Finite Group Databasebeta

← #10798413

Subgroups — #10798413

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#135C211yesmaximal, characteristic
2#82513C211yesmaximal, characteristic, complemented
3C4×#948C211yesmaximal, characteristic, complemented
4#72087C212yesmaximal
5C4×#837C211yesmaximal, characteristic, complemented
6C4×#837C211yesmaximal, characteristic, complemented
7#70661C212yesmaximal, complemented
8#70292C212yesmaximal, complemented
9#70291C212yesmaximal, complemented
10C4×#819C211yesmaximal, characteristic, complemented
11#69750C211yesmaximal, characteristic, complemented
12#21137—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14C3×C4×#41C2211yescharacteristic
15C3×C4×#41C2211yescharacteristic
16C2×#948C2212yes—
17C3×C4×Q16C411yescharacteristic, complemented
18C3×C4×Q16C411yescharacteristic, complemented
19C3×C4×SD16C411yescharacteristic, complemented
20C3×C4×SD16C411yescharacteristic, complemented
21C3×C4×SD16C411yescharacteristic, complemented
22C3×C4×SD16C411yescharacteristic, complemented
23C3×C4×D8C411yescharacteristic, complemented
24C3×C4×D8C411yescharacteristic, complemented
25C3×#427C2211yescharacteristic
26C3×#416C2211yescharacteristic
27C3×#416C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C2×#837C2212yes—
30C2×#837C2212yes—
31#5234C2212yes—
32C2×#819C2212yes—
33#5000—22no—
34#4999C2212yes—
35#4997—22no—
36#4997C2212yes—
37C42×Dic3C2211yescharacteristic
38#4925—21no—
39#4925—21no—
40C4×#208—21no—
41C4×#208C2211yescharacteristic
42C4×#208C2211yescharacteristic
43C2×C4×C3⋊C8—21no—
44C2×C4×C3⋊C8C2211yescharacteristic
45#4552C2212yes—
46#4550C2212yes—
47#4550C2212yes—
48#4532C2212yes—
49#4532C2212yes—
50C4×#135D311yescharacteristic
51#2922—31no—
52C4×#131—31no—
53#1929—32no—
54#1922—32no—
55#1876—32no—
56#1866—32no—
57C4×#108—31no—
58C4×#104—31no—
59C4×#104—31no—
60#1744—31no—
61C2×C3×#41C2311yescharacteristic
62C2×C3×#41C2311yescharacteristic
63C3×#135—24no—
64C2×C3×Q16C2×C411yescharacteristic
65C2×C3×SD16C2×C411yescharacteristic
66C2×C3×SD16C2×C411yescharacteristic
67C2×C3×D8C2×C411yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×C4×Q8C2×C411yescharacteristic
71C3×C4×Q8C2×C411yescharacteristic
72C3×C4×Q8C2×C411yescharacteristic
73C3×C4×Q8C2×C411yescharacteristic
74C3×C4×D4—21no—
75C3×C4×D4—21no—
76C3×C4×D4—21no—
77C3×C4×D4—21no—
78C3×C4×D4C2×C411yescharacteristic
79C3×C4×D4C2×C411yescharacteristic
80C3×C4×D4C2×C411yescharacteristic
81C3×C4×D4C2×C411yescharacteristic
82C3×#117—21no—
83C3×#117—21no—
84C3×#117C2311yescharacteristic
85C3×#117C2311yescharacteristic
86C2×C3×C42—21no—
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89#948—24no—
90#948—24no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3C2312yes—
94C2×#208—22no—
95C2×#208C2312yes—
96C2×#208C2312yes—
97C22×C3⋊C8—22no—
98C22×C3⋊C8C2312yes—
99C3×#107C2×C411yescharacteristic
100C3×#107C2×C411yescharacteristic
101C3×#106C2×C411yescharacteristic
102C3×#106C2×C411yescharacteristic
103C3×#103C2×C411yescharacteristic
104C3×#103C2×C411yescharacteristic
105C3×#103C2×C411yescharacteristic
106C3×#103C2×C411yescharacteristic
107C3×#102C2×C411yescharacteristic
108C3×#102C2×C411yescharacteristic
