Finite Group Databasebeta

← #10798407

Subgroups — #10798407

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
1C4×#948C211yesmaximal, characteristic, complemented
2C4×#950C211yesmaximal, characteristic
3#78519C211yesmaximal, characteristic, complemented
4C3×C4×#104C211yesmaximal, characteristic, complemented
5#70377C212yesmaximal
6#70376C212yesmaximal, complemented
7C4×#822C211yesmaximal, characteristic, complemented
8#69388C211yesmaximal, characteristic, complemented
9C4×#805C211yesmaximal, characteristic, complemented
10#68744C212yesmaximal, complemented
11#68743C212yesmaximal, complemented
12#21137—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14C2×#950C2211yescharacteristic
15C2×#948C2212yes—
16C4×#235C2211yescharacteristic
17C2×C3×#104C2212yes—
18C3×C4×M16C2211yescharacteristic
19C3×C43C2211yescharacteristic
20#5233C2211yescharacteristic
21#5129C2212yes—
22C2×#822C2212yes—
23#5119C2212yes—
24C4×#216C411yescharacteristic, complemented
25C4×#216C411yescharacteristic, complemented
26C4×#215C411yescharacteristic, complemented
27C4×#215C411yescharacteristic, complemented
28C4×#214C411yescharacteristic, complemented
29C4×#214C411yescharacteristic, complemented
30C4×#213C411yescharacteristic, complemented
31C4×#213C411yescharacteristic, complemented
32#5002C2211yescharacteristic
33#4998C2211yescharacteristic
34#4997—22no—
35#4997C2212yes—
36C2×#805C2212yes—
37C4×#208—21no—
38C4×#208C2211yescharacteristic
39C2×C4×C3⋊C8—21no—
40C2×C4×C3⋊C8C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4709—22no—
43#4685—21no—
44#4685—21no—
45C3×#339C2212yes—
46#4558C2212yes—
47#4524C2212yes—
48#4524C2212yes—
49#4523C2212yes—
50C4×#135—31no—
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×#104D311yescharacteristic, complemented
60#1744—31no—
61C2×C3×#41C2311yescharacteristic
62C2×#235C2311yescharacteristic
63C2×C3×M16C2312yes—
64C3×C4×Q8C2×C411yescharacteristic
65C3×C4×Q8C2×C411yescharacteristic
66C3×C4×D4—21no—
67C3×C4×D4—21no—
68C3×C4×D4C2×C411yescharacteristic
69C3×C4×D4C2×C411yescharacteristic
70C3×#117—21no—
71C3×#117C2311yescharacteristic
72C2×C3×C42—21no—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77#950—24no—
78#948—24no—
79#948—24no—
80C2×#216C2×C411yescharacteristic
81C2×#215C2×C411yescharacteristic
82C2×#214C2×C411yescharacteristic
83C2×#213C2×C411yescharacteristic
84C4×#67—21no—
85C4×#67—21no—
86#926C2311yescharacteristic
87C2×#208—22no—
88C2×#208C2312yes—
89C22×C3⋊C8—22no—
90C22×C3⋊C8C2311yescharacteristic
91C22×C3⋊C8C2311yescharacteristic
92C22×C3⋊C8C2312yes—
93C4×D12C2×C411yescharacteristic
94C4×D12C2×C411yescharacteristic
95#872—21no—
96C42×D3—21no—
97C4×Dic6C2×C411yescharacteristic
98C4×Dic6C2×C411yescharacteristic
99C3×#104—24no—
100C3×#97—21no—
101C3×C4×C8—21no—
102#835C2×C411yescharacteristic
103#835C2×C411yescharacteristic
104#832C2×C411yescharacteristic
105#832C2×C411yescharacteristic
106#822—24no—
107(C2×D3)⋊C8—22no—
108(C2×D3)⋊C8—22no—
109Dic3⋊C8—22no—
110Dic3⋊C8—22no—
111#810C2×C411yescharacteristic
112#810C2×C411yescharacteristic
113#809C2×C411yescharacteristic
114#809C2×C411yescharacteristic
115#808C2×C411yescharacteristic
116#808C2×C411yescharacteristic
117#807C2×C411yescharacteristic
118#807C2×C411yescharacteristic
119#805—24no—
120#803—21no—
121#803—22no—
122C4×C3⋊C8—21no—
123C4×C3⋊C8—22no—
124C4×C3⋊C8C2×C411yescharacteristic
125C4×C3⋊C8C2×C411yescharacteristic
126C4×C3⋊C8C2×C411yescharacteristic
127C4×C3⋊C8C2×C411yescharacteristic
128C4×C3⋊C8D411yescharacteristic
129C4×C3⋊C8D411yescharacteristic
130C4×C3⋊C8D412yes—
131C2×#135—31no—
132C2×#131—32no—
133C4×#41—31no—
134C4×#41C2×D311yescharacteristic
135C4×Q16—31no—
136C4×Q16—31no—
137C4×SD16—31no—
138C4×SD16—31no—
139C4×SD16—31no—
140C4×SD16—31no—
141C4×D8—31no—
142C4×D8—31no—
143#433—62no—
144#431—61no—
145#431—61no—
146C2×#108—32no—
