C4C2×D3×D411yescharacteristic
461C2×C4C2×#23711yescharacteristic
462C2×C4C2×#23911yescharacteristic
463C2×C4D3×#4111yescharacteristic
464C2×C4D3×#4111yescharacteristic
465C8—22no—
466C8—62no—
467C8—62no—
468C8#67⋊C414yes—
469C2×C3—42no—
470C2×C3C4×#12111yescharacteristic
471C2×C3#294811yescharacteristic
472C2×C3#295111yescharacteristic
473C2×C3#305811yescharacteristic
474C2×C3#306111yescharacteristic
475C2×C3#317511yescharacteristic
476C2×C3#317611yescharacteristic
477D3—84no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—124no—
486C22—124no—
487C22—124no—
488C22—242no—
489C22—242no—
490C22—242no—
491C22—242no—
492C22C2×#89311yescharacteristic
493C22#559611yescharacteristic
494C22D3×#12111yescharacteristic
495C22#575711yescharacteristic
496C22#576611yescharacteristic
497C22#576911yescharacteristic
498C22#576911yescharacteristic
499C4—22no—
500C4—62no—
501C4—62no—
502C4—82no—
503C4—82no—
504C4—124no—
505C4—121no—
506C4—121no—
507C4C2×#67⋊C411yescharacteristic
508C4#552211yescharacteristic
509C4#562411yescharacteristic
510C4#562711yescharacteristic
511C3#2152411yescharacteristic, complemented, Sylow
512C2—42no—
513C2—244no—
514C2#7386511yescharacteristic
515C2#7913611yescharacteristic
516C2#7916711yescharacteristic
517C2#8069911yescharacteristic
518C2#8071611yescharacteristic
519C2#8074211yescharacteristic
520C2#8076111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1808C2
socleC23×C3C22×D4
prime radicalC2×#5115C2
cosocleC22×C3×C4C24
hypercenter#1808D3
hypocenterC3#21524
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. C2×#5101C2semidirect
3. C2×#4872C2semidirect
4. #73872C2semidirect
5. C3×#1808C2semidirect
6. #70628C2semidirect
7. #70611C2semidirect
8. Dic3×#107C2semidirect
9. #69517C2semidirect
10. #68856C2semidirect
11. #68782C2semidirect
22. #5115C4semidirect
51. #1808D3semidirect
60. C22×C3×C8C2×C4semidirect
69. #930C2×C4semidirect
98. #821C2×C4semidirect
99. #818C2×C4semidirect
103. C8×Dic3D4semidirect
215. C22×C8C4×D3semidirect
511. C3#21524semidirect

Maximal subgroups

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

subgroupindex
C2×#51152
C2×#51012
C2×#48722
#738722
C3×#18082
#706282
#706112
Dic3×#1072
#695172
#688562
#687822
#215243