109C3×#102C2×C411yescharacteristic
110C3×#102C2×C411yescharacteristic
111C3×C4×C8C2×C411yescharacteristic
112C3×C4×C8C2×C411yescharacteristic
113C3×C4×C8C2×C411yescharacteristic
114C3×C4×C8C2×C411yescharacteristic
115#837—24no—
116#837—24no—
117#830—22no—
118#830—22no—
119#819—24no—
120C4⋊(C3⋊C8)—22no—
121C4⋊(C3⋊C8)—22no—
122#803—21no—
123#803—21no—
124#803—22no—
125C4×C3⋊C8—21no—
126C4×C3⋊C8—21no—
127C4×C3⋊C8—22no—
128C4×C3⋊C8D411yescharacteristic
129C4×C3⋊C8D411yescharacteristic
130C4×C3⋊C8D412yes—
131C2×#135C2×D311yescharacteristic
132C2×#131—32no—
133C4×#41C2×D311yescharacteristic
134C4×#41C2×D311yescharacteristic
135C4×Q16Dic311yescharacteristic, complemented
136C4×Q16Dic311yescharacteristic, complemented
137C4×SD16Dic311yescharacteristic, complemented
138C4×SD16Dic311yescharacteristic, complemented
139C4×SD16Dic311yescharacteristic, complemented
140C4×SD16Dic311yescharacteristic, complemented
141C4×D8Dic311yescharacteristic, complemented
142C4×D8Dic311yescharacteristic, complemented
143#433—62no—
144#431—61no—
145#431—61no—
146C2×#108—32no—
147#427C2×D311yescharacteristic
148#424—32no—
149C2×#104—32no—
150C2×#104—32no—
151#416C2×D311yescharacteristic
152#416C2×D311yescharacteristic
153#408—32no—
154#405—32no—
155#405—62no—
156C4×M16—31no—
157C4×M16—31no—
158C4×M16—61no—
159C2×C4×C8—31no—
160C2×C4×C8—61no—
161C2×C4×C8C2×D311yescharacteristic
162C43—31no—
163#341—32no—
164#339—32no—
165#339—32no—
166#336—32no—
167#336—32no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41#4114yes—
172C2×C3×Q8C22×C411yescharacteristic
173C2×C3×Q8C22×C411yescharacteristic
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4C22×C411yescharacteristic
177C2×C3×D4C22×C411yescharacteristic
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4#4111yescharacteristic
185C22×C3×C4#4111yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—22no—
188C3×Q16—22no—
189C3×SD16—22no—
190C3×SD16—22no—
191C3×D8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8C22×C411yescharacteristic
195C2×C3×C8C22×C411yescharacteristic
196C2×C3×C8C22×C411yescharacteristic
197C2×C3×C8C22×C411yescharacteristic
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4C22×C411yescharacteristic
203C3×C4⋊C4C22×C411yescharacteristic
204C3×C4⋊C4C22×C411yescharacteristic
205C3×C4⋊C4C22×C411yescharacteristic
206C3×C22⋊C4—21no—
207C3×C22⋊C4—21no—
208C3×C22⋊C4—21no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—21no—
211C3×C22⋊C4—21no—
212C3×C42—21no—
213C3×C42—21no—
214C3×C42—21no—
215C3×C42—21no—
216C3×C42—21no—
217C3×C42—21no—
218C3×C42C22×C411yescharacteristic
219C3×C42C22×C411yescharacteristic
220C3×C42C2×D411yescharacteristic
221C3×C42C2×D411yescharacteristic
222C4×Dic3—24no—
223C4×Dic3—22no—
224C4×Dic3—22no—
225C4×Dic3—42no—
226#208—28no—
227#208—22no—
228#208—22no—
229#208—22no—
230#208#4114yes—
231C2×C3⋊C8—28no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8C2×D412yes—
237C2×C3⋊C8C2×D412yes—
238C2×#41C22×D311yescharacteristic
239C2×#41C22×D311yescharacteristic
240#135—24no—
241C2×Q16C2×Dic311yescharacteristic
242C2×SD16C2×Dic311yescharacteristic
243C2×SD16C2×Dic311yescharacteristic
244C2×D8C2×Dic311yescharacteristic