147#427—31no—
148#424—32no—
149C2×#104—32no—
150C2×#104C2×D312yes—
151#416—31no—
152#416—31no—
153#408—32no—
154#405—32no—
155#405—62no—
156C4×M16—31no—
157C4×M16—61no—
158C4×M16C2×D311yescharacteristic
159C2×C4×C8—31no—
160C2×C4×C8—31no—
161C2×C4×C8—61no—
162C43C2×D311yescharacteristic
163#341—32no—
164#339—32no—
165#339C2×D312yes—
166#336—32no—
167#336—32no—
168C3×#41—22no—
169C3×#41#4114yes—
170C2×C3×Q8C22×C411yescharacteristic
171C2×C3×D4—21no—
172C2×C3×D4C22×C411yescharacteristic
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4#4111yescharacteristic
178C22×C3×C4#4111yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C2×#67—21no—
181#235—22no—
182#235—22no—
183C2×D12C22×C411yescharacteristic
184C2×C4×D3—21no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×Dic6C22×C411yescharacteristic
188C3×M16—22no—
189C3×M16—22no—
190C2×C3×C8—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4C22×C411yescharacteristic
194C3×C4⋊C4C22×C411yescharacteristic
195C3×C22⋊C4—21no—
196C3×C22⋊C4—21no—
197C3×C22⋊C4—21no—
198C3×C42—24no—
199C3×C42—21no—
200C3×C42—21no—
201C3×C42—21no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—42no—
205C3×C42C22×C411yescharacteristic
206C3×C42C22×C411yescharacteristic
207C3×C42C2×D411yescharacteristic
208C3×C42C2×D411yescharacteristic
209#217—21no—
210#216—22no—
211#215—22no—
212#214—22no—
213#213—22no—
214(C2×D3)⋊C4—21no—
215(C2×D3)⋊C4—21no—
216C4⋊Dic3C22×C411yescharacteristic
217C4⋊Dic3C22×C411yescharacteristic
218Dic3⋊C4—21no—
219Dic3⋊C4—21no—
220C4×Dic3—21no—
221C4×Dic3—21no—
222C4×Dic3—21no—
223#208—28no—
224#208—22no—
225#208#4114yes—
226C2×C3⋊C8—28no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8C22×C411yescharacteristic
233C2×C3⋊C8C22×C411yescharacteristic
234C2×C3⋊C8C22×C411yescharacteristic
235C2×C3⋊C8C22×C411yescharacteristic
236C2×C3⋊C8C2×D412yes—
237C2×C3⋊C8C2×D412yes—
238C2×#41—31no—
239C2×#41C22×D311yescharacteristic
240#135—64no—
241C2×Q16—31no—
242C2×SD16—31no—
243C2×SD16—31no—
244C2×D8—31no—
245#131—64no—
246#131—64no—
247C2×M16—32no—
248C2×M16—62no—
249C2×M16C22×D312yes—
250C22×C8—32no—
251C22×C8—31no—
252C22×C8—31no—
253C22×C8—62no—
254C4×Q8—31no—
255C4×Q8—31no—
256C4×Q8C4×D311yescharacteristic
257C4×Q8C4×D311yescharacteristic
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—31no—
261C4×D4—31no—
262C4×D4—61no—
263C4×D4—61no—
264C4×D4C4×D311yescharacteristic
265C4×D4C4×D311yescharacteristic
266#117—21no—
267#117—31no—
268#117—61no—
269#117C22×D311yescharacteristic
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—61no—
274C2×C42C22×D312yes—
275C2×C42C22×D311yescharacteristic
276#108—64no—
277#107—31no—
278#107—31no—
279#106—31no—
280#106—31no—
281C4⋊C8—62no—
282C4⋊C8—62no—
283#104—24no—
284#104—64no—
285#103—31no—
286#103—31no—
287#103—31no—
288#103—31no—
289#102—31no—
290#102—31no—
291#102—31no—
292#102—31no—
293C22⋊C8—62no—
294C22⋊C8—62no—
295#97—21no—
296#97—62no—
297#97—61no—
298C4×C8—21no—
299C4×C8—31no—
300C4×C8—31no—
301C4×C8—31no—
302C4×C8—31no—
303C4×C8—32no—
304C4×C8—31no—
305C4×C8—31no—
306C4×C8—62no—
307C4×C8—61no—
308C23×C3—21no—
309C23×C3C2×#4111yescharacteristic
310C22×D3—21no—
311C3×Q8—21no—
312C3×Q8#11712yes—
313C3×D4—24no—
314C3×D4—21no—
315C3×D4#11712yes—
316C2×C3×C4—21no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4C4×D411yescharacteristic
328C2×C3×C4C4×D411yescharacteristic
329C2×C3×C4C4×D411yescharacteristic
330C2×C3×C4C4×D411yescharacteristic
331C2×C3×C4C23×C411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4112yes—
334C2×C3×C4C2×#4112yes—
335#67—42no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339D12—21no—
340D12—21no—
341C4×D3—42no—
342C4×D3—42no—
343C4×D3—42no—
344Dic6—21no—
345Dic6—21no—
346C3×C8—42no—
347C3⋊C8—21no—
348C3⋊C8—21no—
349C3⋊C8—21no—
350C3⋊C8—21no—
351C3⋊C8—22no—
352C3⋊C8—24no—
353C3⋊C8—22no—
354#41—22no—
355#41—62no—
356#41—62no—
357#41#23514yes—
358C2×Q8—31no—
359C2×Q8C2×C4×D311yescharacteristic
360C2×D4—21no—
361C2×D4—31no—
362C2×D4—61no—
363C2×D4C2×C4×D311yescharacteristic
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4#23711yescharacteristic
372C22×C4#23711yescharacteristic
373C22×C4C23×D311yescharacteristic
374Q16—62no—
375SD16—62no—
376SD16—62no—
377D8—62no—
378M16—22no—
379M16—22no—
380M16—34no—
381M16—68no—
382M16—62no—
383C2×C8—22no—
384C2×C8—31no—
385C2×C8—31no—
386C2×C8—31no—
387C2×C8—31no—
388C2×C8—32no—
389C2×C8—32no—
390C2×C8—68no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—31no—
399C4⋊C4—31no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4C2×C4×D311yescharacteristic
403C4⋊C4C2×C4×D311yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—61no—
408C22⋊C4—61no—
409C22⋊C4—61no—
410C42—24no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—22no—
415C42—22no—
416C42—42no—
417C42—61no—
418C42—61no—
419C42—61no—
420C42C2×C4×D311yescharacteristic
421C42C2×C4×D311yescharacteristic
422C42D3×D411yescharacteristic
423C42D3×D411yescharacteristic
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—22no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3C2×C4×D411yescharacteristic
430C22×C3#52912yes—
431C2×D3—41no—
432C2×D3—41no—
433C2×D3—41no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—22no—
437C3×C4—44no—
438C3×C4—41no—
439C3×C4—41no—
440C3×C4#44312yes—
441C3×C4#44412yes—
442C3×C4C2×#11711yescharacteristic
443C3×C4C4×#4111yescharacteristic
444C3×C4#52512yes—
445Dic3—41no—
446Dic3—41no—
447Dic3—41no—
448C23—21no—
449C23—61no—
450C23C2×#23711yescharacteristic
451Q8—21no—
452Q8—61no—
453Q8—61no—
454Q8#87212yes—
455D4—24no—
456D4—21no—
457D4—61no—
458D4—61no—
459D4—122no—
460D4#87212yes—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4#67⋊C411yescharacteristic
479C2×C4#67⋊C411yescharacteristic
480C2×C4#67⋊C411yescharacteristic
481C2×C4#67⋊C411yescharacteristic
482C2×C4C22×C4×D311yescharacteristic
483C2×C4C2×#23512yes—
484C2×C4C2×D3×D411yescharacteristic
485C2×C4D3×#4112yes—
486C8—42no—
487C8—61no—
488C8—61no—
489C8—61no—
490C8—61no—
491C8—62no—
492C8—62no—
493C8—64no—
494C2×C3—22no—
495C2×C3—42no—
496C2×C3#229011yescharacteristic
497C2×C3C2×#44311yescharacteristic
498C2×C3C2×#44411yescharacteristic
499D3—81no—
500C22—21no—
501C22—21no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—121no—
506C22—121no—
507C22—121no—
508C22C2×#67⋊C411yescharacteristic
509C22#553812yes—
510C4—21no—
511C4—21no—
512C4—22no—
513C4—44no—
514C4—41no—
515C4—41no—
516C4—121no—
517C4—121no—
518C4—121no—
519C4#482312yes—
520C4#482412yes—
521C4C2×#87211yescharacteristic
522C4#552211yescharacteristic
523C4#554712yes—
524C3#2113711yescharacteristic, complemented, Sylow
525C2—22no—
526C2—42no—
527C2—241no—
528C2#7343811yescharacteristic
529C2C2×#482411yescharacteristic
530C2C2×#482311yescharacteristic
subgroupG/H
centerC42D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#104C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenterC4×#104D3
hypocenterC3#21137
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C42 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
1. C4×#948C2semidirect
3. #78519C2semidirect
4. C3×C4×#104C2semidirect
6. #70376C2semidirect
7. C4×#822C2semidirect
8. #69388C2semidirect
9. C4×#805C2semidirect
10. #68744C2semidirect
11. #68743C2semidirect
24. C4×#216C4semidirect
25. C4×#216C4semidirect
26. C4×#215C4semidirect
27. C4×#215C4semidirect
28. C4×#214C4semidirect
29. C4×#214C4semidirect
30. C4×#213C4semidirect
31. C4×#213C4semidirect
59. C4×#104D3semidirect
524. C3#21137semidirect

Maximal subgroups

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

subgroupindex
C4×#9482
C4×#9502
#785192
C3×C4×#1042
#703772
#703762
C4×#8222
#693882
C4×#8052
#687442
#687432
#211373