245#131—64no—
246#131—64no—
247C2×M16—32no—
248C2×M16—32no—
249C2×M16—62no—
250C22×C8—32no—
251C22×C8—62no—
252C22×C8C22×D311yescharacteristic
253C22×C8C22×D311yescharacteristic
254C4×Q8C2×Dic311yescharacteristic
255C4×Q8C2×Dic311yescharacteristic
256C4×Q8C2×Dic311yescharacteristic
257C4×Q8C2×Dic311yescharacteristic
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4C2×Dic311yescharacteristic
263C4×D4C2×Dic311yescharacteristic
264C4×D4C2×Dic311yescharacteristic
265C4×D4C2×Dic311yescharacteristic
266#117—21no—
267#117—21no—
268#117C22×D311yescharacteristic
269#117C22×D311yescharacteristic
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—32no—
273C2×C42—61no—
274C2×C42—61no—
275C2×C42C22×D311yescharacteristic
276#108—64no—
277#107C2×Dic311yescharacteristic
278#107C2×Dic311yescharacteristic
279#106C2×Dic311yescharacteristic
280#106C2×Dic311yescharacteristic
281C4⋊C8—62no—
282C4⋊C8—62no—
283#104—64no—
284#104—64no—
285#103C2×Dic311yescharacteristic
286#103C2×Dic311yescharacteristic
287#103C2×Dic311yescharacteristic
288#103C2×Dic311yescharacteristic
289#102C2×Dic311yescharacteristic
290#102C2×Dic311yescharacteristic
291#102C2×Dic311yescharacteristic
292#102C2×Dic311yescharacteristic
293C22⋊C8—62no—
294C22⋊C8—62no—
295#97—62no—
296#97—61no—
297#97—61no—
298C4×C8—32no—
299C4×C8—31no—
300C4×C8—31no—
301C4×C8—62no—
302C4×C8—61no—
303C4×C8—61no—
304C4×C8C2×Dic311yescharacteristic
305C4×C8C2×Dic311yescharacteristic
306C4×C8C2×Dic311yescharacteristic
307C4×C8C2×Dic311yescharacteristic
308C23×C3—21no—
309C23×C3—21no—
310C23×C3C2×#4111yescharacteristic
311C3×Q8—21no—
312C3×Q8—21no—
313C3×Q8—21no—
314C3×Q8#11712yes—
315C3×D4—24no—
316C3×D4—21no—
317C3×D4—21no—
318C3×D4—21no—
319C3×D4—42no—
320C3×D4#11712yes—
321C2×C3×C4—21no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4C4×D411yescharacteristic
336C2×C3×C4C4×D411yescharacteristic
337C2×C3×C4C4×D411yescharacteristic
338C2×C3×C4C4×D411yescharacteristic
339C2×C3×C4C23×C411yescharacteristic
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C2×#4112yes—
342C2×C3×C4C2×#4112yes—
343C2×Dic3—24no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C3×C8—21no—
347C3×C8—21no—
348C3×C8—21no—
349C3×C8—21no—
350C3⋊C8—22no—
351C3⋊C8—24no—
352C3⋊C8—22no—
353C3⋊C8—42no—
354#41—22no—
355#41—22no—
356#41—22no—
357#41#23514yes—
358C2×Q8C22×Dic311yescharacteristic
359C2×Q8C22×Dic311yescharacteristic
360C2×D4—21no—
361C2×D4—21no—
362C2×D4C22×Dic311yescharacteristic
363C2×D4C22×Dic311yescharacteristic
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—62no—
371C22×C4#23711yescharacteristic
372C22×C4#23711yescharacteristic
373C22×C4C23×D311yescharacteristic
374Q16—22no—
375SD16—22no—
376SD16—22no—
377D8—22no—
378M16—34no—
379M16—68no—
380M16—62no—
381M16—62no—
382M16—62no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—32no—
386C2×C8—32no—
387C2×C8—68no—
388C2×C8—62no—
389C2×C8—62no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8C22×Dic311yescharacteristic
393C2×C8C22×Dic311yescharacteristic
394C2×C8C22×Dic311yescharacteristic
395C2×C8C22×Dic311yescharacteristic
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4C22×Dic311yescharacteristic
401C4⋊C4C22×Dic311yescharacteristic
402C4⋊C4C22×Dic311yescharacteristic
403C4⋊C4C22×Dic311yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—64no—
417C42—62no—
418C42—62no—
419C42—122no—
420C42D3×D411yescharacteristic
421C42D3×D411yescharacteristic
422C42C22×Dic311yescharacteristic
423C42C22×Dic311yescharacteristic
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—22no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3C2×C4×D411yescharacteristic
433C22×C3#52912yes—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—22no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—41no—
440C3×C4—41no—
441C3×C4—41no—
442C3×C4#44312yes—
443C3×C4#44412yes—
444C3×C4C2×#11711yescharacteristic
445C3×C4C4×#4111yescharacteristic
446C3×C4#52512yes—
447Dic3—44no—
448C23—21no—
449C23—21no—
450C23C2×#23711yescharacteristic
451Q8—21no—
452Q8—21no—
453Q8—21no—
454Q8#92612yes—
455D4—24no—
456D4—21no—
457D4—21no—
458D4—21no—
459D4—42no—
460D4#92612yes—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—64no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4D4×Dic311yescharacteristic
479C2×C4D4×Dic311yescharacteristic
480C2×C4D4×Dic311yescharacteristic
481C2×C4D4×Dic311yescharacteristic
482C2×C4C2×#23512yes—
483C2×C4C2×D3×D411yescharacteristic
484C2×C4D3×#4112yes—
485C2×C4C23×Dic311yescharacteristic
486C8—21no—
487C8—21no—
488C8—21no—
489C8—21no—
490C8—62no—
491C8—62no—
492C8—64no—
493C8—122no—
494C2×C3—22no—
495C2×C3—42no—
496C2×C3—81no—
497C2×C3#229011yescharacteristic
498C2×C3C2×#44311yescharacteristic
499C2×C3C2×#44411yescharacteristic
500C22—21no—
501C22—21no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22#553812yes—
509C22C2×D4×Dic311yescharacteristic
510C4—22no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—124no—
519C4#519612yes—
520C4#519712yes—
521C4#554712yes—
522C4C2×#92611yescharacteristic
523C4Dic3×#4111yescharacteristic
524C3#2113711yescharacteristic, complemented, Sylow
525C2—22no—
526C2—42no—
527C2—81no—
528C2#8231811yescharacteristic
529C2C2×#519611yescharacteristic
530C2C2×#519711yescharacteristic
subgroupG/H
centerC42D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#135C2
socleC22×C3C2×C4×D4
prime radicalC2×C3×#135C22
cosocleC22×C3×C4C24
hypercenterC4×#135D3
hypocenterC3#21137
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C42 ◁ C2×C42 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
2. #82513C2semidirect
3. C4×#948C2semidirect
5. C4×#837C2semidirect
6. C4×#837C2semidirect
7. #70661C2semidirect
8. #70292C2semidirect
9. #70291C2semidirect
10. C4×#819C2semidirect
11. #69750C2semidirect
17. C3×C4×Q16C4semidirect
18. C3×C4×Q16C4semidirect
19. C3×C4×SD16C4semidirect
20. C3×C4×SD16C4semidirect
21. C3×C4×SD16C4semidirect
22. C3×C4×SD16C4semidirect
23. C3×C4×D8C4semidirect
24. C3×C4×D8C4semidirect
135. C4×Q16Dic3semidirect
136. C4×Q16Dic3semidirect
137. C4×SD16Dic3semidirect
138. C4×SD16Dic3semidirect
139. C4×SD16Dic3semidirect
140. C4×SD16Dic3semidirect
141. C4×D8Dic3semidirect
142. C4×D8Dic3semidirect
524. C3#21137semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1352
#825132
C4×#9482
#720872
C4×#8372
#706612
#702922
#702912
C4×#8192
#697502
#211373